Namespaces
namespace	detail

namespace	suffixes

Classes
class	BoundingBox

struct	constants

struct	HullAngle

class	Icosphere
	The class Icosphere represents an icosphere where the subdivision and radius are configurable. More...

struct	MapsToManifoldPoint

class	Mesh
	Base class for meshes. More...

class	Random

struct	RealVectorSpace

struct	SE3Space

struct	SO3Space

class	TriMesh
	This class represents triangle meshes. More...

Typedefs
using	NullSpace = RealVectorSpace< 0u >

using	R1Space = RealVectorSpace< 1u >

using	R2Space = RealVectorSpace< 2u >

using	R3Space = RealVectorSpace< 3u >

using	constantsf = constants< float >

using	constantsd = constants< double >

typedef std::vector< Eigen::Vector3d >	SupportGeometry

typedef common::aligned_vector< Eigen::Vector2d >	SupportPolygon

using	Icospheref = Icosphere< float >

using	Icosphered = Icosphere< double >

using	Inertia = Eigen::Matrix6d

using	LinearJacobian = Eigen::Matrix< double, 3, Eigen::Dynamic >

using	AngularJacobian = Eigen::Matrix< double, 3, Eigen::Dynamic >

using	Jacobian = Eigen::Matrix< double, 6, Eigen::Dynamic >

using	Meshf = Mesh< float >

using	Meshd = Mesh< double >

using	TriMeshf = TriMesh< float >

using	TriMeshd = TriMesh< double >

Enumerations
enum	AxisType { AXIS_X = 0 , AXIS_Y = 1 , AXIS_Z = 2 }

enum	IntersectionResult { INTERSECTING = 0 , PARALLEL , BEYOND_ENDPOINTS }
	Intersection_t is returned by the computeIntersection() function to indicate whether there was a valid intersection between the two line segments. More...

Functions
template<typename SpaceT >
SpaceT::Matrix	inverse (const typename SpaceT::Matrix &mat)

template<typename SpaceT >
SpaceT::EuclideanPoint	toEuclideanPoint (const typename SpaceT::Point &point)

template<typename SpaceT >
SpaceT::Point	toManifoldPoint (const typename SpaceT::EuclideanPoint &point)

template<typename SpaceT >
SpaceT::Point	integratePosition (const typename SpaceT::Point &pos, const typename SpaceT::Vector &vel, double dt)

template<typename SpaceT >
SpaceT::Vector	integrateVelocity (const typename SpaceT::Vector &vel, const typename SpaceT::Vector &acc, double dt)

template<typename S , typename Index >
std::tuple< std::vector< Eigen::Matrix< S, 3, 1 > >, std::vector< Eigen::Matrix< Index, 3, 1 > > >	discardUnusedVertices (const std::vector< Eigen::Matrix< S, 3, 1 > > &vertices, const std::vector< Eigen::Matrix< Index, 3, 1 > > &triangles)

template<typename S , typename Index >
std::tuple< std::vector< Eigen::Matrix< S, 3, 1 > >, std::vector< Eigen::Matrix< Index, 3, 1 > > >	computeConvexHull3D (const std::vector< Eigen::Matrix< S, 3, 1 > > &vertices, bool optimize=true)
	Generates a 3D convex hull given vertices and indices.

Eigen::Quaterniond	expToQuat (const Eigen::Vector3d &_v)

Eigen::Vector3d	quatToExp (const Eigen::Quaterniond &_q)

Eigen::Vector3d	matrixToEulerXYX (const Eigen::Matrix3d &_R)
	get the Euler XYX angle from R

Eigen::Vector3d	matrixToEulerXYZ (const Eigen::Matrix3d &_R)
	get the Euler XYZ angle from R

Eigen::Vector3d	matrixToEulerZYX (const Eigen::Matrix3d &_R)
	get the Euler ZYX angle from R

Eigen::Vector3d	matrixToEulerXZY (const Eigen::Matrix3d &_R)
	get the Euler XZY angle from R

Eigen::Vector3d	matrixToEulerYZX (const Eigen::Matrix3d &_R)
	get the Euler YZX angle from R

Eigen::Vector3d	matrixToEulerZXY (const Eigen::Matrix3d &_R)
	get the Euler ZXY angle from R

Eigen::Vector3d	matrixToEulerYXZ (const Eigen::Matrix3d &_R)
	get the Euler YXZ angle from R

Eigen::Matrix3d	quatDeriv (const Eigen::Quaterniond &_q, int _el)

Eigen::Matrix3d	quatSecondDeriv (const Eigen::Quaterniond &_q, int _el1, int _el2)

Eigen::Vector3d	rotatePoint (const Eigen::Quaterniond &_q, const Eigen::Vector3d &_pt)

Eigen::Vector3d	rotatePoint (const Eigen::Quaterniond &_q, double _x, double _y, double _z)

Eigen::Matrix3d	expMapRot (const Eigen::Vector3d &_expmap)
	Computes the Rotation matrix from a given expmap vector.

Eigen::Matrix3d	expMapJac (const Eigen::Vector3d &_expmap)
	Computes the Jacobian of the expmap.

Eigen::Matrix3d	expMapJacDot (const Eigen::Vector3d &_expmap, const Eigen::Vector3d &_qdot)
	Computes the time derivative of the expmap Jacobian.

Eigen::Matrix3d	expMapJacDeriv (const Eigen::Vector3d &_expmap, int _qi)
	computes the derivative of the Jacobian of the expmap wrt to _qi indexed dof; _qi \( \in \) {0,1,2}

Eigen::Vector3d	logMap (const Eigen::Matrix3d &_R)
	Log mapping.

Eigen::Vector6d	logMap (const Eigen::Isometry3d &_T)
	Log mapping.

Eigen::Vector6d	AdT (const Eigen::Isometry3d &_T, const Eigen::Vector6d &_V)
	Rectify the rotation part so as that it satifies the orthogonality condition.

Eigen::Matrix6d	getAdTMatrix (const Eigen::Isometry3d &T)
	Get linear transformation matrix of Adjoint mapping.

