OpenVDB: openvdb::v5_2::math Namespace Reference

OpenVDB 5.2.0

Namespaces | Classes | Typedefs | Enumerations | Functions
openvdb::v5_2::math Namespace Reference

## Namespaces
| internal
| mat3_internal
| pcg

## Classes
class | AffineMap
| A general linear transform using homogeneous coordinates to perform rotation, scaling, shear and translation. More...
class | BaseStencil
class | BBox
| Axis-aligned bounding box. More...
struct | BIAS_SCHEME
struct | BIAS_SCHEME< FIRST_BIAS >
struct | BIAS_SCHEME< HJWENO5_BIAS >
struct | BIAS_SCHEME< SECOND_BIAS >
struct | BIAS_SCHEME< THIRD_BIAS >
struct | BIAS_SCHEME< WENO5_BIAS >
class | BoxStencil
class | CompoundMap
| Creates the composition of two maps, each of which could be a composition. In the case that each component of the composition classified as linear an acceleration AffineMap is stored. More...
struct | Conversion
| Dummy class for tag dispatch of conversion constructors. More...
class | Coord
| Signed (x, y, z) 32-bit integer coordinates. More...
class | CoordBBox
| Axis-aligned bounding box of signed integer coordinates. More...
struct | CPT
| Compute the closest-point transform to a level set. More...
struct | CPT_RANGE
| Compute the closest-point transform to a level set. More...
struct | Curl
| Compute the curl of a vector-valued grid using differencing of various orders in the space defined by the range of the map. More...
struct | Curl< UniformScaleMap, CD_2ND >
| Full template specialization of Curl. More...
struct | Curl< UniformScaleMap, DiffScheme >
| Partial template specialization of Curl. More...
struct | Curl< UniformScaleTranslateMap, CD_2ND >
| Full template specialization of Curl. More...
struct | Curl< UniformScaleTranslateMap, DiffScheme >
| Partial template specialization of Curl. More...
class | CurvatureStencil
struct | D1
struct | D1< BD_1ST >
struct | D1< BD_2ND >
struct | D1< BD_3RD >
struct | D1< BD_HJWENO5 >
struct | D1< BD_WENO5 >
struct | D1< CD_2ND >
struct | D1< CD_2NDT >
struct | D1< CD_4TH >
struct | D1< CD_6TH >
struct | D1< FD_1ST >
struct | D1< FD_2ND >
struct | D1< FD_3RD >
struct | D1< FD_HJWENO5 >
struct | D1< FD_WENO5 >
struct | D1Vec
struct | D1Vec< CD_2ND >
struct | D1Vec< CD_2NDT >
struct | D1Vec< CD_4TH >
struct | D1Vec< CD_6TH >
struct | D2
struct | D2< CD_FOURTH >
struct | D2< CD_SECOND >
struct | D2< CD_SIXTH >
class | DDA
| A Digital Differential Analyzer specialized for OpenVDB grids. More...
struct | Delta
| Delta for small floating-point offsets. More...
struct | Delta< double >
struct | Delta< float >
class | DenseStencil
| Dense stencil of a given width. More...
struct | Divergence
| Compute the divergence of a vector-valued grid using differencing of various orders, the result defined with respect to the range-space of the map. More...
struct | Divergence< ScaleMap, CD_2ND >
struct | Divergence< ScaleMap, DiffScheme >
struct | Divergence< ScaleTranslateMap, CD_2ND >
struct | Divergence< ScaleTranslateMap, DiffScheme >
struct | Divergence< TranslationMap, DiffScheme >
struct | Divergence< UniformScaleMap, CD_2ND >
struct | Divergence< UniformScaleMap, DiffScheme >
struct | Divergence< UniformScaleTranslateMap, CD_2ND >
struct | Divergence< UniformScaleTranslateMap, DiffScheme >
class | Extrema
| This class computes the minimum and maximum values of a population of floating-point values. More...
class | FourthOrderDenseStencil
class | GenericMap
| A wrapper that holds a MapBase::ConstPtr and exposes a reduced set of functionality needed by the mathematical operators. More...
struct | Gradient
| Center difference gradient operators, defined with respect to the range-space of the `map`. More...
struct | Gradient< ScaleMap, CD_2ND >
struct | Gradient< ScaleTranslateMap, CD_2ND >
struct | Gradient< TranslationMap, DiffScheme >
struct | Gradient< UniformScaleMap, CD_2ND >
struct | Gradient< UniformScaleTranslateMap, CD_2ND >
struct | GradientBiased
| Biased gradient operators, defined with respect to the range-space of the map. More...
struct | GradientNormSqrd
struct | GradientNormSqrd< UniformScaleMap, GradScheme >
| Partial template specialization of GradientNormSqrd. More...
struct | GradientNormSqrd< UniformScaleTranslateMap, GradScheme >
| Partial template specialization of GradientNormSqrd. More...
class | GradStencil
class | Histogram
| This class computes a histogram, with a fixed interval width, of a population of floating-point values. More...
struct | is_diagonal_jacobian
struct | is_double
struct | is_double< double >
struct | is_linear
| Map traits. More...
struct | is_linear< AffineMap >
struct | is_linear< CompoundMap< T1, T2 > >
struct | is_linear< ScaleMap >
struct | is_linear< ScaleTranslateMap >
struct | is_linear< TranslationMap >
struct | is_linear< UniformScaleMap >
struct | is_linear< UniformScaleTranslateMap >
struct | is_linear< UnitaryMap >
struct | is_scale
struct | is_scale< ScaleMap >
struct | is_scale_translate
struct | is_scale_translate< ScaleTranslateMap >
struct | is_uniform_diagonal_jacobian
struct | is_uniform_scale
struct | is_uniform_scale< UniformScaleMap >
struct | is_uniform_scale_translate
struct | is_uniform_scale_translate< TranslationMap >
struct | is_uniform_scale_translate< UniformScaleTranslateMap >
struct | is_vec3d
struct | is_vec3d< Vec3d >
struct | ISCurl
| Curl operator defined in index space using various first derivative schemes. More...
struct | ISDivergence
| Divergence operator defined in index space using various first derivative schemes. More...
struct | ISGradient
| Gradient operators defined in index space of various orders. More...
struct | ISGradientBiased
| Biased Gradient Operators, using upwinding defined by the `Vec3Bias` input. More...
struct | ISGradientNormSqrd
struct | ISLaplacian
| Laplacian defined in index space, using various center-difference stencils. More...
struct | ISLaplacian< CD_FOURTH >
struct | ISLaplacian< CD_SECOND >
struct | ISLaplacian< CD_SIXTH >
struct | ISMeanCurvature
| Compute the mean curvature in index space. More...
struct | ISOpMagnitude
| Adapter for vector-valued index-space operators to return the vector magnitude. More...
struct | Laplacian
| Compute the Laplacian at a given location in a grid using finite differencing of various orders. The result is defined in the range of the map. More...
struct | Laplacian< ScaleMap, DiffScheme >
struct | Laplacian< ScaleTranslateMap, DiffScheme >
struct | Laplacian< TranslationMap, DiffScheme >
struct | Laplacian< UniformScaleMap, DiffScheme >
struct | Laplacian< UniformScaleTranslateMap, DiffScheme >
struct | Laplacian< UnitaryMap, DiffScheme >
struct | LevelSetHDDA
| Helper class that implements Hierarchical Digital Differential Analyzers and is specialized for ray intersections with level sets. More...
struct | LevelSetHDDA< TreeT, -1 >
| Specialization of Hierarchical Digital Differential Analyzer class that intersects a ray against the voxels of a level set. More...
struct | MapAdapter
| Adapter to associate a map with a world-space operator, giving it the same call signature as an index-space operator. More...
class | MapBase
| Abstract base class for maps. More...
class | MapRegistry
| Threadsafe singleton object for accessing the map type-name dictionary. Associates a map type-name with a factory function. More...
class | Mat
class | Mat2
class | Mat3
| 3x3 matrix class. More...
class | Mat4
| 4x4 -matrix class. More...
struct | MeanCurvature
| Compute the mean curvature. More...
struct | MeanCurvature< TranslationMap, DiffScheme2, DiffScheme1 >
struct | MeanCurvature< UniformScaleMap, DiffScheme2, DiffScheme1 >
struct | MeanCurvature< UniformScaleTranslateMap, DiffScheme2, DiffScheme1 >
class | MinMax
| Templated class to compute the minimum and maximum values. More...
class | NineteenPointStencil
class | NonlinearFrustumMap
| This map is composed of three steps. First it will take a box of size (Lx X Ly X Lz) defined by a member data bounding box and map it into a frustum with near plane (1 X Ly/Lx) and prescribed depth Then this frustum is transformed by an internal second map: most often a uniform scale, but other effects can be achieved by accumulating translation, shear and rotation: these are all applied to the second map. More...
struct | OpMagnitude
| Adapter for vector-valued world-space operators to return the vector magnitude. More...
struct | promote
class | QuantizedUnitVec
| Unit vector occupying only 16 bits. More...
class | Quat
class | Rand01
| Simple generator of random numbers over the range [0, 1) More...
class | RandInt
| Simple random integer generator. More...
class | Ray
class | ScaleMap
| A specialized Affine transform that scales along the principal axis the scaling need not be uniform in the three-directions. More...
class | ScaleTranslateMap
| A specialized Affine transform that scales along the principal axis the scaling need not be uniform in the three-directions, and then translates the result. More...
class | SecondOrderDenseStencil
class | SevenPointStencil
class | SixthOrderDenseStencil
class | Stats
| This class computes statistics (minimum value, maximum value, mean, variance and standard deviation) of a population of floating-point values. More...
class | ThirteenPointStencil
struct | Tolerance
| Tolerance for floating-point comparison. More...
struct | Tolerance< double >
struct | Tolerance< float >
class | Transform
class | TranslationMap
| A specialized linear transform that performs a translation. More...
class | Tuple
class | UniformScaleMap
| A specialized Affine transform that scales along the principal axis the scaling is uniform in the three-directions. More...
class | UniformScaleTranslateMap
| A specialized Affine transform that uniformaly scales along the principal axis and then translates the result. More...
class | UnitaryMap
| A specialized linear transform that performs a unitary maping i.e. rotation and or reflection. More...
class | Vec2
class | Vec3
class | Vec4
class | VolumeHDDA
| Helper class that implements Hierarchical Digital Differential Analyzers for ray intersections against a generic volume. More...
class | VolumeHDDA< TreeT, RayT, 0 >
| Specialization of Hierarchical Digital Differential Analyzer class that intersects against the leafs or tiles of a generic volume. More...
class | WenoStencil
| This is a special 19-point stencil that supports optimal fifth-order WENO upwinding, second-order central differencing, Laplacian, and zero-crossing test. More...

## Typedefs
using | UnitaryAndTranslationMap = CompoundMap< UnitaryMap, TranslationMap >
using | SpectralDecomposedMap = CompoundMap< CompoundMap< UnitaryMap, ScaleMap >, UnitaryMap >
using | SymmetricMap = SpectralDecomposedMap
using | FullyDecomposedMap = CompoundMap< SymmetricMap, UnitaryAndTranslationMap >
using | PolarDecomposedMap = CompoundMap< SymmetricMap, UnitaryMap >
using | Mat3s = Mat3< float >
using | Mat3d = Mat3< double >
using | Mat3f = Mat3d
using | Mat4s = Mat4< float >
using | Mat4d = Mat4< double >
using | Mat4f = Mat4d
using | Random01 = Rand01< double, std::mt19937 >
using | RandomInt = RandInt< int, std::mt19937 >
using | Quats = Quat< float >
using | Quatd = Quat< double >
using | Vec2i = Vec2< int32_t >
using | Vec2ui = Vec2< uint32_t >
using | Vec2s = Vec2< float >
using | Vec2d = Vec2< double >
using | Vec3i = Vec3< int32_t >
using | Vec3ui = Vec3< uint32_t >
using | Vec3s = Vec3< float >
using | Vec3d = Vec3< double >
using | Vec4i = Vec4< int32_t >
using | Vec4ui = Vec4< uint32_t >
using | Vec4s = Vec4< float >
using | Vec4d = Vec4< double >

## Enumerations
enum | DScheme { UNKNOWN_DS = -1, CD_2NDT = 0, CD_2ND, CD_4TH, CD_6TH, FD_1ST, FD_2ND, FD_3RD, BD_1ST, BD_2ND, BD_3RD, FD_WENO5, BD_WENO5, FD_HJWENO5, BD_HJWENO5 }
| Different discrete schemes used in the first derivatives. More...
enum | { NUM_DS_SCHEMES = BD_HJWENO5 + 1 }
enum | DDScheme { UNKNOWN_DD = -1, CD_SECOND = 0, CD_FOURTH, CD_SIXTH }
| Different discrete schemes used in the second derivatives. More...
enum | { NUM_DD_SCHEMES = CD_SIXTH + 1 }
enum | BiasedGradientScheme { UNKNOWN_BIAS = -1, FIRST_BIAS = 0, SECOND_BIAS, THIRD_BIAS, WENO5_BIAS, HJWENO5_BIAS }
| Biased Gradients are limited to non-centered differences. More...
enum | { NUM_BIAS_SCHEMES = HJWENO5_BIAS + 1 }
enum | TemporalIntegrationScheme { UNKNOWN_TIS = -1, TVD_RK1, TVD_RK2, TVD_RK3 }
| Temporal integration schemes. More...
enum | { NUM_TEMPORAL_SCHEMES = TVD_RK3 + 1 }
enum | Axis { X_AXIS = 0, Y_AXIS = 1, Z_AXIS = 2 }
enum | RotationOrder { XYZ_ROTATION = 0, XZY_ROTATION, YXZ_ROTATION, YZX_ROTATION, ZXY_ROTATION, ZYX_ROTATION, XZX_ROTATION, ZXZ_ROTATION }