Eigen::Vector6d	AdR (const Eigen::Isometry3d &_T, const Eigen::Vector6d &_V)
	Fast version of Ad([R 0; 0 1], V)

Eigen::Vector6d	AdTAngular (const Eigen::Isometry3d &_T, const Eigen::Vector3d &_w)
	fast version of Ad(T, se3(w, 0))

Eigen::Vector6d	AdTLinear (const Eigen::Isometry3d &_T, const Eigen::Vector3d &_v)
	fast version of Ad(T, se3(0, v))

Eigen::Vector6d	AdInvT (const Eigen::Isometry3d &_T, const Eigen::Vector6d &_V)
	fast version of Ad(Inv(T), V)

Eigen::Vector6d	AdInvRLinear (const Eigen::Isometry3d &_T, const Eigen::Vector3d &_v)
	Fast version of Ad(Inv([R 0; 0 1]), se3(0, v))

Eigen::Vector6d	ad (const Eigen::Vector6d &_X, const Eigen::Vector6d &_Y)
	adjoint mapping

Eigen::Vector6d	dAdT (const Eigen::Isometry3d &_T, const Eigen::Vector6d &_F)
	dual adjoint mapping

Eigen::Vector6d	dAdInvT (const Eigen::Isometry3d &_T, const Eigen::Vector6d &_F)
	fast version of dAd(Inv(T), F)

Eigen::Vector6d	dAdInvR (const Eigen::Isometry3d &_T, const Eigen::Vector6d &_F)
	fast version of dAd(Inv([R 0; 0 1]), F)

Eigen::Matrix3d	eulerXYXToMatrix (const Eigen::Vector3d &_angle)
	Given Euler XYX angles, return a 3x3 rotation matrix, which is equivalent to RotX(angle(0)) * RotY(angle(1)) * RotX(angle(2)).

Eigen::Matrix3d	eulerXYZToMatrix (const Eigen::Vector3d &_angle)
	Given EulerXYZ angles, return a 3x3 rotation matrix, which is equivalent to RotX(angle(0)) * RotY(angle(1)) * RotZ(angle(2)).

Eigen::Matrix3d	eulerXZXToMatrix (const Eigen::Vector3d &_angle)
	Given EulerXZX angles, return a 3x3 rotation matrix, which is equivalent to RotX(angle(0)) * RotZ(angle(1)) * RotX(angle(2)).

Eigen::Matrix3d	eulerXZYToMatrix (const Eigen::Vector3d &_angle)
	Given EulerXZY angles, return a 3x3 rotation matrix, which is equivalent to RotX(angle(0)) * RotZ(angle(1)) * RotY(angle(2)).

Eigen::Matrix3d	eulerYXYToMatrix (const Eigen::Vector3d &_angle)
	Given EulerYXY angles, return a 3x3 rotation matrix, which is equivalent to RotY(angle(0)) * RotX(angle(1)) * RotY(angle(2)).

Eigen::Matrix3d	eulerYXZToMatrix (const Eigen::Vector3d &_angle)
	Given EulerYXZ angles, return a 3x3 rotation matrix, which is equivalent to RotY(angle(0)) * RotX(angle(1)) * RotZ(angle(2)).

Eigen::Matrix3d	eulerYZXToMatrix (const Eigen::Vector3d &_angle)
	Given EulerYZX angles, return a 3x3 rotation matrix, which is equivalent to RotY(angle(0)) * RotZ(angle(1)) * RotX(angle(2)).

Eigen::Matrix3d	eulerYZYToMatrix (const Eigen::Vector3d &_angle)
	Given EulerYZY angles, return a 3x3 rotation matrix, which is equivalent to RotY(angle(0)) * RotZ(angle(1)) * RotY(angle(2)).

Eigen::Matrix3d	eulerZXYToMatrix (const Eigen::Vector3d &_angle)
	Given EulerZXY angles, return a 3x3 rotation matrix, which is equivalent to RotZ(angle(0)) * RotX(angle(1)) * RotY(angle(2)).

Eigen::Matrix3d	eulerZYXToMatrix (const Eigen::Vector3d &_angle)
	Given EulerZYX angles, return a 3x3 rotation matrix, which is equivalent to RotZ(angle(0)) * RotY(angle(1)) * RotX(angle(2)).

Eigen::Matrix3d	eulerZXZToMatrix (const Eigen::Vector3d &_angle)
	Given EulerZXZ angles, return a 3x3 rotation matrix, which is equivalent to RotZ(angle(0)) * RotX(angle(1)) * RotZ(angle(2)).

Eigen::Matrix3d	eulerZYZToMatrix (const Eigen::Vector3d &_angle)
	Given EulerZYZ angles, return a 3x3 rotation matrix, which is equivalent to RotZ(angle(0)) * RotY(angle(1)) * RotZ(angle(2)).

Eigen::Isometry3d	expMap (const Eigen::Vector6d &_S)
	Exponential mapping.

Eigen::Isometry3d	expAngular (const Eigen::Vector3d &_s)
	fast version of Exp(se3(s, 0))

Eigen::Vector6d	dad (const Eigen::Vector6d &_s, const Eigen::Vector6d &_t)
	fast version of ad(se3(Eigen_Vec3(0), v), S)

Inertia	transformInertia (const Eigen::Isometry3d &_T, const Inertia &_AI)

Eigen::Matrix3d	parallelAxisTheorem (const Eigen::Matrix3d &_original, const Eigen::Vector3d &_comShift, double _mass)
	Use the Parallel Axis Theorem to compute the moment of inertia of a body whose center of mass has been shifted from the origin.

bool	verifyRotation (const Eigen::Matrix3d &_R)
	Check if determinant of _R is equat to 1 and all the elements are not NaN values.

bool	verifyTransform (const Eigen::Isometry3d &_T)
	Check if determinant of the rotational part of _T is equat to 1 and all the elements are not NaN values.

Eigen::Vector3d	fromSkewSymmetric (const Eigen::Matrix3d &_m)

Eigen::Matrix3d	makeSkewSymmetric (const Eigen::Vector3d &_v)

Eigen::Matrix3d	computeRotation (const Eigen::Vector3d &axis, AxisType axisType=AxisType::AXIS_X)
	Compute a rotation matrix from a vector.

Eigen::Isometry3d	computeTransform (const Eigen::Vector3d &axis, const Eigen::Vector3d &translation, AxisType axisType=AxisType::AXIS_X)
	Compute a transform from a vector and a position.

Eigen::Isometry3d	getFrameOriginAxisZ (const Eigen::Vector3d &_origin, const Eigen::Vector3d &_axisZ)
	Generate frame given origin and z-axis.

SupportPolygon	computeSupportPolgyon (const SupportGeometry &_geometry, const Eigen::Vector3d &_axis1=Eigen::Vector3d::UnitX(), const Eigen::Vector3d &_axis2=Eigen::Vector3d::UnitY())
	Project the support geometry points onto a plane with the given axes and then compute their convex hull, which will take the form of a polgyon.

SupportPolygon	computeSupportPolgyon (std::vector< std::size_t > &_originalIndices, const SupportGeometry &_geometry, const Eigen::Vector3d &_axis1=Eigen::Vector3d::UnitX(), const Eigen::Vector3d &_axis2=Eigen::Vector3d::UnitY())
	Same as computeSupportPolgyon, except you can pass in a std::vector<std::size_t> which will have the same size as the returned SupportPolygon, and each entry will contain the original index of each point in the SupportPolygon.

static bool	HullAngleComparison (const HullAngle &a, const HullAngle &b)

SupportPolygon	computeConvexHull (const SupportPolygon &_points)
	Computes the convex hull of a set of 2D points.

Eigen::Vector2d	computeCentroidOfHull (const SupportPolygon &_convexHull)
	Compute the centroid of a polygon, assuming the polygon is a convex hull.

static bool	isLeftTurn (const Eigen::Vector2d &p1, const Eigen::Vector2d &p2, const Eigen::Vector2d &p3)

SupportPolygon	computeConvexHull (std::vector< std::size_t > &_originalIndices, const SupportPolygon &_points)
	Computes the convex hull of a set of 2D points and fills in _originalIndices with the original index of each entry in the returned SupportPolygon.

IntersectionResult	computeIntersection (Eigen::Vector2d &_intersectionPoint, const Eigen::Vector2d &a1, const Eigen::Vector2d &a2, const Eigen::Vector2d &b1, const Eigen::Vector2d &b2)
	Compute the intersection between a line segment that goes from a1 -> a2 and a line segment that goes from b1 -> b2.

double	cross (const Eigen::Vector2d &_v1, const Eigen::Vector2d &_v2)
	Compute a 2D cross product.

bool	isInsideSupportPolygon (const Eigen::Vector2d &_p, const SupportPolygon &_support, bool _includeEdge=true)
	Returns true if the point _p is inside the support polygon.

Eigen::Vector2d	computeClosestPointOnLineSegment (const Eigen::Vector2d &_p, const Eigen::Vector2d &_s1, const Eigen::Vector2d &_s2)
	Returns the point which is closest to _p that also lays on the line segment that goes from _s1 -> _s2.

Eigen::Vector2d	computeClosestPointOnSupportPolygon (const Eigen::Vector2d &_p, const SupportPolygon &_support)
	Returns the point which is closest to _p that also lays on the edge of the support polygon.

Eigen::Vector2d	computeClosestPointOnSupportPolygon (std::size_t &_index1, std::size_t &_index2, const Eigen::Vector2d &_p, const SupportPolygon &_support)
	Same as closestPointOnSupportPolygon, but also fills in _index1 and _index2 with the indices of the line segment.

template<typename Derived >
Derived::PlainObject	AdTJac (const Eigen::Isometry3d &_T, const Eigen::MatrixBase< Derived > &_J)
	Adjoint mapping for dynamic size Jacobian.

template<typename Derived >
Derived::PlainObject	AdTJacFixed (const Eigen::Isometry3d &_T, const Eigen::MatrixBase< Derived > &_J)
	Adjoint mapping for fixed size Jacobian.

template<typename Derived >
Derived::PlainObject	AdRJac (const Eigen::Isometry3d &_T, const Eigen::MatrixBase< Derived > &_J)
	Change coordinate Frame of a Jacobian.

template<typename Derived >
Derived::PlainObject	AdRInvJac (const Eigen::Isometry3d &_T, const Eigen::MatrixBase< Derived > &_J)

template<typename Derived >
Derived::PlainObject	adJac (const Eigen::Vector6d &_V, const Eigen::MatrixBase< Derived > &_J)

template<typename Derived >
Derived::PlainObject	AdInvTJac (const Eigen::Isometry3d &_T, const Eigen::MatrixBase< Derived > &_J)
	Adjoint mapping for dynamic size Jacobian.

template<typename Derived >
Derived::PlainObject	AdInvTJacFixed (const Eigen::Isometry3d &_T, const Eigen::MatrixBase< Derived > &_J)
	Adjoint mapping for fixed size Jacobian.

double	wrapToPi (double angle)
	Compute the angle (in the range of -pi to +pi) which ignores any full rotations.

template<typename MatrixType , typename ReturnType >
void	extractNullSpace (const Eigen::JacobiSVD< MatrixType > &_SVD, ReturnType &_NS)

template<typename MatrixType , typename ReturnType >
void	computeNullSpace (const MatrixType &_M, ReturnType &_NS)

template<typename T >
constexpr T	toRadian (const T &degree)

template<typename T >
constexpr T	toDegree (const T &radian)

const Eigen::Matrix2d	CR ((Eigen::Matrix2d()<< 0.0, -1.0, 1.0, 0.0).finished())
	a cross b = (CR*a) dot b const Matd CR(2,2,0.0,-1.0,1.0,0.0);

int	delta (int _i, int _j)

template<typename T >
constexpr int	sign (T x, std::false_type)

template<typename T >
constexpr int	sign (T x, std::true_type)

template<typename T >
constexpr int	sign (T x)

double	sqr (double _x)

double	Tsinc (double _theta)

bool	isZero (double _theta)

double	asinh (double _X)

double	acosh (double _X)

double	atanh (double _X)

double	asech (double _X)

double	acosech (double _X)

double	acotanh (double _X)

double	round (double _x)

double	round2 (double _x)

template<typename T >
T	clip (const T &val, const T &lower, const T &upper)

template<typename DerivedA , typename DerivedB >
DerivedA::PlainObject	clip (const Eigen::MatrixBase< DerivedA > &val, const Eigen::MatrixBase< DerivedB > &lower, const Eigen::MatrixBase< DerivedB > &upper)

bool	isEqual (double _x, double _y)

bool	isInt (double _x)

bool	isNan (double _v)
	Returns whether _v is a NaN (Not-A-Number) value.

bool	isNan (const Eigen::MatrixXd &_m)
	Returns whether _m is a NaN (Not-A-Number) matrix.