## Functions
template<typename Vec3T >
std::ostream & | operator<< (std::ostream &os, const BBox< Vec3T > &b)
std::ostream & | operator<< (std::ostream &os, const Coord &xyz)
Coord | Abs (const Coord &xyz)
std::ostream & | operator<< (std::ostream &os, const CoordBBox &b)
template<typename RayT , Index Log2Dim>
std::ostream & | operator<< (std::ostream &os, const DDA< RayT, Log2Dim > &dda)
| Output streaming of the Ray class. More...
std::string | dsSchemeToString (DScheme dss)
DScheme | stringToDScheme (const std::string &s)
std::string | dsSchemeToMenuName (DScheme dss)
std::string | biasedGradientSchemeToString (BiasedGradientScheme bgs)
BiasedGradientScheme | stringToBiasedGradientScheme (const std::string &s)
std::string | biasedGradientSchemeToMenuName (BiasedGradientScheme bgs)
std::string | temporalIntegrationSchemeToString (TemporalIntegrationScheme tis)
TemporalIntegrationScheme | stringToTemporalIntegrationScheme (const std::string &s)
std::string | temporalIntegrationSchemeToMenuName (TemporalIntegrationScheme tis)
template<typename ValueType >
ValueType | WENO5 (const ValueType &v1, const ValueType &v2, const ValueType &v3, const ValueType &v4, const ValueType &v5, float scale2=0.01f)
| Implementation of nominally fifth-order finite-difference WENO. More...
template<typename Real >
Real | GodunovsNormSqrd (bool isOutside, Real dP_xm, Real dP_xp, Real dP_ym, Real dP_yp, Real dP_zm, Real dP_zp)
template<typename Real >
Real | GodunovsNormSqrd (bool isOutside, const Vec3< Real > &gradient_m, const Vec3< Real > &gradient_p)
OPENVDB_API SharedPtr< SymmetricMap > | createSymmetricMap (const Mat3d &m)
| Utility methods. More...
OPENVDB_API SharedPtr< FullyDecomposedMap > | createFullyDecomposedMap (const Mat4d &m)
| General decomposition of a Matrix into a Unitary (e.g. rotation) following a Symmetric (e.g. stretch & shear) More...
OPENVDB_API SharedPtr< PolarDecomposedMap > | createPolarDecomposedMap (const Mat3d &m)
| Decomposes a general linear into translation following polar decomposition. More...
OPENVDB_API SharedPtr< MapBase > | simplify (SharedPtr< AffineMap > affine)
| reduces an AffineMap to a ScaleMap or a ScaleTranslateMap when it can More...
OPENVDB_API Mat4d | approxInverse (const Mat4d &mat)
| Returns the left pseudoInverse of the input matrix when the 3x3 part is symmetric otherwise it zeros the 3x3 and reverses the translation. More...
template<class MatType >
MatType | rotation (const Quat< typename MatType::value_type > &q, typename MatType::value_type eps=static_cast< typename MatType::value_type >(1.0e-8))
| Return the rotation matrix specified by the given quaternion. More...
template<class MatType >
MatType | rotation (Axis axis, typename MatType::value_type angle)
| Return a matrix for rotation by angle radians about the given axis. More...
template<class MatType >
MatType | rotation (const Vec3< typename MatType::value_type > &_axis, typename MatType::value_type angle)
| Return a matrix for rotation by angle radians about the given axis. More...
template<class MatType >
Vec3< typename MatType::value_type > | eulerAngles (const MatType &mat, RotationOrder rotationOrder, typename MatType::value_type eps=static_cast< typename MatType::value_type >(1.0e-8))
| Return the Euler angles composing the given rotation matrix. More...
template<typename MatType , typename ValueType1 , typename ValueType2 >
MatType | rotation (const Vec3< ValueType1 > &_v1, const Vec3< ValueType2 > &_v2, typename MatType::value_type eps=static_cast< typename MatType::value_type >(1.0e-8))
| Return a rotation matrix that maps v1 onto v2 about the cross product of v1 and v2. More...
template<class MatType >
MatType | scale (const Vec3< typename MatType::value_type > &s)
| Return a matrix that scales by s. More...
template<class MatType >
Vec3< typename MatType::value_type > | getScale (const MatType &mat)
| Return a Vec3 representing the lengths of the passed matrix's upper 3×3's rows. More...
template<class MatType >
MatType | unit (const MatType &mat, typename MatType::value_type eps=1.0e-8)
| Return a copy of the given matrix with its upper 3×3 rows normalized. More...
template<class MatType >
MatType | unit (const MatType &in, typename MatType::value_type eps, Vec3< typename MatType::value_type > &scaling)
| Return a copy of the given matrix with its upper 3×3 rows normalized, and return the length of each of these rows in scaling. More...
template<class MatType >
MatType | shear (Axis axis0, Axis axis1, typename MatType::value_type shear)
| Set the matrix to a shear along axis0 by a fraction of axis1. More...
template<class MatType >
MatType | skew (const Vec3< typename MatType::value_type > &skew)
| Return a matrix as the cross product of the given vector. More...
template<class MatType >
MatType | aim (const Vec3< typename MatType::value_type > &direction, const Vec3< typename MatType::value_type > &vertical)
| Return an orientation matrix such that z points along direction, and y is along the direction / vertical plane. More...
template<class MatType >
MatType | snapMatBasis (const MatType &source, Axis axis, const Vec3< typename MatType::value_type > &direction)
| This function snaps a specific axis to a specific direction, preserving scaling. More...
template<class MatType >
static MatType & | padMat4 (MatType &dest)
| Write 0s along Mat4's last row and column, and a 1 on its diagonal. More...
template<typename MatType >
void | sqrtSolve (const MatType &aA, MatType &aB, double aTol=0.01)
| Solve for A=B*B, given A. More...
template<typename MatType >
void | powSolve (const MatType &aA, MatType &aB, double aPower, double aTol=0.01)
template<typename MatType >
bool | isIdentity (const MatType &m)
| Determine if a matrix is an identity matrix. More...
template<typename MatType >
bool | isInvertible (const MatType &m)
| Determine if a matrix is invertible. More...
template<typename MatType >
bool | isSymmetric (const MatType &m)
| Determine if a matrix is symmetric. More...
template<typename MatType >
bool | isUnitary (const MatType &m)
| Determine if a matrix is unitary (i.e., rotation or reflection). More...
template<typename MatType >
bool | isDiagonal (const MatType &mat)
| Determine if a matrix is diagonal. More...
template<typename MatType >
MatType::ValueType | lInfinityNorm (const MatType &matrix)
| Return the L∞ norm of an N×N matrix. More...
template<typename MatType >
MatType::ValueType | lOneNorm (const MatType &matrix)
| Return the L1 norm of an N×N matrix. More...
template<typename MatType >
bool | polarDecomposition (const MatType &input, MatType &unitary, MatType &positive_hermitian, unsigned int MAX_ITERATIONS=100)
| Decompose an invertible 3×3 matrix into a unitary matrix followed by a symmetric matrix (positive semi-definite Hermitian), i.e., M = U * S. More...
template<typename T0 , typename T1 >
Mat3< typename promote< T0, T1 >::type > | operator* (const Mat3< T0 > &m0, const Mat3< T1 > &m1)
| Multiply m0 by m1 and return the resulting matrix. More...
template<typename T >
Mat3< T > | outerProduct (const Vec3< T > &v1, const Vec3< T > &v2)
template<typename T , typename T0 >
Mat3< T > | powLerp (const Mat3< T0 > &m1, const Mat3< T0 > &m2, T t)
template<typename T >
bool | diagonalizeSymmetricMatrix (const Mat3< T > &input, Mat3< T > &Q, Vec3< T > &D, unsigned int MAX_ITERATIONS=250)
| Use Jacobi iterations to decompose a symmetric 3x3 matrix (diagonalize and compute eigenvectors) More...
template<typename T0 , typename T1 >
Vec3< T1 > | transformNormal (const Mat4< T0 > &m, const Vec3< T1 > &n)
template<typename T >
bool | isAffine (const Mat4< T > &m)
template<typename T >
bool | hasTranslation (const Mat4< T > &m)
template<typename T >
T | negative (const T &val)
| Return the unary negation of the given value. More...
template<>
bool | negative (const bool &val)
| Return the negation of the given boolean. More...
template<>
std::string | negative (const std::string &val)
| Return the "negation" of the given string. More...
template<typename Type >
Type | Clamp (Type x, Type min, Type max)
| Return x clamped to [min, max]. More...
template<typename Type >
Type | Clamp01 (Type x)
| Return x clamped to [0, 1]. More...
template<typename Type >
bool | ClampTest01 (Type &x)
| Return `true` if x is outside [0,1]. More...
template<typename Type >
Type | SmoothUnitStep (Type x)
| Return 0 if x < 0, 1 if x > 1 or else (3 − 2 x) x². More...
template<typename Type >
Type | SmoothUnitStep (Type x, Type min, Type max)
| Return 0 if x < min, 1 if x > max or else (3 − 2 t) t², where t = (x − min)/(max − min). More...
template<typename Type >
bool | isZero (const Type &x)
| Return `true` if x is exactly equal to zero. More...
template<typename Type >
bool | isApproxZero (const Type &x)
| Return `true` if x is equal to zero to within the default floating-point comparison tolerance. More...
template<typename Type >
bool | isApproxZero (const Type &x, const Type &tolerance)
| Return `true` if x is equal to zero to within the given tolerance. More...
template<typename Type >
bool | isNegative (const Type &x)
| Return `true` if x is less than zero. More...
template<>
bool | isNegative< bool > (const bool &)
bool | isFinite (const float x)
| Return `true` if x is finite. More...
template<typename Type , typename std::enable_if< std::is_arithmetic< Type >::value, int >::type = 0>
bool | isFinite (const Type &x)
| Return `true` if x is finite. More...
template<typename Type >
bool | isApproxEqual (const Type &a, const Type &b)
| Return `true` if a is equal to b to within the default floating-point comparison tolerance. More...
template<typename Type >
bool | isApproxEqual (const Type &a, const Type &b, const Type &tolerance)
| Return `true` if a is equal to b to within the given tolerance. More...
template<>
bool | isApproxEqual< bool > (const bool &a, const bool &b)
template<>
bool | isApproxEqual< bool > (const bool &a, const bool &b, const bool &)
template<>
bool | isApproxEqual< std::string > (const std::string &a, const std::string &b)
template<>
bool | isApproxEqual< std::string > (const std::string &a, const std::string &b, const std::string &)
template<typename Type >
bool | isApproxLarger (const Type &a, const Type &b, const Type &tolerance)
| Return `true` if a is larger than b to within the given tolerance, i.e., if b - a < tolerance. More...
template<typename T0 , typename T1 >
bool | isExactlyEqual (const T0 &a, const T1 &b)
| Return `true` if a is exactly equal to b. More...
template<typename Type >
bool | isRelOrApproxEqual (const Type &a, const Type &b, const Type &absTol, const Type &relTol)
template<>
bool | isRelOrApproxEqual (const bool &a, const bool &b, const bool &, const bool &)
int32_t | floatToInt32 (const float aFloatValue)
int64_t | doubleToInt64 (const double aDoubleValue)
bool | isUlpsEqual (const double aLeft, const double aRight, const int64_t aUnitsInLastPlace)
bool | isUlpsEqual (const float aLeft, const float aRight, const int32_t aUnitsInLastPlace)
template<typename Type >
Type | Pow2 (Type x)
| Return x2. More...
template<typename Type >
Type | Pow3 (Type x)
| Return x3. More...
template<typename Type >
Type | Pow4 (Type x)
| Return x4. More...
template<typename Type >
Type | Pow (Type x, int n)
| Return xn. More...
template<typename Type >
const Type & | Max (const Type &a, const Type &b)
| Return the maximum of two values. More...
template<typename Type >
const Type & | Max (const Type &a, const Type &b, const Type &c)
| Return the maximum of three values. More...
template<typename Type >
const Type & | Max (const Type &a, const Type &b, const Type &c, const Type &d)
| Return the maximum of four values. More...
template<typename Type >
const Type & | Max (const Type &a, const Type &b, const Type &c, const Type &d, const Type &e)
| Return the maximum of five values. More...
template<typename Type >
const Type & | Max (const Type &a, const Type &b, const Type &c, const Type &d, const Type &e, const Type &f)
| Return the maximum of six values. More...
template<typename Type >
const Type & | Max (const Type &a, const Type &b, const Type &c, const Type &d, const Type &e, const Type &f, const Type &g)
| Return the maximum of seven values. More...
template<typename Type >
const Type & | Max (const Type &a, const Type &b, const Type &c, const Type &d, const Type &e, const Type &f, const Type &g, const Type &h)
| Return the maximum of eight values. More...
template<typename Type >
const Type & | Min (const Type &a, const Type &b)
| Return the minimum of two values. More...
template<typename Type >
const Type & | Min (const Type &a, const Type &b, const Type &c)
| Return the minimum of three values. More...
template<typename Type >
const Type & | Min (const Type &a, const Type &b, const Type &c, const Type &d)
| Return the minimum of four values. More...
template<typename Type >
const Type & | Min (const Type &a, const Type &b, const Type &c, const Type &d, const Type &e)
| Return the minimum of five values. More...
template<typename Type >
const Type & | Min (const Type &a, const Type &b, const Type &c, const Type &d, const Type &e, const Type &f)
| Return the minimum of six values. More...
template<typename Type >
const Type & | Min (const Type &a, const Type &b, const Type &c, const Type &d, const Type &e, const Type &f, const Type &g)
| Return the minimum of seven values. More...
template<typename Type >
const Type & | Min (const Type &a, const Type &b, const Type &c, const Type &d, const Type &e, const Type &f, const Type &g, const Type &h)
| Return the minimum of eight values. More...
template<typename Type >
Type | Exp (const Type &x)
| Return ex. More...
template<typename Type >
int | Sign (const Type &x)
| Return the sign of the given value as an integer (either -1, 0 or 1). More...
template<typename Type >
bool | SignChange (const Type &a, const Type &b)
| Return `true` if a and b have different signs. More...
template<typename Type >
bool | ZeroCrossing (const Type &a, const Type &b)
| Return `true` if the interval [a, b] includes zero, i.e., if either a or b is zero or if they have different signs. More...
template<typename Type >
Type | RoundUp (Type x, Type base)
| Return x rounded up to the nearest multiple of base. More...
template<typename Type >
Type | RoundDown (Type x, Type base)
| Return x rounded down to the nearest multiple of base. More...
template<typename Type >
Type | EuclideanRemainder (Type x)
template<typename Type >
Type | IntegerPart (Type x)
| Return the integer part of x. More...
template<typename Type >
Type | FractionalPart (Type x)
| Return the fractional part of x. More...
template<typename Type >
Type | Chop (Type x, Type delta)
| Return x if it is greater or equal in magnitude than delta. Otherwise, return zero. More...
template<typename Type >
Type | Truncate (Type x, unsigned int digits)
| Return x truncated to the given number of decimal digits. More...
template<typename T >
auto | PrintCast (const T &val) -> typename std::enable_if<!std::is_same< T, int8_t >::value &&!std::is_same< T, uint8_t >::value, const T &>::type
| 8-bit integer values print to std::ostreams as characters. Cast them so that they print as integers instead. More...
int32_t | PrintCast (int8_t val)
uint32_t | PrintCast (uint8_t val)
template<typename Type >
Type | Inv (Type x)
| Return the inverse of x. More...
template<typename Vec3T >
size_t | MinIndex (const Vec3T &v)
| Return the index [0,1,2] of the smallest value in a 3D vector. More...
template<typename Vec3T >
size_t | MaxIndex (const Vec3T &v)
| Return the index [0,1,2] of the largest value in a 3D vector. More...
OPENVDB_API Vec3d | closestPointOnTriangleToPoint (const Vec3d &a, const Vec3d &b, const Vec3d &c, const Vec3d &p, Vec3d &uvw)
| Closest Point on Triangle to Point. Given a triangle `abc` and a point `p`, return the point on `abc` closest to `p` and the corresponding barycentric coordinates. More...
OPENVDB_API Vec3d | closestPointOnSegmentToPoint (const Vec3d &a, const Vec3d &b, const Vec3d &p, double &t)
| Closest Point on Line Segment to Point. Given segment `ab` and point `p`, return the point on `ab` closest to `p` and `t` the parametric distance to `b`. More...
template<typename T >