bool	isInf (double _v)
	Returns whether _v is an infinity value (either positive infinity or negative infinity).

bool	isInf (const Eigen::MatrixXd &_m)
	Returns whether _m is an infinity matrix (either positive infinity or negative infinity).

bool	isSymmetric (const Eigen::MatrixXd &_m, double _tol=1e-6)
	Returns whether _m is symmetric or not.

unsigned	seedRand ()

double	random (double _min, double _max)

template<int N>
Eigen::Matrix< double, N, 1 >	randomVector (double _min, double _max)

template<int N>
Eigen::Matrix< double, N, 1 >	randomVector (double _limit)

Eigen::VectorXd	randomVectorXd (std::size_t size, double min, double max)

Eigen::VectorXd	randomVectorXd (std::size_t size, double limit)

Typedef Documentation

◆ AngularJacobian

using dart::math::AngularJacobian = typedef Eigen::Matrix<double, 3, Eigen::Dynamic>

◆ constantsd

using dart::math::constantsd = typedef constants<double>

◆ constantsf

using dart::math::constantsf = typedef constants<float>

◆ Icosphered

using dart::math::Icosphered = typedef Icosphere<double>

◆ Icospheref

using dart::math::Icospheref = typedef Icosphere<float>

◆ Inertia

using dart::math::Inertia = typedef Eigen::Matrix6d

◆ Jacobian

using dart::math::Jacobian = typedef Eigen::Matrix<double, 6, Eigen::Dynamic>

◆ LinearJacobian

using dart::math::LinearJacobian = typedef Eigen::Matrix<double, 3, Eigen::Dynamic>

◆ Meshd

using dart::math::Meshd = typedef Mesh<double>

◆ Meshf

using dart::math::Meshf = typedef Mesh<float>

◆ NullSpace

using dart::math::NullSpace = typedef RealVectorSpace<0u>

◆ R1Space

using dart::math::R1Space = typedef RealVectorSpace<1u>

◆ R2Space

using dart::math::R2Space = typedef RealVectorSpace<2u>

◆ R3Space

using dart::math::R3Space = typedef RealVectorSpace<3u>

◆ SupportGeometry

typedef std::vector<Eigen::Vector3d> dart::math::SupportGeometry

◆ SupportPolygon

typedef common::aligned_vector<Eigen::Vector2d> dart::math::SupportPolygon

◆ TriMeshd

using dart::math::TriMeshd = typedef TriMesh<double>

◆ TriMeshf

using dart::math::TriMeshf = typedef TriMesh<float>

Enumeration Type Documentation

◆ AxisType

enum dart::math::AxisType

Enumerator
AXIS_X
AXIS_Y
AXIS_Z

◆ IntersectionResult

enum dart::math::IntersectionResult

Intersection_t is returned by the computeIntersection() function to indicate whether there was a valid intersection between the two line segments.

Enumerator
INTERSECTING	An intersection was found.
PARALLEL	The line segments are parallel.
BEYOND_ENDPOINTS	There is no intersection because the end points do not expand far enough.

Function Documentation

◆ acosech()

double dart::math::acosech ( double _X )

inline

◆ acosh()

double dart::math::acosh ( double _X )

inline

◆ acotanh()

double dart::math::acotanh ( double _X )

inline

◆ ad()

Eigen::Vector6d dart::math::ad	(	const Eigen::Vector6d &	_X,
		const Eigen::Vector6d &	_Y
	)

adjoint mapping

Note: \(ad_X Y = ( w_X @times w_Y@,,~w_X @times v_Y - w_Y @times v_X),\), where \(X=(w_X,v_X)@in se(3), @quad Y=(w_Y,v_Y)@in se(3) \).

◆ AdInvRLinear()

Eigen::Vector6d dart::math::AdInvRLinear	(	const Eigen::Isometry3d &	_T,
		const Eigen::Vector3d &	_v
	)

Fast version of Ad(Inv([R 0; 0 1]), se3(0, v))

◆ AdInvT()

Eigen::Vector6d dart::math::AdInvT	(	const Eigen::Isometry3d &	_T,
		const Eigen::Vector6d &	_V
	)

fast version of Ad(Inv(T), V)

◆ AdInvTJac()

template<typename Derived >

Derived::PlainObject dart::math::AdInvTJac	(	const Eigen::Isometry3d &	_T,
		const Eigen::MatrixBase< Derived > &	_J
	)

Adjoint mapping for dynamic size Jacobian.

◆ AdInvTJacFixed()

template<typename Derived >

Derived::PlainObject dart::math::AdInvTJacFixed	(	const Eigen::Isometry3d &	_T,
		const Eigen::MatrixBase< Derived > &	_J
	)

Adjoint mapping for fixed size Jacobian.

◆ adJac()

template<typename Derived >

Derived::PlainObject dart::math::adJac	(	const Eigen::Vector6d &	_V,
		const Eigen::MatrixBase< Derived > &	_J
	)

◆ AdR()

Eigen::Vector6d dart::math::AdR	(	const Eigen::Isometry3d &	_T,
		const Eigen::Vector6d &	_V
	)

Fast version of Ad([R 0; 0 1], V)

◆ AdRInvJac()

template<typename Derived >

Derived::PlainObject dart::math::AdRInvJac	(	const Eigen::Isometry3d &	_T,
		const Eigen::MatrixBase< Derived > &	_J
	)

◆ AdRJac()

template<typename Derived >

Derived::PlainObject dart::math::AdRJac	(	const Eigen::Isometry3d &	_T,
		const Eigen::MatrixBase< Derived > &	_J
	)

Change coordinate Frame of a Jacobian.

◆ AdT()

Eigen::Vector6d dart::math::AdT	(	const Eigen::Isometry3d &	_T,
		const Eigen::Vector6d &	_V
	)

Rectify the rotation part so as that it satifies the orthogonality condition.

It is one step of \(R_{i_1}=1/2(R_i + R_i^{-T})\). Hence by calling this function iterativley, you can make the rotation part closer to SO(3).

reparameterize such as ||s'|| < M_PI and Exp(s) == Epx(s')

adjoint mapping

Note: \(Ad_TV = ( Rw@,, ~p @times Rw + Rv)\), where \(T=(R,p)@in SE(3), @quad V=(w,v)@in se(3) \).

◆ AdTAngular()

Eigen::Vector6d dart::math::AdTAngular	(	const Eigen::Isometry3d &	_T,
		const Eigen::Vector3d &	_w
	)

fast version of Ad(T, se3(w, 0))

◆ AdTJac()

template<typename Derived >

Derived::PlainObject dart::math::AdTJac	(	const Eigen::Isometry3d &	_T,
		const Eigen::MatrixBase< Derived > &	_J
	)

Adjoint mapping for dynamic size Jacobian.

◆ AdTJacFixed()

template<typename Derived >

Derived::PlainObject dart::math::AdTJacFixed	(	const Eigen::Isometry3d &	_T,
		const Eigen::MatrixBase< Derived > &	_J
	)

Adjoint mapping for fixed size Jacobian.

◆ AdTLinear()

Eigen::Vector6d dart::math::AdTLinear	(	const Eigen::Isometry3d &	_T,
		const Eigen::Vector3d &	_v
	)

fast version of Ad(T, se3(0, v))

◆ asech()

double dart::math::asech ( double _X )

inline

◆ asinh()

double dart::math::asinh ( double _X )

inline

◆ atanh()

double dart::math::atanh ( double _X )

inline

◆ clip() [1/2]

template<typename DerivedA , typename DerivedB >

DerivedA::PlainObject dart::math::clip	(	const Eigen::MatrixBase< DerivedA > &	val,
		const Eigen::MatrixBase< DerivedB > &	lower,
		const Eigen::MatrixBase< DerivedB > &	upper
	)

inline

◆ clip() [2/2]

template<typename T >

T dart::math::clip	(	const T &	val,
		const T &	lower,
		const T &	upper
	)

inline

◆ computeCentroidOfHull()

Eigen::Vector2d dart::math::computeCentroidOfHull ( const SupportPolygon & _convexHull )

Compute the centroid of a polygon, assuming the polygon is a convex hull.

◆ computeClosestPointOnLineSegment()

Eigen::Vector2d dart::math::computeClosestPointOnLineSegment	(	const Eigen::Vector2d &	_p,
		const Eigen::Vector2d &	_s1,
		const Eigen::Vector2d &	_s2
	)

Returns the point which is closest to _p that also lays on the line segment that goes from _s1 -> _s2.

◆ computeClosestPointOnSupportPolygon() [1/2]

Eigen::Vector2d dart::math::computeClosestPointOnSupportPolygon	(	const Eigen::Vector2d &	_p,
		const SupportPolygon &	_support
	)

Returns the point which is closest to _p that also lays on the edge of the support polygon.

◆ computeClosestPointOnSupportPolygon() [2/2]

Eigen::Vector2d dart::math::computeClosestPointOnSupportPolygon	(	std::size_t &	_index1,
		std::size_t &	_index2,
		const Eigen::Vector2d &	_p,
		const SupportPolygon &	_support
	)

Same as closestPointOnSupportPolygon, but also fills in _index1 and _index2 with the indices of the line segment.

◆ computeConvexHull() [1/2]

SupportPolygon dart::math::computeConvexHull ( const SupportPolygon & _points )

Computes the convex hull of a set of 2D points.

◆ computeConvexHull() [2/2]

SupportPolygon dart::math::computeConvexHull	(	std::vector< std::size_t > &	_originalIndices,
		const SupportPolygon &	_points
	)

Computes the convex hull of a set of 2D points and fills in _originalIndices with the original index of each entry in the returned SupportPolygon.

◆ computeConvexHull3D()

template<typename S , typename Index >

std::tuple< std::vector< Eigen::Matrix< S, 3, 1 > >, std::vector< Eigen::Matrix< Index, 3, 1 > > > dart::math::computeConvexHull3D	(	const std::vector< Eigen::Matrix< S, 3, 1 > > &	vertices,
		bool	optimize = `true`
	)

Generates a 3D convex hull given vertices and indices.

Template Parameters

S	The scalar type of the vertices.
Index	The index type of the triangles.

Parameters

[in]	vertices	The given vertices to generate a convex hull from.
[in]	optimize	(Optional) Whether to discard vertices that are not referred to in the resulted convex hull. The resulted indices will be updated accordingly.

Returns: A tuple of the vertices and indices of the resulted convex hull.

◆ computeIntersection()

IntersectionResult dart::math::computeIntersection	(	Eigen::Vector2d &	_intersectionPoint,
		const Eigen::Vector2d &	a1,
		const Eigen::Vector2d &	a2,
		const Eigen::Vector2d &	b1,
		const Eigen::Vector2d &	b2
	)

Compute the intersection between a line segment that goes from a1 -> a2 and a line segment that goes from b1 -> b2.

◆ computeNullSpace()

template<typename MatrixType , typename ReturnType >

void dart::math::computeNullSpace	(	const MatrixType &	_M,
		ReturnType &	_NS
	)

◆ computeRotation()

Eigen::Matrix3d dart::math::computeRotation	(	const Eigen::Vector3d &	axis,
		AxisType	axisType = `AxisType::AXIS_X`
	)

Compute a rotation matrix from a vector.

One axis of the rotated coordinates by the rotation matrix matches the input axis where the axis is specified by axisType.

◆ computeSupportPolgyon() [1/2]

SupportPolygon dart::math::computeSupportPolgyon	(	const SupportGeometry &	_geometry,
		const Eigen::Vector3d &	_axis1 = `Eigen::Vector3d::UnitX()`,
		const Eigen::Vector3d &	_axis2 = `Eigen::Vector3d::UnitY()`
	)

Project the support geometry points onto a plane with the given axes and then compute their convex hull, which will take the form of a polgyon.

_axis1 and _axis2 must both have unit length for this function to work correctly.

◆ computeSupportPolgyon() [2/2]

SupportPolygon dart::math::computeSupportPolgyon	(	std::vector< std::size_t > &	_originalIndices,
		const SupportGeometry &	_geometry,
		const Eigen::Vector3d &	_axis1,
		const Eigen::Vector3d &	_axis2
	)

Same as computeSupportPolgyon, except you can pass in a std::vector<std::size_t> which will have the same size as the returned SupportPolygon, and each entry will contain the original index of each point in the SupportPolygon.

◆ computeTransform()

Eigen::Isometry3d dart::math::computeTransform	(	const Eigen::Vector3d &	axis,
		const Eigen::Vector3d &	translation,
		AxisType	axisType = `AxisType::AXIS_X`
	)

Compute a transform from a vector and a position.