Quat< T > | slerp (const Quat< T > &q1, const Quat< T > &q2, T t, T tolerance=0.00001)
| Linear interpolation between the two quaternions. More...
template<typename S , typename T >
Quat< T > | operator* (S scalar, const Quat< T > &q)
| Multiply each element of the given quaternion by scalar and return the result. More...
template<typename T , typename T0 >
Mat3< T > | slerp (const Mat3< T0 > &m1, const Mat3< T0 > &m2, T t)
| Interpolate between m1 and m2. Converts to quaternion form and uses slerp m1 and m2 must be rotation matrices! More...
template<typename T , typename T0 >
Mat3< T > | bezLerp (const Mat3< T0 > &m1, const Mat3< T0 > &m2, const Mat3< T0 > &m3, const Mat3< T0 > &m4, T t)
template<typename RealT >
std::ostream & | operator<< (std::ostream &os, const Ray< RealT > &r)
| Output streaming of the Ray class. More...
OPENVDB_API void | calculateBounds (const Transform &t, const Vec3d &minWS, const Vec3d &maxWS, Vec3d &minIS, Vec3d &maxIS)
| Calculate an axis-aligned bounding box in index space from an axis-aligned bounding box in world space. More...
OPENVDB_API std::ostream & | operator<< (std::ostream &, const Transform &)
template<typename ResolvedMapType , typename OpType >
void | doProcessTypedMap (Transform &transform, OpType &op)
| Helper function used internally by processTypedMap() More...
template<typename ResolvedMapType , typename OpType >
void | doProcessTypedMap (const Transform &transform, OpType &op)
| Helper function used internally by processTypedMap() More...
template<typename TransformType , typename OpType >
bool | processTypedMap (TransformType &transform, OpType &op)
| Utility function that, given a generic map pointer, calls a functor on the fully-resoved map. More...
template<int SIZE, typename T0 , typename T1 >
bool | operator< (const Tuple< SIZE, T0 > &t0, const Tuple< SIZE, T1 > &t1)
template<int SIZE, typename T0 , typename T1 >
bool | operator> (const Tuple< SIZE, T0 > &t0, const Tuple< SIZE, T1 > &t1)
template<int SIZE, typename T >
Tuple< SIZE, T > | Abs (const Tuple< SIZE, T > &t)
template<int SIZE, typename T >
bool | isNan (const Tuple< SIZE, T > &t)
| Return `true` if a Nan is present in the tuple. More...
template<int SIZE, typename T >
bool | isInfinite (const Tuple< SIZE, T > &t)
| Return `true` if an Inf is present in the tuple. More...
template<int SIZE, typename T >
bool | isFinite (const Tuple< SIZE, T > &t)
| Return `true` if no Nan or Inf values are present. More...
template<int SIZE, typename T >
bool | isZero (const Tuple< SIZE, T > &t)
| Return `true` if all elements are exactly equal to zero. More...
template<int SIZE, typename T >
std::ostream & | operator<< (std::ostream &ostr, const Tuple< SIZE, T > &classname)
| Write a Tuple to an output stream. More...
template<typename S , typename T >
Vec2< typename promote< S, T >::type > | operator* (S scalar, const Vec2< T > &v)
| Multiply each element of the given vector by scalar and return the result. More...
template<typename S , typename T >
Vec2< typename promote< S, T >::type > | operator* (const Vec2< T > &v, S scalar)
| Multiply each element of the given vector by scalar and return the result. More...
template<typename T0 , typename T1 >
Vec2< typename promote< T0, T1 >::type > | operator* (const Vec2< T0 > &v0, const Vec2< T1 > &v1)
| Multiply corresponding elements of v0 and v1 and return the result. More...
template<typename S , typename T >
Vec2< typename promote< S, T >::type > | operator/ (S scalar, const Vec2< T > &v)
| Divide scalar by each element of the given vector and return the result. More...
template<typename S , typename T >
Vec2< typename promote< S, T >::type > | operator/ (const Vec2< T > &v, S scalar)
| Divide each element of the given vector by scalar and return the result. More...
template<typename T0 , typename T1 >
Vec2< typename promote< T0, T1 >::type > | operator/ (const Vec2< T0 > &v0, const Vec2< T1 > &v1)
| Divide corresponding elements of v0 and v1 and return the result. More...
template<typename T0 , typename T1 >
Vec2< typename promote< T0, T1 >::type > | operator+ (const Vec2< T0 > &v0, const Vec2< T1 > &v1)
| Add corresponding elements of v0 and v1 and return the result. More...
template<typename S , typename T >
Vec2< typename promote< S, T >::type > | operator+ (const Vec2< T > &v, S scalar)
| Add scalar to each element of the given vector and return the result. More...
template<typename T0 , typename T1 >
Vec2< typename promote< T0, T1 >::type > | operator- (const Vec2< T0 > &v0, const Vec2< T1 > &v1)
| Subtract corresponding elements of v0 and v1 and return the result. More...
template<typename S , typename T >
Vec2< typename promote< S, T >::type > | operator- (const Vec2< T > &v, S scalar)
| Subtract scalar from each element of the given vector and return the result. More...
template<typename T >
T | angle (const Vec2< T > &v1, const Vec2< T > &v2)
template<typename T >
bool | isApproxEqual (const Vec2< T > &a, const Vec2< T > &b)
template<typename T >
bool | isApproxEqual (const Vec2< T > &a, const Vec2< T > &b, const Vec2< T > &eps)
template<typename T >
Vec2< T > | Abs (const Vec2< T > &v)
template<typename T >
void | orthonormalize (Vec2< T > &v1, Vec2< T > &v2)
template<typename T >
Vec2< T > | minComponent (const Vec2< T > &v1, const Vec2< T > &v2)
| Return component-wise minimum of the two vectors. More...
template<typename T >
Vec2< T > | maxComponent (const Vec2< T > &v1, const Vec2< T > &v2)
| Return component-wise maximum of the two vectors. More...
template<typename T >
Vec2< T > | Exp (Vec2< T > v)
| Return a vector with the exponent applied to each of the components of the input vector. More...
template<typename T >
Vec2< T > | Log (Vec2< T > v)
| Return a vector with log applied to each of the components of the input vector. More...
template<typename T0 , typename T1 >
bool | operator== (const Vec3< T0 > &v0, const Vec3< T1 > &v1)
| Equality operator, does exact floating point comparisons. More...
template<typename T0 , typename T1 >
bool | operator!= (const Vec3< T0 > &v0, const Vec3< T1 > &v1)
| Inequality operator, does exact floating point comparisons. More...
template<typename S , typename T >
Vec3< typename promote< S, T >::type > | operator* (S scalar, const Vec3< T > &v)
| Multiply each element of the given vector by scalar and return the result. More...
template<typename S , typename T >
Vec3< typename promote< S, T >::type > | operator* (const Vec3< T > &v, S scalar)
| Multiply each element of the given vector by scalar and return the result. More...
template<typename T0 , typename T1 >
Vec3< typename promote< T0, T1 >::type > | operator* (const Vec3< T0 > &v0, const Vec3< T1 > &v1)
| Multiply corresponding elements of v0 and v1 and return the result. More...
template<typename S , typename T >
Vec3< typename promote< S, T >::type > | operator/ (S scalar, const Vec3< T > &v)
| Divide scalar by each element of the given vector and return the result. More...
template<typename S , typename T >
Vec3< typename promote< S, T >::type > | operator/ (const Vec3< T > &v, S scalar)
| Divide each element of the given vector by scalar and return the result. More...
template<typename T0 , typename T1 >
Vec3< typename promote< T0, T1 >::type > | operator/ (const Vec3< T0 > &v0, const Vec3< T1 > &v1)
| Divide corresponding elements of v0 and v1 and return the result. More...
template<typename T0 , typename T1 >
Vec3< typename promote< T0, T1 >::type > | operator+ (const Vec3< T0 > &v0, const Vec3< T1 > &v1)
| Add corresponding elements of v0 and v1 and return the result. More...
template<typename S , typename T >
Vec3< typename promote< S, T >::type > | operator+ (const Vec3< T > &v, S scalar)
| Add scalar to each element of the given vector and return the result. More...
template<typename T0 , typename T1 >
Vec3< typename promote< T0, T1 >::type > | operator- (const Vec3< T0 > &v0, const Vec3< T1 > &v1)
| Subtract corresponding elements of v0 and v1 and return the result. More...
template<typename S , typename T >
Vec3< typename promote< S, T >::type > | operator- (const Vec3< T > &v, S scalar)
| Subtract scalar from each element of the given vector and return the result. More...
template<typename T >
T | angle (const Vec3< T > &v1, const Vec3< T > &v2)
template<typename T >
bool | isApproxEqual (const Vec3< T > &a, const Vec3< T > &b)
template<typename T >
bool | isApproxEqual (const Vec3< T > &a, const Vec3< T > &b, const Vec3< T > &eps)
template<typename T >
Vec3< T > | Abs (const Vec3< T > &v)
template<typename T >
void | orthonormalize (Vec3< T > &v1, Vec3< T > &v2, Vec3< T > &v3)
template<typename T >
Vec3< T > | minComponent (const Vec3< T > &v1, const Vec3< T > &v2)
| Return component-wise minimum of the two vectors. More...
template<typename T >
Vec3< T > | maxComponent (const Vec3< T > &v1, const Vec3< T > &v2)
| Return component-wise maximum of the two vectors. More...
template<typename T >
Vec3< T > | Exp (Vec3< T > v)
| Return a vector with the exponent applied to each of the components of the input vector. More...
template<typename T >
Vec3< T > | Log (Vec3< T > v)
| Return a vector with log applied to each of the components of the input vector. More...
template<typename T0 , typename T1 >
bool | operator== (const Vec4< T0 > &v0, const Vec4< T1 > &v1)
| Equality operator, does exact floating point comparisons. More...
template<typename T0 , typename T1 >
bool | operator!= (const Vec4< T0 > &v0, const Vec4< T1 > &v1)
| Inequality operator, does exact floating point comparisons. More...
template<typename S , typename T >
Vec4< typename promote< S, T >::type > | operator* (S scalar, const Vec4< T > &v)
| Multiply each element of the given vector by scalar and return the result. More...
template<typename S , typename T >
Vec4< typename promote< S, T >::type > | operator* (const Vec4< T > &v, S scalar)
| Multiply each element of the given vector by scalar and return the result. More...
template<typename T0 , typename T1 >
Vec4< typename promote< T0, T1 >::type > | operator* (const Vec4< T0 > &v0, const Vec4< T1 > &v1)
| Multiply corresponding elements of v0 and v1 and return the result. More...
template<typename S , typename T >
Vec4< typename promote< S, T >::type > | operator/ (S scalar, const Vec4< T > &v)
| Divide scalar by each element of the given vector and return the result. More...
template<typename S , typename T >
Vec4< typename promote< S, T >::type > | operator/ (const Vec4< T > &v, S scalar)
| Divide each element of the given vector by scalar and return the result. More...
template<typename T0 , typename T1 >
Vec4< typename promote< T0, T1 >::type > | operator/ (const Vec4< T0 > &v0, const Vec4< T1 > &v1)
| Divide corresponding elements of v0 and v1 and return the result. More...
template<typename T0 , typename T1 >
Vec4< typename promote< T0, T1 >::type > | operator+ (const Vec4< T0 > &v0, const Vec4< T1 > &v1)
| Add corresponding elements of v0 and v1 and return the result. More...
template<typename S , typename T >
Vec4< typename promote< S, T >::type > | operator+ (const Vec4< T > &v, S scalar)
| Add scalar to each element of the given vector and return the result. More...
template<typename T0 , typename T1 >
Vec4< typename promote< T0, T1 >::type > | operator- (const Vec4< T0 > &v0, const Vec4< T1 > &v1)
| Subtract corresponding elements of v0 and v1 and return the result. More...
template<typename S , typename T >
Vec4< typename promote< S, T >::type > | operator- (const Vec4< T > &v, S scalar)
| Subtract scalar from each element of the given vector and return the result. More...
template<typename T >
bool | isApproxEqual (const Vec4< T > &a, const Vec4< T > &b)
template<typename T >
bool | isApproxEqual (const Vec4< T > &a, const Vec4< T > &b, const Vec4< T > &eps)
template<typename T >
Vec4< T > | Abs (const Vec4< T > &v)
template<typename T >
Vec4< T > | minComponent (const Vec4< T > &v1, const Vec4< T > &v2)
| Return component-wise minimum of the two vectors. More...
template<typename T >
Vec4< T > | maxComponent (const Vec4< T > &v1, const Vec4< T > &v2)
| Return component-wise maximum of the two vectors. More...
template<typename T >
Vec4< T > | Exp (Vec4< T > v)
| Return a vector with the exponent applied to each of the components of the input vector. More...
template<typename T >
Vec4< T > | Log (Vec4< T > v)
| Return a vector with log applied to each of the components of the input vector. More...
template<typename T >
Vec3< typename promote< T, typename Coord::ValueType >::type > | operator+ (const Vec3< T > &v0, const Coord &v1)
| Allow a Coord to be added to or subtracted from a Vec3. More...
template<typename T >
Vec3< typename promote< T, typename Coord::ValueType >::type > | operator+ (const Coord &v1, const Vec3< T > &v0)
| Allow a Coord to be added to or subtracted from a Vec3. More...
template<typename T >
Vec3< typename promote< T, Coord::ValueType >::type > | operator- (const Vec3< T > &v0, const Coord &v1)
| Allow a Coord to be subtracted from a Vec3. More...
template<typename T >
Vec3< typename promote< T, Coord::ValueType >::type > | operator- (const Coord &v1, const Vec3< T > &v0)
| Allow a Coord to be subtracted from a Vec3. More...
int32_t | Abs (int32_t i)
| Return the absolute value of the given quantity. More...
int64_t | Abs (int64_t i)
| Return the absolute value of the given quantity. More...
float | Abs (float x)
| Return the absolute value of the given quantity. More...
double | Abs (double x)
| Return the absolute value of the given quantity. More...
long double | Abs (long double x)
| Return the absolute value of the given quantity. More...
uint32_t | Abs (uint32_t i)
| Return the absolute value of the given quantity. More...
uint64_t | Abs (uint64_t i)
| Return the absolute value of the given quantity. More...
bool | Abs (bool b)
| Return the absolute value of the given quantity. More...
float | Pow (float b, float e)
| Return be. More...
double | Pow (double b, double e)
| Return be. More...
float | Sin (const float &x)
| Return sin x. More...
double | Sin (const double &x)
| Return sin x. More...
float | Cos (const float &x)
| Return cos x. More...
double | Cos (const double &x)
| Return cos x. More...
float | Sqrt (float x)
| Return the square root of a floating-point value. More...
double | Sqrt (double x)
| Return the square root of a floating-point value. More...
long double | Sqrt (long double x)
| Return the square root of a floating-point value. More...
float | Cbrt (float x)
| Return the cube root of a floating-point value. More...
double | Cbrt (double x)
| Return the cube root of a floating-point value. More...
long double | Cbrt (long double x)
| Return the cube root of a floating-point value. More...
int | Mod (int x, int y)
| Return the remainder of x / y. More...
float | Mod (float x, float y)
| Return the remainder of x / y. More...
double | Mod (double x, double y)
| Return the remainder of x / y. More...
long double | Mod (long double x, long double y)
| Return the remainder of x / y. More...
template<typename Type >
Type | Remainder (Type x, Type y)
| Return the remainder of x / y. More...
float | RoundUp (float x)
| Return x rounded up to the nearest integer. More...
double | RoundUp (double x)
| Return x rounded up to the nearest integer. More...
long double | RoundUp (long double x)
| Return x rounded up to the nearest integer. More...
float | RoundDown (float x)
| Return x rounded down to the nearest integer. More...
double | RoundDown (double x)
| Return x rounded down to the nearest integer. More...
long double | RoundDown (long double x)
| Return x rounded down to the nearest integer. More...
float | Round (float x)
| Return x rounded to the nearest integer. More...
double | Round (double x)
| Return x rounded to the nearest integer. More...
long double | Round (long double x)
| Return x rounded to the nearest integer. More...
int | Floor (float x)
| Return the floor of x. More...
int | Floor (double x)
| Return the floor of x. More...
int | Floor (long double x)
| Return the floor of x. More...
int | Ceil (float x)
| Return the ceiling of x. More...
int | Ceil (double x)
| Return the ceiling of x. More...
int | Ceil (long double x)
| Return the ceiling of x. More...