The rotation of the result transform is computed by computeRotationMatrix(), and the translation is just the input translation.

◆ CR()

const Eigen::Matrix2d dart::math::CR ( (Eigen::Matrix2d()<< 0.0, -1.0, 1.0, 0.0).finished() )

a cross b = (CR*a) dot b const Matd CR(2,2,0.0,-1.0,1.0,0.0);

◆ cross()

double dart::math::cross	(	const Eigen::Vector2d &	_v1,
		const Eigen::Vector2d &	_v2
	)

Compute a 2D cross product.

◆ dad()

Eigen::Vector6d dart::math::dad	(	const Eigen::Vector6d &	_s,
		const Eigen::Vector6d &	_t
	)

fast version of ad(se3(Eigen_Vec3(0), v), S)

fast version of ad(se3(w, 0), se3(v, 0)) -> check

dual adjoint mapping

Note: \(ad^{@,*}_V F = (m @times w + f @times v@,,~ f @times w),\) , where \(F=(m,f)@in se^{@,*}(3), @quad V=(w,v)@in se(3) \).

◆ dAdInvR()

Eigen::Vector6d dart::math::dAdInvR	(	const Eigen::Isometry3d &	_T,
		const Eigen::Vector6d &	_F
	)

fast version of dAd(Inv([R 0; 0 1]), F)

◆ dAdInvT()

Eigen::Vector6d dart::math::dAdInvT	(	const Eigen::Isometry3d &	_T,
		const Eigen::Vector6d &	_F
	)

fast version of dAd(Inv(T), F)

◆ dAdT()

Eigen::Vector6d dart::math::dAdT	(	const Eigen::Isometry3d &	_T,
		const Eigen::Vector6d &	_F
	)

dual adjoint mapping

Note: \(Ad^{@,*}_TF = ( R^T (m - p@times f)@,,~ R^T f)\), where \(T=(R,p)@in SE(3), F=(m,f)@in se(3)^*\).

◆ delta()

int dart::math::delta	(	int	_i,
		int	_j
	)

inline

◆ discardUnusedVertices()

template<typename S , typename Index >

std::tuple< std::vector< Eigen::Matrix< S, 3, 1 > >, std::vector< Eigen::Matrix< Index, 3, 1 > > > dart::math::discardUnusedVertices	(	const std::vector< Eigen::Matrix< S, 3, 1 > > &	vertices,
		const std::vector< Eigen::Matrix< Index, 3, 1 > > &	triangles
	)

◆ eulerXYXToMatrix()

Eigen::Matrix3d dart::math::eulerXYXToMatrix ( const Eigen::Vector3d & _angle )

Given Euler XYX angles, return a 3x3 rotation matrix, which is equivalent to RotX(angle(0)) * RotY(angle(1)) * RotX(angle(2)).

◆ eulerXYZToMatrix()

Eigen::Matrix3d dart::math::eulerXYZToMatrix ( const Eigen::Vector3d & _angle )

Given EulerXYZ angles, return a 3x3 rotation matrix, which is equivalent to RotX(angle(0)) * RotY(angle(1)) * RotZ(angle(2)).

◆ eulerXZXToMatrix()

Eigen::Matrix3d dart::math::eulerXZXToMatrix ( const Eigen::Vector3d & _angle )

Given EulerXZX angles, return a 3x3 rotation matrix, which is equivalent to RotX(angle(0)) * RotZ(angle(1)) * RotX(angle(2)).

◆ eulerXZYToMatrix()

Eigen::Matrix3d dart::math::eulerXZYToMatrix ( const Eigen::Vector3d & _angle )

Given EulerXZY angles, return a 3x3 rotation matrix, which is equivalent to RotX(angle(0)) * RotZ(angle(1)) * RotY(angle(2)).

◆ eulerYXYToMatrix()

Eigen::Matrix3d dart::math::eulerYXYToMatrix ( const Eigen::Vector3d & _angle )

Given EulerYXY angles, return a 3x3 rotation matrix, which is equivalent to RotY(angle(0)) * RotX(angle(1)) * RotY(angle(2)).

◆ eulerYXZToMatrix()

Eigen::Matrix3d dart::math::eulerYXZToMatrix ( const Eigen::Vector3d & _angle )

Given EulerYXZ angles, return a 3x3 rotation matrix, which is equivalent to RotY(angle(0)) * RotX(angle(1)) * RotZ(angle(2)).

◆ eulerYZXToMatrix()

Eigen::Matrix3d dart::math::eulerYZXToMatrix ( const Eigen::Vector3d & _angle )

Given EulerYZX angles, return a 3x3 rotation matrix, which is equivalent to RotY(angle(0)) * RotZ(angle(1)) * RotX(angle(2)).

◆ eulerYZYToMatrix()

Eigen::Matrix3d dart::math::eulerYZYToMatrix ( const Eigen::Vector3d & _angle )

Given EulerYZY angles, return a 3x3 rotation matrix, which is equivalent to RotY(angle(0)) * RotZ(angle(1)) * RotY(angle(2)).

◆ eulerZXYToMatrix()

Eigen::Matrix3d dart::math::eulerZXYToMatrix ( const Eigen::Vector3d & _angle )

Given EulerZXY angles, return a 3x3 rotation matrix, which is equivalent to RotZ(angle(0)) * RotX(angle(1)) * RotY(angle(2)).

◆ eulerZXZToMatrix()

Eigen::Matrix3d dart::math::eulerZXZToMatrix ( const Eigen::Vector3d & _angle )

Given EulerZXZ angles, return a 3x3 rotation matrix, which is equivalent to RotZ(angle(0)) * RotX(angle(1)) * RotZ(angle(2)).

◆ eulerZYXToMatrix()

Eigen::Matrix3d dart::math::eulerZYXToMatrix ( const Eigen::Vector3d & _angle )

Given EulerZYX angles, return a 3x3 rotation matrix, which is equivalent to RotZ(angle(0)) * RotY(angle(1)) * RotX(angle(2)).

singularity : angle[1] = -+ 0.5*PI

◆ eulerZYZToMatrix()

Eigen::Matrix3d dart::math::eulerZYZToMatrix ( const Eigen::Vector3d & _angle )

Given EulerZYZ angles, return a 3x3 rotation matrix, which is equivalent to RotZ(angle(0)) * RotY(angle(1)) * RotZ(angle(2)).

singularity : angle[1] = 0, PI

◆ expAngular()

Eigen::Isometry3d dart::math::expAngular ( const Eigen::Vector3d & _s )

fast version of Exp(se3(s, 0))

◆ expMap()

Eigen::Isometry3d dart::math::expMap ( const Eigen::Vector6d & _S )

Exponential mapping.

◆ expMapJac()

Eigen::Matrix3d dart::math::expMapJac ( const Eigen::Vector3d & _q )

Computes the Jacobian of the expmap.

◆ expMapJacDeriv()

Eigen::Matrix3d dart::math::expMapJacDeriv	(	const Eigen::Vector3d &	_q,
		int	_qi
	)

computes the derivative of the Jacobian of the expmap wrt to _qi indexed dof; _qi \( \in \) {0,1,2}

◆ expMapJacDot()

Eigen::Matrix3d dart::math::expMapJacDot	(	const Eigen::Vector3d &	_q,
		const Eigen::Vector3d &	_qdot
	)

Computes the time derivative of the expmap Jacobian.

◆ expMapRot()

Eigen::Matrix3d dart::math::expMapRot ( const Eigen::Vector3d & _q )

Computes the Rotation matrix from a given expmap vector.

◆ expToQuat()

Eigen::Quaterniond dart::math::expToQuat ( const Eigen::Vector3d & _v )

◆ extractNullSpace()

template<typename MatrixType , typename ReturnType >

void dart::math::extractNullSpace	(	const Eigen::JacobiSVD< MatrixType > &	_SVD,
		ReturnType &	_NS
	)

◆ fromSkewSymmetric()

Eigen::Vector3d dart::math::fromSkewSymmetric ( const Eigen::Matrix3d & _m )

◆ getAdTMatrix()

Eigen::Matrix6d dart::math::getAdTMatrix ( const Eigen::Isometry3d & T )

Get linear transformation matrix of Adjoint mapping.

◆ getFrameOriginAxisZ()

Eigen::Isometry3d dart::math::getFrameOriginAxisZ	(	const Eigen::Vector3d &	_origin,
		const Eigen::Vector3d &	_axisZ
	)

Generate frame given origin and z-axis.

◆ HullAngleComparison()

static bool dart::math::HullAngleComparison	(	const HullAngle &	a,
		const HullAngle &	b
	)

static

◆ integratePosition()

template<typename SpaceT >

SpaceT::Point dart::math::integratePosition	(	const typename SpaceT::Point &	pos,
		const typename SpaceT::Vector &	vel,
		double	dt
	)

◆ integrateVelocity()

template<typename SpaceT >

SpaceT::Vector dart::math::integrateVelocity	(	const typename SpaceT::Vector &	vel,
		const typename SpaceT::Vector &	acc,
		double	dt
	)

◆ inverse()

template<typename SpaceT >

SpaceT::Matrix dart::math::inverse ( const typename SpaceT::Matrix & mat )

◆ isEqual()

bool dart::math::isEqual	(	double	_x,
		double	_y
	)

inline

◆ isInf() [1/2]

bool dart::math::isInf ( const Eigen::MatrixXd & _m )

inline

Returns whether _m is an infinity matrix (either positive infinity or negative infinity).

◆ isInf() [2/2]

bool dart::math::isInf ( double _v )

inline

Returns whether _v is an infinity value (either positive infinity or negative infinity).

◆ isInsideSupportPolygon()

bool dart::math::isInsideSupportPolygon	(	const Eigen::Vector2d &	_p,
		const SupportPolygon &	_support,
		bool	_includeEdge
	)

Returns true if the point _p is inside the support polygon.

◆ isInt()

bool dart::math::isInt ( double _x )

inline

◆ isLeftTurn()

static bool dart::math::isLeftTurn	(	const Eigen::Vector2d &	p1,
		const Eigen::Vector2d &	p2,
		const Eigen::Vector2d &	p3
	)

static

◆ isNan() [1/2]

bool dart::math::isNan ( const Eigen::MatrixXd & _m )

inline

Returns whether _m is a NaN (Not-A-Number) matrix.

◆ isNan() [2/2]

bool dart::math::isNan ( double _v )

inline

Returns whether _v is a NaN (Not-A-Number) value.

◆ isSymmetric()

bool dart::math::isSymmetric	(	const Eigen::MatrixXd &	_m,
		double	_tol = `1e-6`
	)

inline

Returns whether _m is symmetric or not.

◆ isZero()

bool dart::math::isZero ( double _theta )

inline

◆ logMap() [1/2]

Eigen::Vector6d dart::math::logMap ( const Eigen::Isometry3d & _T )

Log mapping.

◆ logMap() [2/2]

Eigen::Vector3d dart::math::logMap ( const Eigen::Matrix3d & _R )

Log mapping.

Note: When \(|Log(R)| = @pi\), Exp(LogR(R) = Exp(-Log(R)). The implementation returns only the positive one.

◆ makeSkewSymmetric()

Eigen::Matrix3d dart::math::makeSkewSymmetric ( const Eigen::Vector3d & _v )

◆ matrixToEulerXYX()

Eigen::Vector3d dart::math::matrixToEulerXYX ( const Eigen::Matrix3d & _R )

get the Euler XYX angle from R

◆ matrixToEulerXYZ()

Eigen::Vector3d dart::math::matrixToEulerXYZ ( const Eigen::Matrix3d & _R )

get the Euler XYZ angle from R

◆ matrixToEulerXZY()

Eigen::Vector3d dart::math::matrixToEulerXZY ( const Eigen::Matrix3d & _R )

get the Euler XZY angle from R

◆ matrixToEulerYXZ()

Eigen::Vector3d dart::math::matrixToEulerYXZ ( const Eigen::Matrix3d & _R )

get the Euler YXZ angle from R

◆ matrixToEulerYZX()

Eigen::Vector3d dart::math::matrixToEulerYZX ( const Eigen::Matrix3d & _R )

get the Euler YZX angle from R

◆ matrixToEulerZXY()

Eigen::Vector3d dart::math::matrixToEulerZXY ( const Eigen::Matrix3d & _R )

get the Euler ZXY angle from R

◆ matrixToEulerZYX()

Eigen::Vector3d dart::math::matrixToEulerZYX ( const Eigen::Matrix3d & _R )

get the Euler ZYX angle from R

◆ parallelAxisTheorem()

Eigen::Matrix3d dart::math::parallelAxisTheorem	(	const Eigen::Matrix3d &	_original,
		const Eigen::Vector3d &	_comShift,
		double	_mass
	)

Use the Parallel Axis Theorem to compute the moment of inertia of a body whose center of mass has been shifted from the origin.