## Typedef Documentation

## ◆ FullyDecomposedMap

using FullyDecomposedMap = CompoundMap<SymmetricMap, UnitaryAndTranslationMap>

## ◆ Mat3d

using Mat3d = Mat3<double>

## ◆ Mat3f

using Mat3f = Mat3d

## ◆ Mat3s

using Mat3s = Mat3<float>

## ◆ Mat4d

using Mat4d = Mat4<double>

## ◆ Mat4f

using Mat4f = Mat4d

## ◆ Mat4s

using Mat4s = Mat4<float>

## ◆ PolarDecomposedMap

using PolarDecomposedMap = CompoundMap<SymmetricMap, UnitaryMap>

## ◆ Quatd

using Quatd = Quat<double>

## ◆ Quats

using Quats = Quat<float>

## ◆ Random01

using Random01 = Rand01<double, std::mt19937>

## ◆ RandomInt

using RandomInt = RandInt<int, std::mt19937>

## ◆ SpectralDecomposedMap

using SpectralDecomposedMap = CompoundMap<CompoundMap<UnitaryMap, ScaleMap>, UnitaryMap>

## ◆ SymmetricMap

using SymmetricMap = SpectralDecomposedMap

## ◆ UnitaryAndTranslationMap

using UnitaryAndTranslationMap = CompoundMap<UnitaryMap, TranslationMap>

## ◆ Vec2d

using Vec2d = Vec2<double>

## ◆ Vec2i

using Vec2i = Vec2<int32_t>

## ◆ Vec2s

using Vec2s = Vec2<float>

## ◆ Vec2ui

using Vec2ui = Vec2<uint32_t>

## ◆ Vec3d

using Vec3d = Vec3<double>

## ◆ Vec3i

using Vec3i = Vec3<int32_t>

## ◆ Vec3s

using Vec3s = Vec3<float>

## ◆ Vec3ui

using Vec3ui = Vec3<uint32_t>

## ◆ Vec4d

using Vec4d = Vec4<double>

## ◆ Vec4i

using Vec4i = Vec4<int32_t>

## ◆ Vec4s

using Vec4s = Vec4<float>

## ◆ Vec4ui

using Vec4ui = Vec4<uint32_t>

## Enumeration Type Documentation

## ◆ anonymous enum

anonymous enum

Enumerator
NUM_DS_SCHEMES |

## ◆ anonymous enum

anonymous enum

Enumerator
NUM_DD_SCHEMES |

## ◆ anonymous enum

anonymous enum

Enumerator
NUM_BIAS_SCHEMES |

## ◆ anonymous enum

anonymous enum

Enumerator
NUM_TEMPORAL_SCHEMES |

## ◆ Axis

enum Axis

Enumerator
X_AXIS |
Y_AXIS |
Z_AXIS |

## ◆ BiasedGradientScheme

enum BiasedGradientScheme

Biased Gradients are limited to non-centered differences.

Enumerator
UNKNOWN_BIAS |
FIRST_BIAS |
SECOND_BIAS |
THIRD_BIAS |
WENO5_BIAS |
HJWENO5_BIAS |

## ◆ DDScheme

enum DDScheme

Different discrete schemes used in the second derivatives.

Enumerator
UNKNOWN_DD |
CD_SECOND |
CD_FOURTH |
CD_SIXTH |

## ◆ DScheme

enum DScheme

Different discrete schemes used in the first derivatives.

Enumerator
UNKNOWN_DS |
CD_2NDT |
CD_2ND |
CD_4TH |
CD_6TH |
FD_1ST |
FD_2ND |
FD_3RD |
BD_1ST |
BD_2ND |
BD_3RD |
FD_WENO5 |
BD_WENO5 |
FD_HJWENO5 |
BD_HJWENO5 |

## ◆ RotationOrder

enum RotationOrder

Enumerator
XYZ_ROTATION |
XZY_ROTATION |
YXZ_ROTATION |
YZX_ROTATION |
ZXY_ROTATION |
ZYX_ROTATION |
XZX_ROTATION |
ZXZ_ROTATION |

## ◆ TemporalIntegrationScheme

enum TemporalIntegrationScheme

Temporal integration schemes.

Enumerator
UNKNOWN_TIS |
TVD_RK1 |
TVD_RK2 |
TVD_RK3 |

## Function Documentation

## ◆ Abs() [1/13]

Tuple<SIZE, T> openvdb::v5_2::math::Abs | ( | const Tuple< SIZE, T > & | t | ) |

Returns
  the absolute value of the given Tuple.

## ◆ Abs() [2/13]

int32_t openvdb::v5_2::math::Abs | ( | int32_t | i | ) |
| inline

Return the absolute value of the given quantity.

## ◆ Abs() [3/13]

int64_t openvdb::v5_2::math::Abs | ( | int64_t | i | ) |
| inline

Return the absolute value of the given quantity.

## ◆ Abs() [4/13]

float openvdb::v5_2::math::Abs | ( | float | x | ) |
| inline

Return the absolute value of the given quantity.

## ◆ Abs() [5/13]

double openvdb::v5_2::math::Abs | ( | double | x | ) |
| inline

Return the absolute value of the given quantity.

## ◆ Abs() [6/13]

long double openvdb::v5_2::math::Abs | ( | long double | x | ) |
| inline

Return the absolute value of the given quantity.

## ◆ Abs() [7/13]

uint32_t openvdb::v5_2::math::Abs | ( | uint32_t | i | ) |
| inline

Return the absolute value of the given quantity.

## ◆ Abs() [8/13]

uint64_t openvdb::v5_2::math::Abs | ( | uint64_t | i | ) |
| inline

Return the absolute value of the given quantity.

## ◆ Abs() [9/13]

bool openvdb::v5_2::math::Abs | ( | bool | b | ) |
| inline

Return the absolute value of the given quantity.

## ◆ Abs() [10/13]

Vec2<T> openvdb::v5_2::math::Abs | ( | const Vec2< T > & | v | ) |
| inline

## ◆ Abs() [11/13]

Coord openvdb::v5_2::math::Abs | ( | const Coord & | xyz | ) |
| inline

## ◆ Abs() [12/13]

Vec4<T> openvdb::v5_2::math::Abs | ( | const Vec4< T > & | v | ) |
| inline

## ◆ Abs() [13/13]

Vec3<T> openvdb::v5_2::math::Abs | ( | const Vec3< T > & | v | ) |
| inline

## ◆ aim()

MatType openvdb::v5_2::math::aim | ( | const Vec3< typename MatType::value_type > & | direction,
|  | const Vec3< typename MatType::value_type > & | vertical
| ) |  |

Return an orientation matrix such that z points along direction, and y is along the direction / vertical plane.

## ◆ angle() [1/2]

T openvdb::v5_2::math::angle | ( | const Vec2< T > & | v1,
|  | const Vec2< T > & | v2
| ) |  |
| inline

Angle between two vectors, the result is between [0, pi], e.g. float a = Vec2f::angle(v1,v2);

## ◆ angle() [2/2]

T openvdb::v5_2::math::angle | ( | const Vec3< T > & | v1,
|  | const Vec3< T > & | v2
| ) |  |
| inline

Angle between two vectors, the result is between [0, pi], e.g. double a = Vec3d::angle(v1,v2);

## ◆ approxInverse()

OPENVDB_API Mat4d openvdb::v5_2::math::approxInverse | ( | const Mat4d & | mat | ) |

Returns the left pseudoInverse of the input matrix when the 3x3 part is symmetric otherwise it zeros the 3x3 and reverses the translation.

## ◆ bezLerp()

Mat3<T> openvdb::v5_2::math::bezLerp | ( | const Mat3< T0 > & | m1,
|  | const Mat3< T0 > & | m2,
|  | const Mat3< T0 > & | m3,
|  | const Mat3< T0 > & | m4,
|  | T | t
| ) |  |

Interpolate between m1 and m4 by converting m1 ... m4 into quaternions and treating them as control points of a Bezier curve using slerp in place of lerp in the De Castlejeau evaluation algorithm. Just like a cubic Bezier curve, this will interpolate m1 at t = 0 and m4 at t = 1 but in general will not pass through m2 and m3. Unlike a standard Bezier curve this curve will not have the convex hull property. m1 ... m4 must be rotation matrices!

## ◆ biasedGradientSchemeToMenuName()

std::string openvdb::v5_2::math::biasedGradientSchemeToMenuName | ( | BiasedGradientScheme | bgs | ) |
| inline

## ◆ biasedGradientSchemeToString()

std::string openvdb::v5_2::math::biasedGradientSchemeToString | ( | BiasedGradientScheme | bgs | ) |
| inline

## ◆ calculateBounds()

OPENVDB_API void openvdb::v5_2::math::calculateBounds | ( | const Transform & | t,
|  | const Vec3d & | minWS,
|  | const Vec3d & | maxWS,
|  | Vec3d & | minIS,
|  | Vec3d & | maxIS
| ) |  |

Calculate an axis-aligned bounding box in index space from an axis-aligned bounding box in world space.

See also
  Transform::worldToIndex(const BBoxd&) const

## ◆ Cbrt() [1/3]

float openvdb::v5_2::math::Cbrt | ( | float | x | ) |
| inline

Return the cube root of a floating-point value.

## ◆ Cbrt() [2/3]

double openvdb::v5_2::math::Cbrt | ( | double | x | ) |
| inline

Return the cube root of a floating-point value.

## ◆ Cbrt() [3/3]

long double openvdb::v5_2::math::Cbrt | ( | long double | x | ) |
| inline

Return the cube root of a floating-point value.

## ◆ Ceil() [1/3]

int openvdb::v5_2::math::Ceil | ( | float | x | ) |
| inline

Return the ceiling of x.

## ◆ Ceil() [2/3]

int openvdb::v5_2::math::Ceil | ( | double | x | ) |
| inline

Return the ceiling of x.

## ◆ Ceil() [3/3]

int openvdb::v5_2::math::Ceil | ( | long double | x | ) |
| inline

Return the ceiling of x.

## ◆ Chop()

Type openvdb::v5_2::math::Chop | ( | Type | x,
|  | Type | delta
| ) |  |
| inline

Return x if it is greater or equal in magnitude than delta. Otherwise, return zero.

## ◆ Clamp()

Type openvdb::v5_2::math::Clamp | ( | Type | x,
|  | Type | min,
|  | Type | max
| ) |  |
| inline

Return x clamped to [min, max].

## ◆ Clamp01()

Type openvdb::v5_2::math::Clamp01 | ( | Type | x | ) |
| inline

Return x clamped to [0, 1].

## ◆ ClampTest01()

bool openvdb::v5_2::math::ClampTest01 | ( | Type & | x | ) |
| inline

Return `true` if x is outside [0,1].

## ◆ closestPointOnSegmentToPoint()

OPENVDB_API Vec3d openvdb::v5_2::math::closestPointOnSegmentToPoint | ( | const Vec3d & | a,
|  | const Vec3d & | b,
|  | const Vec3d & | p,
|  | double & | t
| ) |  |

Closest Point on Line Segment to Point. Given segment `ab` and point `p`, return the point on `ab` closest to `p` and `t` the parametric distance to `b`.

Parameters

  a | The segment's first vertex point.
  b | The segment's second vertex point.
  p | Point to compute the closest point on `ab` for.
  t | Parametric distance to `b`.

## ◆ closestPointOnTriangleToPoint()

OPENVDB_API Vec3d openvdb::v5_2::math::closestPointOnTriangleToPoint | ( | const Vec3d & | a,
|  | const Vec3d & | b,
|  | const Vec3d & | c,
|  | const Vec3d & | p,
|  | Vec3d & | uvw
| ) |  |

Closest Point on Triangle to Point. Given a triangle `abc` and a point `p`, return the point on `abc` closest to `p` and the corresponding barycentric coordinates.

Algorithms from "Real-Time Collision Detection" pg 136 to 142 by Christer Ericson. The closest point is obtained by first determining which of the triangles' Voronoi feature regions `p` is in and then computing the orthogonal projection of `p` onto the corresponding feature.

Parameters

  a | The triangle's first vertex point.
  b | The triangle's second vertex point.
  c | The triangle's third vertex point.
  p | Point to compute the closest point on `abc` for.
  uvw | Barycentric coordinates, computed and returned.

## ◆ Cos() [1/2]

float openvdb::v5_2::math::Cos | ( | const float & | x | ) |
| inline

Return cos x.

## ◆ Cos() [2/2]

double openvdb::v5_2::math::Cos | ( | const double & | x | ) |
| inline

Return cos x.

## ◆ createFullyDecomposedMap()

OPENVDB_API SharedPtr<FullyDecomposedMap> openvdb::v5_2::math::createFullyDecomposedMap | ( | const Mat4d & | m | ) |

General decomposition of a Matrix into a Unitary (e.g. rotation) following a Symmetric (e.g. stretch & shear)

## ◆ createPolarDecomposedMap()

OPENVDB_API SharedPtr<PolarDecomposedMap> openvdb::v5_2::math::createPolarDecomposedMap | ( | const Mat3d & | m | ) |

Decomposes a general linear into translation following polar decomposition.

T U S where:

T: Translation U: Unitary (rotation or reflection) S: Symmetric

Note
  : the Symmetric is automatically decomposed into Q D Q^T, where Q is rotation and D is diagonal.

## ◆ createSymmetricMap()

OPENVDB_API SharedPtr<SymmetricMap> openvdb::v5_2::math::createSymmetricMap | ( | const Mat3d & | m | ) |

Utility methods.

Create a SymmetricMap from a symmetric matrix. Decomposes the map into Rotation Diagonal Rotation^T

## ◆ diagonalizeSymmetricMatrix()

bool openvdb::v5_2::math::diagonalizeSymmetricMatrix | ( | const Mat3< T > & | input,
|  | Mat3< T > & | Q,
|  | Vec3< T > & | D,
|  | unsigned int | MAX_ITERATIONS = `250`
| ) |  |
| inline

Use Jacobi iterations to decompose a symmetric 3x3 matrix (diagonalize and compute eigenvectors)

This is based on the "Efficient numerical diagonalization of Hermitian 3x3 matrices" Joachim Kopp. arXiv.org preprint: physics/0610206 with the addition of largest pivot

## ◆ doProcessTypedMap() [1/2]

void openvdb::v5_2::math::doProcessTypedMap | ( | Transform & | transform,
|  | OpType & | op
| ) |  |
| inline

Helper function used internally by processTypedMap()

## ◆ doProcessTypedMap() [2/2]

void openvdb::v5_2::math::doProcessTypedMap | ( | const Transform & | transform,
|  | OpType & | op
| ) |  |
| inline

Helper function used internally by processTypedMap()

## ◆ doubleToInt64()

int64_t openvdb::v5_2::math::doubleToInt64 | ( | const double | aDoubleValue | ) |
| inline

## ◆ dsSchemeToMenuName()

std::string openvdb::v5_2::math::dsSchemeToMenuName | ( | DScheme | dss | ) |
| inline

## ◆ dsSchemeToString()

std::string openvdb::v5_2::math::dsSchemeToString | ( | DScheme | dss | ) |
| inline

## ◆ EuclideanRemainder()

Type openvdb::v5_2::math::EuclideanRemainder | ( | Type | x | ) |
| inline

Return the euclidean remainder of x. Note unlike % operator this will always return a positive result

## ◆ eulerAngles()

Vec3<typename MatType::value_type> openvdb::v5_2::math::eulerAngles | ( | const MatType & | mat,
|  | RotationOrder | rotationOrder,
|  | typename MatType::value_type | eps = `static_cast<typename MatType::value_type>(1.0e-8)`
| ) |  |

Return the Euler angles composing the given rotation matrix.

Optional axes arguments describe in what order elementary rotations are applied. Note that in our convention, XYZ means Rz * Ry * Rx. Because we are using rows rather than columns to represent the local axes of a coordinate frame, the interpretation from a local reference point of view is to first rotate about the x axis, then about the newly rotated y axis, and finally by the new local z axis. From a fixed reference point of view, the interpretation is to rotate about the stationary world z, y, and x axes respectively.

Irrespective of the Euler angle convention, in the case of distinct axes, eulerAngles() returns the x, y, and z angles in the corresponding x, y, z components of the returned Vec3. For the XZX convention, the left X value is returned in Vec3.x, and the right X value in Vec3.y. For the ZXZ convention the left Z value is returned in Vec3.z and the right Z value in Vec3.y

Examples of reconstructing r from its Euler angle decomposition

v = eulerAngles(r, ZYX_ROTATION); rx.setToRotation(Vec3d(1,0,0), v[0]); ry.setToRotation(Vec3d(0,1,0), v[1]); rz.setToRotation(Vec3d(0,0,1), v[2]); r = rx * ry * rz;

v = eulerAngles(r, ZXZ_ROTATION); rz1.setToRotation(Vec3d(0,0,1), v[2]); rx.setToRotation (Vec3d(1,0,0), v[0]); rz2.setToRotation(Vec3d(0,0,1), v[1]); r = rz2 * rx * rz1;

v = eulerAngles(r, XZX_ROTATION); rx1.setToRotation (Vec3d(1,0,0), v[0]); rx2.setToRotation (Vec3d(1,0,0), v[1]); rz.setToRotation (Vec3d(0,0,1), v[2]); r = rx2 * rz * rx1;

## ◆ Exp() [1/4]

Vec2<T> openvdb::v5_2::math::Exp | ( | Vec2< T > | v | ) |
| inline

Return a vector with the exponent applied to each of the components of the input vector.

## ◆ Exp() [2/4]

Vec4<T> openvdb::v5_2::math::Exp | ( | Vec4< T > | v | ) |
| inline

Return a vector with the exponent applied to each of the components of the input vector.

## ◆ Exp() [3/4]

Type openvdb::v5_2::math::Exp | ( | const Type & | x | ) |
| inline

Return ex.

## ◆ Exp() [4/4]

Vec3<T> openvdb::v5_2::math::Exp | ( | Vec3< T > | v | ) |
| inline

Return a vector with the exponent applied to each of the components of the input vector.

## ◆ floatToInt32()

int32_t openvdb::v5_2::math::floatToInt32 | ( | const float | aFloatValue | ) |
| inline

## ◆ Floor() [1/3]

int openvdb::v5_2::math::Floor | ( | float | x | ) |
| inline

Return the floor of x.

## ◆ Floor() [2/3]

int openvdb::v5_2::math::Floor | ( | double | x | ) |
| inline

Return the floor of x.

## ◆ Floor() [3/3]

int openvdb::v5_2::math::Floor | ( | long double | x | ) |
| inline

Return the floor of x.

## ◆ FractionalPart()

Type openvdb::v5_2::math::FractionalPart | ( | Type | x | ) |
| inline

Return the fractional part of x.

## ◆ getScale()

Vec3<typename MatType::value_type> openvdb::v5_2::math::getScale | ( | const MatType & | mat | ) |

Return a Vec3 representing the lengths of the passed matrix's upper 3×3's rows.

## ◆ GodunovsNormSqrd() [1/2]

Real openvdb::v5_2::math::GodunovsNormSqrd | ( | bool | isOutside,
|  | Real | dP_xm,
|  | Real | dP_xp,
|  | Real | dP_ym,
|  | Real | dP_yp,
|  | Real | dP_zm,
|  | Real | dP_zp
| ) |  |
| inline

## ◆ GodunovsNormSqrd() [2/2]

Real openvdb::v5_2::math::GodunovsNormSqrd | ( | bool | isOutside,
|  | const Vec3< Real > & | gradient_m,
|  | const Vec3< Real > & | gradient_p
| ) |  |
| inline

## ◆ hasTranslation()

bool openvdb::v5_2::math::hasTranslation | ( | const Mat4< T > & | m | ) |
| inline

## ◆ IntegerPart()

Type openvdb::v5_2::math::IntegerPart | ( | Type | x | ) |
| inline

Return the integer part of x.

## ◆ Inv()

Type openvdb::v5_2::math::Inv | ( | Type | x | ) |
| inline

Return the inverse of x.

## ◆ isAffine()

bool openvdb::v5_2::math::isAffine | ( | const Mat4< T > & | m | ) |
| inline

## ◆ isApproxEqual() [1/8]

bool openvdb::v5_2::math::isApproxEqual | ( | const Type & | a,
|  | const Type & | b
| ) |  |
| inline

Return `true` if a is equal to b to within the default floating-point comparison tolerance.

## ◆ isApproxEqual() [2/8]

bool openvdb::v5_2::math::isApproxEqual | ( | const Type & | a,
|  | const Type & | b,
|  | const Type & | tolerance
| ) |  |
| inline

Return `true` if a is equal to b to within the given tolerance.

## ◆ isApproxEqual() [3/8]

bool openvdb::v5_2::math::isApproxEqual | ( | const Vec2< T > & | a,
|  | const Vec2< T > & | b
| ) |  |
| inline

## ◆ isApproxEqual() [4/8]

bool openvdb::v5_2::math::isApproxEqual | ( | const Vec2< T > & | a,
|  | const Vec2< T > & | b,
|  | const Vec2< T > & | eps
| ) |  |
| inline

## ◆ isApproxEqual() [5/8]

bool openvdb::v5_2::math::isApproxEqual | ( | const Vec4< T > & | a,
|  | const Vec4< T > & | b
| ) |  |
| inline

## ◆ isApproxEqual() [6/8]

bool openvdb::v5_2::math::isApproxEqual | ( | const Vec4< T > & | a,
|  | const Vec4< T > & | b,
|  | const Vec4< T > & | eps
| ) |  |
| inline

## ◆ isApproxEqual() [7/8]

bool openvdb::v5_2::math::isApproxEqual | ( | const Vec3< T > & | a,
|  | const Vec3< T > & | b
| ) |  |
| inline

## ◆ isApproxEqual() [8/8]

bool openvdb::v5_2::math::isApproxEqual | ( | const Vec3< T > & | a,
|  | const Vec3< T > & | b,
|  | const Vec3< T > & | eps
| ) |  |
| inline

## ◆ isApproxEqual< bool >() [1/2]

bool openvdb::v5_2::math::isApproxEqual< bool > | ( | const bool & | a,
|  | const bool & | b
| ) |  |
| inline

## ◆ isApproxEqual< bool >() [2/2]

bool openvdb::v5_2::math::isApproxEqual< bool > | ( | const bool & | a,
|  | const bool & | b,
|  | const bool & |
| ) |  |
| inline

## ◆ isApproxEqual< std::string >() [1/2]

bool openvdb::v5_2::math::isApproxEqual< std::string > | ( | const std::string & | a,
|  | const std::string & | b
| ) |  |
| inline

## ◆ isApproxEqual< std::string >() [2/2]

bool openvdb::v5_2::math::isApproxEqual< std::string > | ( | const std::string & | a,
|  | const std::string & | b,
|  | const std::string & |
| ) |  |
| inline

## ◆ isApproxLarger()

bool openvdb::v5_2::math::isApproxLarger | ( | const Type & | a,
|  | const Type & | b,
|  | const Type & | tolerance
| ) |  |
| inline

Return `true` if a is larger than b to within the given tolerance, i.e., if b - a < tolerance.