◆ quatDeriv()

Eigen::Matrix3d dart::math::quatDeriv	(	const Eigen::Quaterniond &	_q,
		int	_el
	)

◆ quatSecondDeriv()

Eigen::Matrix3d dart::math::quatSecondDeriv	(	const Eigen::Quaterniond &	,
		int	_el1,
		int	_el2
	)

◆ quatToExp()

Eigen::Vector3d dart::math::quatToExp ( const Eigen::Quaterniond & _q )

◆ random()

double dart::math::random	(	double	_min,
		double	_max
	)

inline

Deprecated:: Please use Random::uniform() instead.

◆ randomVector() [1/2]

template<int N>

Eigen::Matrix< double, N, 1 > dart::math::randomVector ( double _limit )

Deprecated:: Please use Random::uniform() instead.

◆ randomVector() [2/2]

template<int N>

Eigen::Matrix< double, N, 1 > dart::math::randomVector	(	double	_min,
		double	_max
	)

Deprecated:: Please use Random::uniform() instead.

◆ randomVectorXd() [1/2]

Eigen::VectorXd dart::math::randomVectorXd	(	std::size_t	size,
		double	limit
	)

inline

Deprecated:: Please use Random::uniform() instead.

◆ randomVectorXd() [2/2]

Eigen::VectorXd dart::math::randomVectorXd	(	std::size_t	size,
		double	min,
		double	max
	)

inline

Deprecated:: Please use Random::uniform() instead.

◆ rotatePoint() [1/2]

Eigen::Vector3d dart::math::rotatePoint	(	const Eigen::Quaterniond &	_q,
		const Eigen::Vector3d &	_pt
	)

◆ rotatePoint() [2/2]

Eigen::Vector3d dart::math::rotatePoint	(	const Eigen::Quaterniond &	_q,
		double	_x,
		double	_y,
		double	_z
	)

◆ round()

double dart::math::round ( double _x )

inline

◆ round2()

double dart::math::round2 ( double _x )

inline

◆ seedRand()

unsigned dart::math::seedRand ( )

inline

◆ sign() [1/3]

template<typename T >

constexpr int dart::math::sign ( T x )

inlineconstexpr

◆ sign() [2/3]

template<typename T >

constexpr int dart::math::sign	(	T	x,
		std::false_type
	)

inlineconstexpr

◆ sign() [3/3]

template<typename T >

constexpr int dart::math::sign	(	T	x,
		std::true_type
	)

inlineconstexpr

◆ sqr()

double dart::math::sqr ( double _x )

inline

◆ toDegree()

template<typename T >

constexpr T dart::math::toDegree ( const T & radian )

constexpr

◆ toEuclideanPoint()

template<typename SpaceT >

SpaceT::EuclideanPoint dart::math::toEuclideanPoint ( const typename SpaceT::Point & point )

◆ toManifoldPoint()

template<typename SpaceT >

SpaceT::Point dart::math::toManifoldPoint ( const typename SpaceT::EuclideanPoint & point )

◆ toRadian()

template<typename T >

constexpr T dart::math::toRadian ( const T & degree )

constexpr

◆ transformInertia()

Inertia dart::math::transformInertia	(	const Eigen::Isometry3d &	_T,
		const Inertia &	_I
	)

◆ Tsinc()

double dart::math::Tsinc ( double _theta )

inline

◆ verifyRotation()

bool dart::math::verifyRotation ( const Eigen::Matrix3d & _T )

Check if determinant of _R is equat to 1 and all the elements are not NaN values.

◆ verifyTransform()

bool dart::math::verifyTransform ( const Eigen::Isometry3d & _T )

Check if determinant of the rotational part of _T is equat to 1 and all the elements are not NaN values.

◆ wrapToPi()

double dart::math::wrapToPi ( double angle )

inline

Compute the angle (in the range of -pi to +pi) which ignores any full rotations.

Namespaces

Classes

Typedefs

Enumerations

Functions

Typedef Documentation

◆ AngularJacobian

◆ constantsd

◆ constantsf

◆ Icosphered

◆ Icospheref

◆ Inertia

◆ Jacobian

◆ LinearJacobian

◆ Meshd

◆ Meshf

◆ NullSpace

◆ R1Space

◆ R2Space

◆ R3Space

◆ SupportGeometry

◆ SupportPolygon

◆ TriMeshd

◆ TriMeshf

Enumeration Type Documentation

◆ AxisType

◆ IntersectionResult

Function Documentation

◆ acosech()

◆ acosh()

◆ acotanh()

◆ ad()

◆ AdInvRLinear()

◆ AdInvT()

◆ AdInvTJac()

◆ AdInvTJacFixed()

◆ adJac()

◆ AdR()

◆ AdRInvJac()

◆ AdRJac()

◆ AdT()

◆ AdTAngular()

◆ AdTJac()

◆ AdTJacFixed()

◆ AdTLinear()

◆ asech()

◆ asinh()

◆ atanh()

◆ clip() [1/2]

◆ clip() [2/2]

◆ computeCentroidOfHull()

◆ computeClosestPointOnLineSegment()

◆ computeClosestPointOnSupportPolygon() [1/2]

◆ computeClosestPointOnSupportPolygon() [2/2]

◆ computeConvexHull() [1/2]

◆ computeConvexHull() [2/2]

◆ computeConvexHull3D()

◆ computeIntersection()

◆ computeNullSpace()

◆ computeRotation()

◆ computeSupportPolgyon() [1/2]

◆ computeSupportPolgyon() [2/2]

◆ computeTransform()

◆ CR()

◆ cross()

◆ dad()

◆ dAdInvR()

◆ dAdInvT()

◆ dAdT()

◆ delta()

◆ discardUnusedVertices()

◆ eulerXYXToMatrix()

◆ eulerXYZToMatrix()

◆ eulerXZXToMatrix()

◆ eulerXZYToMatrix()

◆ eulerYXYToMatrix()

◆ eulerYXZToMatrix()

◆ eulerYZXToMatrix()

◆ eulerYZYToMatrix()

◆ eulerZXYToMatrix()