## ◆ isApproxZero() [1/2]

bool openvdb::v5_2::math::isApproxZero | ( | const Type & | x | ) |
| inline

Return `true` if x is equal to zero to within the default floating-point comparison tolerance.

## ◆ isApproxZero() [2/2]

bool openvdb::v5_2::math::isApproxZero | ( | const Type & | x,
|  | const Type & | tolerance
| ) |  |
| inline

Return `true` if x is equal to zero to within the given tolerance.

## ◆ isDiagonal()

bool openvdb::v5_2::math::isDiagonal | ( | const MatType & | mat | ) |
| inline

Determine if a matrix is diagonal.

## ◆ isExactlyEqual()

bool openvdb::v5_2::math::isExactlyEqual | ( | const T0 & | a,
|  | const T1 & | b
| ) |  |
| inline

Return `true` if a is exactly equal to b.

## ◆ isFinite() [1/3]

bool openvdb::v5_2::math::isFinite | ( | const Tuple< SIZE, T > & | t | ) |
| inline

Return `true` if no Nan or Inf values are present.

## ◆ isFinite() [2/3]

bool openvdb::v5_2::math::isFinite | ( | const float | x | ) |
| inline

Return `true` if x is finite.

## ◆ isFinite() [3/3]

bool openvdb::v5_2::math::isFinite | ( | const Type & | x | ) |
| inline

Return `true` if x is finite.

## ◆ isIdentity()

bool openvdb::v5_2::math::isIdentity | ( | const MatType & | m | ) |
| inline

Determine if a matrix is an identity matrix.

## ◆ isInfinite()

bool openvdb::v5_2::math::isInfinite | ( | const Tuple< SIZE, T > & | t | ) |
| inline

Return `true` if an Inf is present in the tuple.

## ◆ isInvertible()

bool openvdb::v5_2::math::isInvertible | ( | const MatType & | m | ) |
| inline

Determine if a matrix is invertible.

## ◆ isNan()

bool openvdb::v5_2::math::isNan | ( | const Tuple< SIZE, T > & | t | ) |
| inline

Return `true` if a Nan is present in the tuple.

## ◆ isNegative()

bool openvdb::v5_2::math::isNegative | ( | const Type & | x | ) |
| inline

Return `true` if x is less than zero.

## ◆ isNegative< bool >()

bool openvdb::v5_2::math::isNegative< bool > | ( | const bool & |  | ) |
| inline

## ◆ isRelOrApproxEqual() [1/2]

bool openvdb::v5_2::math::isRelOrApproxEqual | ( | const Type & | a,
|  | const Type & | b,
|  | const Type & | absTol,
|  | const Type & | relTol
| ) |  |
| inline

## ◆ isRelOrApproxEqual() [2/2]

bool openvdb::v5_2::math::isRelOrApproxEqual | ( | const bool & | a,
|  | const bool & | b,
|  | const bool & | ,
|  | const bool & |
| ) |  |
| inline

## ◆ isSymmetric()

bool openvdb::v5_2::math::isSymmetric | ( | const MatType & | m | ) |
| inline

Determine if a matrix is symmetric.

This implicitly uses math::isApproxEqual() to determine equality.

## ◆ isUlpsEqual() [1/2]

bool openvdb::v5_2::math::isUlpsEqual | ( | const double | aLeft,
|  | const double | aRight,
|  | const int64_t | aUnitsInLastPlace
| ) |  |
| inline

## ◆ isUlpsEqual() [2/2]

bool openvdb::v5_2::math::isUlpsEqual | ( | const float | aLeft,
|  | const float | aRight,
|  | const int32_t | aUnitsInLastPlace
| ) |  |
| inline

## ◆ isUnitary()

bool openvdb::v5_2::math::isUnitary | ( | const MatType & | m | ) |
| inline

Determine if a matrix is unitary (i.e., rotation or reflection).

## ◆ isZero() [1/2]

bool openvdb::v5_2::math::isZero | ( | const Tuple< SIZE, T > & | t | ) |
| inline

Return `true` if all elements are exactly equal to zero.

## ◆ isZero() [2/2]

bool openvdb::v5_2::math::isZero | ( | const Type & | x | ) |
| inline

Return `true` if x is exactly equal to zero.

## ◆ lInfinityNorm()

MatType::ValueType openvdb::v5_2::math::lInfinityNorm | ( | const MatType & | matrix | ) |

Return the L∞ norm of an N×N matrix.

## ◆ Log() [1/3]

Vec2<T> openvdb::v5_2::math::Log | ( | Vec2< T > | v | ) |
| inline

Return a vector with log applied to each of the components of the input vector.

## ◆ Log() [2/3]

Vec4<T> openvdb::v5_2::math::Log | ( | Vec4< T > | v | ) |
| inline

Return a vector with log applied to each of the components of the input vector.

## ◆ Log() [3/3]

Vec3<T> openvdb::v5_2::math::Log | ( | Vec3< T > | v | ) |
| inline

Return a vector with log applied to each of the components of the input vector.

## ◆ lOneNorm()

MatType::ValueType openvdb::v5_2::math::lOneNorm | ( | const MatType & | matrix | ) |

Return the L1 norm of an N×N matrix.

## ◆ Max() [1/7]

const Type& openvdb::v5_2::math::Max | ( | const Type & | a,
|  | const Type & | b
| ) |  |
| inline

Return the maximum of two values.

## ◆ Max() [2/7]

const Type& openvdb::v5_2::math::Max | ( | const Type & | a,
|  | const Type & | b,
|  | const Type & | c
| ) |  |
| inline

Return the maximum of three values.

## ◆ Max() [3/7]

const Type& openvdb::v5_2::math::Max | ( | const Type & | a,
|  | const Type & | b,
|  | const Type & | c,
|  | const Type & | d
| ) |  |
| inline

Return the maximum of four values.

## ◆ Max() [4/7]

const Type& openvdb::v5_2::math::Max | ( | const Type & | a,
|  | const Type & | b,
|  | const Type & | c,
|  | const Type & | d,
|  | const Type & | e
| ) |  |
| inline

Return the maximum of five values.

## ◆ Max() [5/7]

const Type& openvdb::v5_2::math::Max | ( | const Type & | a,
|  | const Type & | b,
|  | const Type & | c,
|  | const Type & | d,
|  | const Type & | e,
|  | const Type & | f
| ) |  |
| inline

Return the maximum of six values.

## ◆ Max() [6/7]

const Type& openvdb::v5_2::math::Max | ( | const Type & | a,
|  | const Type & | b,
|  | const Type & | c,
|  | const Type & | d,
|  | const Type & | e,
|  | const Type & | f,
|  | const Type & | g
| ) |  |
| inline

Return the maximum of seven values.

## ◆ Max() [7/7]

const Type& openvdb::v5_2::math::Max | ( | const Type & | a,
|  | const Type & | b,
|  | const Type & | c,
|  | const Type & | d,
|  | const Type & | e,
|  | const Type & | f,
|  | const Type & | g,
|  | const Type & | h
| ) |  |
| inline

Return the maximum of eight values.

## ◆ maxComponent() [1/3]

Vec2<T> openvdb::v5_2::math::maxComponent | ( | const Vec2< T > & | v1,
|  | const Vec2< T > & | v2
| ) |  |
| inline

Return component-wise maximum of the two vectors.

## ◆ maxComponent() [2/3]

Vec4<T> openvdb::v5_2::math::maxComponent | ( | const Vec4< T > & | v1,
|  | const Vec4< T > & | v2
| ) |  |
| inline

Return component-wise maximum of the two vectors.

## ◆ maxComponent() [3/3]

Vec3<T> openvdb::v5_2::math::maxComponent | ( | const Vec3< T > & | v1,
|  | const Vec3< T > & | v2
| ) |  |
| inline

Return component-wise maximum of the two vectors.

## ◆ MaxIndex()

size_t openvdb::v5_2::math::MaxIndex | ( | const Vec3T & | v | ) |

Return the index [0,1,2] of the largest value in a 3D vector.

Note
  This methods assumes operator[] exists and avoids branching.

If two components of the input vector are equal and larger than the third component, the largest index of the two is always returned. If all three vector components are equal the largest index, i.e. 2, is returned. In other words the return value corresponds to the largest index of the largest vector components.

## ◆ Min() [1/7]

const Type& openvdb::v5_2::math::Min | ( | const Type & | a,
|  | const Type & | b
| ) |  |
| inline

Return the minimum of two values.

## ◆ Min() [2/7]

const Type& openvdb::v5_2::math::Min | ( | const Type & | a,
|  | const Type & | b,
|  | const Type & | c
| ) |  |
| inline

Return the minimum of three values.

## ◆ Min() [3/7]

const Type& openvdb::v5_2::math::Min | ( | const Type & | a,
|  | const Type & | b,
|  | const Type & | c,
|  | const Type & | d
| ) |  |
| inline

Return the minimum of four values.

## ◆ Min() [4/7]

const Type& openvdb::v5_2::math::Min | ( | const Type & | a,
|  | const Type & | b,
|  | const Type & | c,
|  | const Type & | d,
|  | const Type & | e
| ) |  |
| inline

Return the minimum of five values.

## ◆ Min() [5/7]

const Type& openvdb::v5_2::math::Min | ( | const Type & | a,
|  | const Type & | b,
|  | const Type & | c,
|  | const Type & | d,
|  | const Type & | e,
|  | const Type & | f
| ) |  |
| inline

Return the minimum of six values.

## ◆ Min() [6/7]

const Type& openvdb::v5_2::math::Min | ( | const Type & | a,
|  | const Type & | b,
|  | const Type & | c,
|  | const Type & | d,
|  | const Type & | e,
|  | const Type & | f,
|  | const Type & | g
| ) |  |
| inline

Return the minimum of seven values.

## ◆ Min() [7/7]

const Type& openvdb::v5_2::math::Min | ( | const Type & | a,
|  | const Type & | b,
|  | const Type & | c,
|  | const Type & | d,
|  | const Type & | e,
|  | const Type & | f,
|  | const Type & | g,
|  | const Type & | h
| ) |  |
| inline

Return the minimum of eight values.

## ◆ minComponent() [1/3]

Vec2<T> openvdb::v5_2::math::minComponent | ( | const Vec2< T > & | v1,
|  | const Vec2< T > & | v2
| ) |  |
| inline

Return component-wise minimum of the two vectors.

Remarks
  We are switching to a more explicit name because the semantics are different from std::min/max. In that case, the function returns a reference to one of the objects based on a comparator. Here, we must fabricate a new object which might not match either of the inputs.

## ◆ minComponent() [2/3]

Vec4<T> openvdb::v5_2::math::minComponent | ( | const Vec4< T > & | v1,
|  | const Vec4< T > & | v2
| ) |  |
| inline

Return component-wise minimum of the two vectors.

Remarks
  We are switching to a more explicit name because the semantics are different from std::min/max. In that case, the function returns a reference to one of the objects based on a comparator. Here, we must fabricate a new object which might not match either of the inputs.

## ◆ minComponent() [3/3]

Vec3<T> openvdb::v5_2::math::minComponent | ( | const Vec3< T > & | v1,
|  | const Vec3< T > & | v2
| ) |  |
| inline

Return component-wise minimum of the two vectors.

Remarks
  We are switching to a more explicit name because the semantics are different from std::min/max. In that case, the function returns a reference to one of the objects based on a comparator. Here, we must fabricate a new object which might not match either of the inputs.

## ◆ MinIndex()

size_t openvdb::v5_2::math::MinIndex | ( | const Vec3T & | v | ) |

Return the index [0,1,2] of the smallest value in a 3D vector.

Note
  This methods assumes operator[] exists and avoids branching.

If two components of the input vector are equal and smaller than the third component, the largest index of the two is always returned. If all three vector components are equal the largest index, i.e. 2, is returned. In other words the return value corresponds to the largest index of the of the smallest vector components.

## ◆ Mod() [1/4]

int openvdb::v5_2::math::Mod | ( | int | x,
|  | int | y
| ) |  |
| inline

Return the remainder of x / y.

## ◆ Mod() [2/4]

float openvdb::v5_2::math::Mod | ( | float | x,
|  | float | y
| ) |  |
| inline

Return the remainder of x / y.

## ◆ Mod() [3/4]

double openvdb::v5_2::math::Mod | ( | double | x,
|  | double | y
| ) |  |
| inline

Return the remainder of x / y.

## ◆ Mod() [4/4]

long double openvdb::v5_2::math::Mod | ( | long double | x,
|  | long double | y
| ) |  |
| inline

Return the remainder of x / y.

## ◆ negative() [1/3]

T openvdb::v5_2::math::negative | ( | const T & | val | ) |
| inline

Return the unary negation of the given value.

Note
  A negative<T>() specialization must be defined for each ValueType T for which unary negation is not defined.

## ◆ negative() [2/3]

bool openvdb::v5_2::math::negative | ( | const bool & | val | ) |
| inline

Return the negation of the given boolean.

## ◆ negative() [3/3]

std::string openvdb::v5_2::math::negative | ( | const std::string & | val | ) |
| inline

Return the "negation" of the given string.

## ◆ operator!=() [1/2]

bool openvdb::v5_2::math::operator!= | ( | const Vec4< T0 > & | v0,
|  | const Vec4< T1 > & | v1
| ) |  |
| inline

Inequality operator, does exact floating point comparisons.

## ◆ operator!=() [2/2]

bool openvdb::v5_2::math::operator!= | ( | const Vec3< T0 > & | v0,
|  | const Vec3< T1 > & | v1
| ) |  |
| inline

Inequality operator, does exact floating point comparisons.

## ◆ operator*() [1/11]

Vec2<typename promote<S, T>::type> openvdb::v5_2::math::operator* | ( | S | scalar,
|  | const Vec2< T > & | v
| ) |  |
| inline

Multiply each element of the given vector by scalar and return the result.

## ◆ operator*() [2/11]

Vec2<typename promote<S, T>::type> openvdb::v5_2::math::operator* | ( | const Vec2< T > & | v,
|  | S | scalar
| ) |  |
| inline

Multiply each element of the given vector by scalar and return the result.

## ◆ operator*() [3/11]

Vec2<typename promote<T0, T1>::type> openvdb::v5_2::math::operator* | ( | const Vec2< T0 > & | v0,
|  | const Vec2< T1 > & | v1
| ) |  |
| inline

Multiply corresponding elements of v0 and v1 and return the result.

## ◆ operator*() [4/11]

Vec4<typename promote<S, T>::type> openvdb::v5_2::math::operator* | ( | S | scalar,
|  | const Vec4< T > & | v
| ) |  |
| inline

Multiply each element of the given vector by scalar and return the result.

## ◆ operator*() [5/11]

Vec4<typename promote<S, T>::type> openvdb::v5_2::math::operator* | ( | const Vec4< T > & | v,
|  | S | scalar
| ) |  |
| inline

Multiply each element of the given vector by scalar and return the result.

## ◆ operator*() [6/11]

Vec4<typename promote<T0, T1>::type> openvdb::v5_2::math::operator* | ( | const Vec4< T0 > & | v0,
|  | const Vec4< T1 > & | v1
| ) |  |
| inline

Multiply corresponding elements of v0 and v1 and return the result.

## ◆ operator*() [7/11]

Vec3<typename promote<S, T>::type> openvdb::v5_2::math::operator* | ( | S | scalar,
|  | const Vec3< T > & | v
| ) |  |
| inline

Multiply each element of the given vector by scalar and return the result.

## ◆ operator*() [8/11]

Vec3<typename promote<S, T>::type> openvdb::v5_2::math::operator* | ( | const Vec3< T > & | v,
|  | S | scalar
| ) |  |
| inline

Multiply each element of the given vector by scalar and return the result.

## ◆ operator*() [9/11]

Vec3<typename promote<T0, T1>::type> openvdb::v5_2::math::operator* | ( | const Vec3< T0 > & | v0,
|  | const Vec3< T1 > & | v1
| ) |  |
| inline

Multiply corresponding elements of v0 and v1 and return the result.

## ◆ operator*() [10/11]

Quat<T> openvdb::v5_2::math::operator* | ( | S | scalar,
|  | const Quat< T > & | q
| ) |  |

Multiply each element of the given quaternion by scalar and return the result.

## ◆ operator*() [11/11]

Mat3<typename promote<T0, T1>::type> openvdb::v5_2::math::operator* | ( | const Mat3< T0 > & | m0,
|  | const Mat3< T1 > & | m1
| ) |  |

Multiply m0 by m1 and return the resulting matrix.

## ◆ operator+() [1/8]

Vec2<typename promote<T0, T1>::type> openvdb::v5_2::math::operator+ | ( | const Vec2< T0 > & | v0,
|  | const Vec2< T1 > & | v1
| ) |  |
| inline

Add corresponding elements of v0 and v1 and return the result.

## ◆ operator+() [2/8]

Vec2<typename promote<S, T>::type> openvdb::v5_2::math::operator+ | ( | const Vec2< T > & | v,
|  | S | scalar
| ) |  |
| inline

Add scalar to each element of the given vector and return the result.

## ◆ operator+() [3/8]

Vec4<typename promote<T0, T1>::type> openvdb::v5_2::math::operator+ | ( | const Vec4< T0 > & | v0,
|  | const Vec4< T1 > & | v1
| ) |  |
| inline

Add corresponding elements of v0 and v1 and return the result.

## ◆ operator+() [4/8]

Vec4<typename promote<S, T>::type> openvdb::v5_2::math::operator+ | ( | const Vec4< T > & | v,
|  | S | scalar
| ) |  |
| inline

Add scalar to each element of the given vector and return the result.

## ◆ operator+() [5/8]

Vec3<typename promote<T, typename Coord::ValueType>::type> openvdb::v5_2::math::operator+ | ( | const Vec3< T > & | v0,
|  | const Coord & | v1
| ) |  |
| inline

Allow a Coord to be added to or subtracted from a Vec3.

## ◆ operator+() [6/8]

Vec3<typename promote<T, typename Coord::ValueType>::type> openvdb::v5_2::math::operator+ | ( | const Coord & | v1,
|  | const Vec3< T > & | v0
| ) |  |
| inline

Allow a Coord to be added to or subtracted from a Vec3.

## ◆ operator+() [7/8]

Vec3<typename promote<T0, T1>::type> openvdb::v5_2::math::operator+ | ( | const Vec3< T0 > & | v0,
|  | const Vec3< T1 > & | v1
| ) |  |
| inline

Add corresponding elements of v0 and v1 and return the result.

## ◆ operator+() [8/8]

Vec3<typename promote<S, T>::type> openvdb::v5_2::math::operator+ | ( | const Vec3< T > & | v,
|  | S | scalar
| ) |  |
| inline

Add scalar to each element of the given vector and return the result.

## ◆ operator-() [1/8]

Vec2<typename promote<T0, T1>::type> openvdb::v5_2::math::operator- | ( | const Vec2< T0 > & | v0,
|  | const Vec2< T1 > & | v1
| ) |  |
| inline

Subtract corresponding elements of v0 and v1 and return the result.

## ◆ operator-() [2/8]

Vec2<typename promote<S, T>::type> openvdb::v5_2::math::operator- | ( | const Vec2< T > & | v,
|  | S | scalar
| ) |  |
| inline

Subtract scalar from each element of the given vector and return the result.

## ◆ operator-() [3/8]

Vec4<typename promote<T0, T1>::type> openvdb::v5_2::math::operator- | ( | const Vec4< T0 > & | v0,
|  | const Vec4< T1 > & | v1
| ) |  |
| inline

Subtract corresponding elements of v0 and v1 and return the result.

## ◆ operator-() [4/8]

Vec4<typename promote<S, T>::type> openvdb::v5_2::math::operator- | ( | const Vec4< T > & | v,
|  | S | scalar
| ) |  |
| inline

Subtract scalar from each element of the given vector and return the result.

## ◆ operator-() [5/8]

Vec3<typename promote<T, Coord::ValueType>::type> openvdb::v5_2::math::operator- | ( | const Vec3< T > & | v0,
|  | const Coord & | v1
| ) |  |
| inline

Allow a Coord to be subtracted from a Vec3.

## ◆ operator-() [6/8]

Vec3<typename promote<T, Coord::ValueType>::type> openvdb::v5_2::math::operator- | ( | const Coord & | v1,
|  | const Vec3< T > & | v0
| ) |  |
| inline

Allow a Coord to be subtracted from a Vec3.

## ◆ operator-() [7/8]

Vec3<typename promote<T0, T1>::type> openvdb::v5_2::math::operator- | ( | const Vec3< T0 > & | v0,
|  | const Vec3< T1 > & | v1
| ) |  |
| inline

Subtract corresponding elements of v0 and v1 and return the result.

## ◆ operator-() [8/8]

Vec3<typename promote<S, T>::type> openvdb::v5_2::math::operator- | ( | const Vec3< T > & | v,
|  | S | scalar
| ) |  |
| inline

Subtract scalar from each element of the given vector and return the result.

## ◆ operator/() [1/9]

Vec2<typename promote<S, T>::type> openvdb::v5_2::math::operator/ | ( | S | scalar,
|  | const Vec2< T > & | v
| ) |  |
| inline

Divide scalar by each element of the given vector and return the result.

## ◆ operator/() [2/9]

Vec2<typename promote<S, T>::type> openvdb::v5_2::math::operator/ | ( | const Vec2< T > & | v,
|  | S | scalar
| ) |  |
| inline

Divide each element of the given vector by scalar and return the result.

## ◆ operator/() [3/9]

Vec2<typename promote<T0, T1>::type> openvdb::v5_2::math::operator/ | ( | const Vec2< T0 > & | v0,
|  | const Vec2< T1 > & | v1
| ) |  |
| inline

Divide corresponding elements of v0 and v1 and return the result.

## ◆ operator/() [4/9]

Vec4<typename promote<S, T>::type> openvdb::v5_2::math::operator/ | ( | S | scalar,
|  | const Vec4< T > & | v
| ) |  |
| inline

Divide scalar by each element of the given vector and return the result.

## ◆ operator/() [5/9]

Vec4<typename promote<S, T>::type> openvdb::v5_2::math::operator/ | ( | const Vec4< T > & | v,
|  | S | scalar
| ) |  |
| inline

Divide each element of the given vector by scalar and return the result.

## ◆ operator/() [6/9]

Vec4<typename promote<T0, T1>::type> openvdb::v5_2::math::operator/ | ( | const Vec4< T0 > & | v0,
|  | const Vec4< T1 > & | v1
| ) |  |
| inline

Divide corresponding elements of v0 and v1 and return the result.

## ◆ operator/() [7/9]

Vec3<typename promote<S, T>::type> openvdb::v5_2::math::operator/ | ( | S | scalar,
|  | const Vec3< T > & | v
| ) |  |
| inline

Divide scalar by each element of the given vector and return the result.

## ◆ operator/() [8/9]

Vec3<typename promote<S, T>::type> openvdb::v5_2::math::operator/ | ( | const Vec3< T > & | v,
|  | S | scalar
| ) |  |
| inline

Divide each element of the given vector by scalar and return the result.

## ◆ operator/() [9/9]

Vec3<typename promote<T0, T1>::type> openvdb::v5_2::math::operator/ | ( | const Vec3< T0 > & | v0,
|  | const Vec3< T1 > & | v1
| ) |  |
| inline

Divide corresponding elements of v0 and v1 and return the result.

## ◆ operator<()

bool openvdb::v5_2::math::operator< | ( | const Tuple< SIZE, T0 > & | t0,
|  | const Tuple< SIZE, T1 > & | t1
| ) |  |

Returns
  true if t0 < t1, comparing components in order of significance.

## ◆ operator<<() [1/7]

std::ostream& openvdb::v5_2::math::operator<< | ( | std::ostream & | os,
|  | const DDA< RayT, Log2Dim > & | dda
| ) |  |
| inline

Output streaming of the Ray class.

Note
  Primarily intended for debugging.

## ◆ operator<<() [2/7]

OPENVDB_API std::ostream& openvdb::v5_2::math::operator<< | ( | std::ostream & | ,
|  | const Transform & |
| ) |  |

## ◆ operator<<() [3/7]

std::ostream& openvdb::v5_2::math::operator<< | ( | std::ostream & | ostr,
|  | const Tuple< SIZE, T > & | classname
| ) |  |

Write a Tuple to an output stream.

## ◆ operator<<() [4/7]

std::ostream& openvdb::v5_2::math::operator<< | ( | std::ostream & | os,
|  | const Ray< RealT > & | r
| ) |  |
| inline

Output streaming of the Ray class.

Note
  Primarily intended for debugging.

## ◆ operator<<() [5/7]

std::ostream& openvdb::v5_2::math::operator<< | ( | std::ostream & | os,
|  | const BBox< Vec3T > & | b
| ) |  |
| inline

## ◆ operator<<() [6/7]

std::ostream& openvdb::v5_2::math::operator<< | ( | std::ostream & | os,
|  | const Coord & | xyz
| ) |  |
| inline

## ◆ operator<<() [7/7]

std::ostream& openvdb::v5_2::math::operator<< | ( | std::ostream & | os,
|  | const CoordBBox & | b
| ) |  |
| inline

## ◆ operator==() [1/2]

bool openvdb::v5_2::math::operator== | ( | const Vec4< T0 > & | v0,
|  | const Vec4< T1 > & | v1
| ) |  |
| inline

Equality operator, does exact floating point comparisons.

## ◆ operator==() [2/2]

bool openvdb::v5_2::math::operator== | ( | const Vec3< T0 > & | v0,
|  | const Vec3< T1 > & | v1
| ) |  |
| inline

Equality operator, does exact floating point comparisons.

## ◆ operator>()

bool openvdb::v5_2::math::operator> | ( | const Tuple< SIZE, T0 > & | t0,
|  | const Tuple< SIZE, T1 > & | t1
| ) |  |

Returns
  true if t0 > t1, comparing components in order of significance.

## ◆ orthonormalize() [1/2]

void openvdb::v5_2::math::orthonormalize | ( | Vec2< T > & | v1,
|  | Vec2< T > & | v2
| ) |  |
| inline

Orthonormalize vectors v1 and v2 and store back the resulting basis e.g. Vec2f::orthonormalize(v1,v2);

## ◆ orthonormalize() [2/2]

void openvdb::v5_2::math::orthonormalize | ( | Vec3< T > & | v1,
|  | Vec3< T > & | v2,
|  | Vec3< T > & | v3
| ) |  |
| inline

Orthonormalize vectors v1, v2 and v3 and store back the resulting basis e.g. Vec3d::orthonormalize(v1,v2,v3);

## ◆ outerProduct()

Mat3<T> openvdb::v5_2::math::outerProduct | ( | const Vec3< T > & | v1,
|  | const Vec3< T > & | v2
| ) |  |

Returns outer product of v1, v2, i.e. v1 v2^T if v1 and v2 are column vectors, e.g. M = Mat3f::outerproduct(v1,v2);

## ◆ padMat4()

static MatType& openvdb::v5_2::math::padMat4 | ( | MatType & | dest | ) |
| static

Write 0s along Mat4's last row and column, and a 1 on its diagonal.

Useful initialization when we're initializing just the 3×3 block.

## ◆ polarDecomposition()

bool openvdb::v5_2::math::polarDecomposition | ( | const MatType & | input,
|  | MatType & | unitary,
|  | MatType & | positive_hermitian,
|  | unsigned int | MAX_ITERATIONS = `100`
| ) |  |

Decompose an invertible 3×3 matrix into a unitary matrix followed by a symmetric matrix (positive semi-definite Hermitian), i.e., M = U * S.

If det(U) = 1 it is a rotation, otherwise det(U) = -1, meaning there is some part reflection. See "Computing the polar decomposition with applications" Higham, N.J. - SIAM J. Sc. Stat Comput 7(4):1160-1174

## ◆ Pow() [1/3]

Type openvdb::v5_2::math::Pow | ( | Type | x,
|  | int | n
| ) |  |

Return xn.

## ◆ Pow() [2/3]

float openvdb::v5_2::math::Pow | ( | float | b,
|  | float | e
| ) |  |
| inline

Return be.

## ◆ Pow() [3/3]

double openvdb::v5_2::math::Pow | ( | double | b,
|  | double | e
| ) |  |
| inline

Return be.

## ◆ Pow2()

Type openvdb::v5_2::math::Pow2 | ( | Type | x | ) |
| inline

Return x2.

## ◆ Pow3()

Type openvdb::v5_2::math::Pow3 | ( | Type | x | ) |
| inline

Return x3.

## ◆ Pow4()

Type openvdb::v5_2::math::Pow4 | ( | Type | x | ) |
| inline

Return x4.

## ◆ powLerp()

Mat3<T> openvdb::v5_2::math::powLerp | ( | const Mat3< T0 > & | m1,
|  | const Mat3< T0 > & | m2,
|  | T | t
| ) |  |

Interpolate the rotation between m1 and m2 using Mat::powSolve. Unlike slerp, translation is not treated independently. This results in smoother animation results.

## ◆ powSolve()

void openvdb::v5_2::math::powSolve | ( | const MatType & | aA,
|  | MatType & | aB,
|  | double | aPower,
|  | double | aTol = `0.01`
| ) |  |
| inline

## ◆ PrintCast() [1/3]

auto openvdb::v5_2::math::PrintCast | ( | const T & | val | ) | -> typename std::enable_if<!std::is_same<T, int8_t>::value && !std::is_same<T, uint8_t>::value, const T&>::type
| inline

8-bit integer values print to std::ostreams as characters. Cast them so that they print as integers instead.

## ◆ PrintCast() [2/3]

int32_t openvdb::v5_2::math::PrintCast | ( | int8_t | val | ) |
| inline

## ◆ PrintCast() [3/3]

uint32_t openvdb::v5_2::math::PrintCast | ( | uint8_t | val | ) |
| inline

## ◆ processTypedMap()

bool openvdb::v5_2::math::processTypedMap | ( | TransformType & | transform,
|  | OpType & | op
| ) |  |

Utility function that, given a generic map pointer, calls a functor on the fully-resoved map.

Usage:

struct Foo {
template<typename MapT>
void operator()(const MapT& map) const { blah }
};
processTypedMap(myMap, Foo());

Returns
  `false` if the grid type is unknown or unhandled.

## ◆ Remainder()

Type openvdb::v5_2::math::Remainder | ( | Type | x,
|  | Type | y
| ) |  |
| inline

Return the remainder of x / y.

## ◆ rotation() [1/4]

MatType openvdb::v5_2::math::rotation | ( | const Quat< typename MatType::value_type > & | q,
|  | typename MatType::value_type | eps = `static_cast<typename MatType::value_type>(1.0e-8)`
| ) |  |

Return the rotation matrix specified by the given quaternion.

The quaternion is normalized and used to construct the matrix. Note that the matrix is transposed to match post-multiplication semantics.

## ◆ rotation() [2/4]

MatType openvdb::v5_2::math::rotation | ( | Axis | axis,
|  | typename MatType::value_type | angle
| ) |  |

Return a matrix for rotation by angle radians about the given axis.

Parameters

  axis | The axis (one of X, Y, Z) to rotate about.
  angle | The rotation angle, in radians.

## ◆ rotation() [3/4]

MatType openvdb::v5_2::math::rotation | ( | const Vec3< typename MatType::value_type > & | _axis,
|  | typename MatType::value_type | angle
| ) |  |

Return a matrix for rotation by angle radians about the given axis.

Note
  The axis must be a unit vector.

## ◆ rotation() [4/4]

MatType openvdb::v5_2::math::rotation | ( | const Vec3< ValueType1 > & | _v1,
|  | const Vec3< ValueType2 > & | _v2,
|  | typename MatType::value_type | eps = `static_cast<typename MatType::value_type>(1.0e-8)`
| ) |  |
| inline

Return a rotation matrix that maps v1 onto v2 about the cross product of v1 and v2.

## ◆ Round() [1/3]

float openvdb::v5_2::math::Round | ( | float | x | ) |
| inline

Return x rounded to the nearest integer.

## ◆ Round() [2/3]

double openvdb::v5_2::math::Round | ( | double | x | ) |
| inline

Return x rounded to the nearest integer.

## ◆ Round() [3/3]

long double openvdb::v5_2::math::Round | ( | long double | x | ) |
| inline

Return x rounded to the nearest integer.

## ◆ RoundDown() [1/4]

float openvdb::v5_2::math::RoundDown | ( | float | x | ) |
| inline

Return x rounded down to the nearest integer.

## ◆ RoundDown() [2/4]

double openvdb::v5_2::math::RoundDown | ( | double | x | ) |
| inline

Return x rounded down to the nearest integer.

## ◆ RoundDown() [3/4]

long double openvdb::v5_2::math::RoundDown | ( | long double | x | ) |
| inline

Return x rounded down to the nearest integer.

## ◆ RoundDown() [4/4]

Type openvdb::v5_2::math::RoundDown | ( | Type | x,
|  | Type | base
| ) |  |
| inline

Return x rounded down to the nearest multiple of base.

## ◆ RoundUp() [1/4]

float openvdb::v5_2::math::RoundUp | ( | float | x | ) |
| inline

Return x rounded up to the nearest integer.

## ◆ RoundUp() [2/4]

double openvdb::v5_2::math::RoundUp | ( | double | x | ) |
| inline

Return x rounded up to the nearest integer.

## ◆ RoundUp() [3/4]

long double openvdb::v5_2::math::RoundUp | ( | long double | x | ) |
| inline

Return x rounded up to the nearest integer.

## ◆ RoundUp() [4/4]

Type openvdb::v5_2::math::RoundUp | ( | Type | x,
|  | Type | base
| ) |  |
| inline

Return x rounded up to the nearest multiple of base.

## ◆ scale()

MatType openvdb::v5_2::math::scale | ( | const Vec3< typename MatType::value_type > & | s | ) |

Return a matrix that scales by s.

## ◆ shear()

MatType openvdb::v5_2::math::shear | ( | Axis | axis0,
|  | Axis | axis1,
|  | typename MatType::value_type | shear
| ) |  |

Set the matrix to a shear along axis0 by a fraction of axis1.

Parameters

  axis0 | The fixed axis of the shear.
  axis1 | The shear axis.
  shear | The shear factor.

## ◆ Sign()

int openvdb::v5_2::math::Sign | ( | const Type & | x | ) |
| inline

Return the sign of the given value as an integer (either -1, 0 or 1).

## ◆ SignChange()

bool openvdb::v5_2::math::SignChange | ( | const Type & | a,
|  | const Type & | b
| ) |  |
| inline

Return `true` if a and b have different signs.

Note
  Zero is considered a positive number.

## ◆ simplify()

OPENVDB_API SharedPtr<MapBase> openvdb::v5_2::math::simplify | ( | SharedPtr< AffineMap > | affine | ) |

reduces an AffineMap to a ScaleMap or a ScaleTranslateMap when it can

## ◆ Sin() [1/2]

float openvdb::v5_2::math::Sin | ( | const float & | x | ) |
| inline

Return sin x.

## ◆ Sin() [2/2]

double openvdb::v5_2::math::Sin | ( | const double & | x | ) |
| inline

Return sin x.

## ◆ skew()

MatType openvdb::v5_2::math::skew | ( | const Vec3< typename MatType::value_type > & | skew | ) |

Return a matrix as the cross product of the given vector.

## ◆ slerp() [1/2]

Quat<T> openvdb::v5_2::math::slerp | ( | const Quat< T > & | q1,
|  | const Quat< T > & | q2,
|  | T | t,
|  | T | tolerance = `0.00001`
| ) |  |

Linear interpolation between the two quaternions.

## ◆ slerp() [2/2]

Mat3<T> openvdb::v5_2::math::slerp | ( | const Mat3< T0 > & | m1,
|  | const Mat3< T0 > & | m2,
|  | T | t
| ) |  |

Interpolate between m1 and m2. Converts to quaternion form and uses slerp m1 and m2 must be rotation matrices!

## ◆ SmoothUnitStep() [1/2]

Type openvdb::v5_2::math::SmoothUnitStep | ( | Type | x | ) |
| inline

Return 0 if x < 0, 1 if x > 1 or else (3 − 2 x) x².

## ◆ SmoothUnitStep() [2/2]

Type openvdb::v5_2::math::SmoothUnitStep | ( | Type | x,
|  | Type | min,
|  | Type | max
| ) |  |
| inline

Return 0 if x < min, 1 if x > max or else (3 − 2 t) t², where t = (x − min)/(max − min).

## ◆ snapMatBasis()

MatType openvdb::v5_2::math::snapMatBasis | ( | const MatType & | source,
|  | Axis | axis,
|  | const Vec3< typename MatType::value_type > & | direction
| ) |  |
| inline

This function snaps a specific axis to a specific direction, preserving scaling.

It does this using minimum energy, thus posing a unique solution if basis & direction aren't parallel.

Note
  direction need not be unit.

## ◆ Sqrt() [1/3]

float openvdb::v5_2::math::Sqrt | ( | float | x | ) |
| inline

Return the square root of a floating-point value.

## ◆ Sqrt() [2/3]

double openvdb::v5_2::math::Sqrt | ( | double | x | ) |
| inline

Return the square root of a floating-point value.

## ◆ Sqrt() [3/3]

long double openvdb::v5_2::math::Sqrt | ( | long double | x | ) |
| inline

Return the square root of a floating-point value.

## ◆ sqrtSolve()

void openvdb::v5_2::math::sqrtSolve | ( | const MatType & | aA,
|  | MatType & | aB,
|  | double | aTol = `0.01`
| ) |  |
| inline

Solve for A=B*B, given A.

Denman-Beavers square root iteration

## ◆ stringToBiasedGradientScheme()

BiasedGradientScheme openvdb::v5_2::math::stringToBiasedGradientScheme | ( | const std::string & | s | ) |
| inline

## ◆ stringToDScheme()

DScheme openvdb::v5_2::math::stringToDScheme | ( | const std::string & | s | ) |
| inline

## ◆ stringToTemporalIntegrationScheme()

TemporalIntegrationScheme openvdb::v5_2::math::stringToTemporalIntegrationScheme | ( | const std::string & | s | ) |
| inline

## ◆ temporalIntegrationSchemeToMenuName()

std::string openvdb::v5_2::math::temporalIntegrationSchemeToMenuName | ( | TemporalIntegrationScheme | tis | ) |
| inline

## ◆ temporalIntegrationSchemeToString()

std::string openvdb::v5_2::math::temporalIntegrationSchemeToString | ( | TemporalIntegrationScheme | tis | ) |
| inline

## ◆ transformNormal()

Vec3<T1> openvdb::v5_2::math::transformNormal | ( | const Mat4< T0 > & | m,
|  | const Vec3< T1 > & | n
| ) |  |

Transform a Vec3 by pre-multiplication, without translation. Presumes this matrix is inverse of coordinate transform Synonymous to "pretransform3x3"

## ◆ Truncate()

Type openvdb::v5_2::math::Truncate | ( | Type | x,
|  | unsigned int | digits
| ) |  |
| inline

Return x truncated to the given number of decimal digits.

## ◆ unit() [1/2]

MatType openvdb::v5_2::math::unit | ( | const MatType & | mat,
|  | typename MatType::value_type | eps = `1.0e-8`
| ) |  |

Return a copy of the given matrix with its upper 3×3 rows normalized.

This can be geometrically interpreted as a matrix with no scaling along its major axes.

## ◆ unit() [2/2]

MatType openvdb::v5_2::math::unit | ( | const MatType & | in,
|  | typename MatType::value_type | eps,
|  | Vec3< typename MatType::value_type > & | scaling
| ) |  |

Return a copy of the given matrix with its upper 3×3 rows normalized, and return the length of each of these rows in scaling.

This can be geometrically interpretted as a matrix with no scaling along its major axes, and the scaling in the input vector

## ◆ WENO5()

ValueType openvdb::v5_2::math::WENO5 | ( | const ValueType & | v1,
|  | const ValueType & | v2,
|  | const ValueType & | v3,
|  | const ValueType & | v4,
|  | const ValueType & | v5,
|  | float | scale2 = `0.01f`
| ) |  |
| inline

Implementation of nominally fifth-order finite-difference WENO.

This function returns the numerical flux. See "High Order Finite Difference and Finite Volume WENO Schemes and Discontinuous Galerkin Methods for CFD" - Chi-Wang Shu ICASE Report No 2001-11 (page 6). Also see ICASE No 97-65 for a more complete reference (Shu, 1997). Given v1 = f(x-2dx), v2 = f(x-dx), v3 = f(x), v4 = f(x+dx) and v5 = f(x+2dx), return an interpolated value f(x+dx/2) with the special property that ( f(x+dx/2) - f(x-dx/2) ) / dx = df/dx (x) + error, where the error is fifth-order in smooth regions: O(dx) <= error <=O(dx^5)

## ◆ ZeroCrossing()

bool openvdb::v5_2::math::ZeroCrossing | ( | const Type & | a,
|  | const Type & | b
| ) |  |
| inline

Return `true` if the interval [a, b] includes zero, i.e., if either a or b is zero or if they have different signs.