Represents a 3D Bezier patch defined by a 2D array of control points. More...

#include </home/docs/checkouts/readthedocs.org/user_builds/axom/checkouts/main/src/axom/primal/geometry/BezierPatch.hpp>

Public Types
using	PointType = Point< T, NDIMS >

using	VectorType = Vector< T, NDIMS >

using	PlaneType = Plane< T, NDIMS >

using	CoordsVec = axom::Array< PointType, 1 >

using	CoordsMat = axom::Array< PointType, 2 >

using	WeightsVec = axom::Array< T, 1 >

using	WeightsMat = axom::Array< T, 2 >

using	BoundingBoxType = BoundingBox< T, NDIMS >

using	OrientedBoundingBoxType = OrientedBoundingBox< T, NDIMS >

using	BezierCurveType = primal::BezierCurve< T, NDIMS >

Public Member Functions
	AXOM_STATIC_ASSERT_MSG ((NDIMS==1)\|\|(NDIMS==2)\|\|(NDIMS==3), "A Bezier Patch object may be defined in 1-, 2-, or 3-D")

	AXOM_STATIC_ASSERT_MSG (std::is_arithmetic< T >::value, "A Bezier Patch must be defined using an arithmetic type")

std::ostream &	print (std::ostream &os) const
	Simple formatted print of a Bezier Patch instance. More...

Constructors for BezierPatch
The constructors allow for flexible initialization of BezierPatch objects from: 1D or 2D Axom arrays of control points and weights, C-style arrays of control points and weights, Specified polynomial orders in each axis, Rational or polynomial (nonrational) patches, depending on the presence of weights. For 1D arrays, the mapping of control points and weights to the patch is lexicographical, i.e. pts[0] -> nodes[0, 0], ..., pts[ord_v] -> nodes[0, ord_v] pts[ord_v+1] -> nodes[1, 0], ..., pts[2ord_v] -> nodes[1, ord_v] ... pts[ord_u(ord_v-1)] -> nodes[ord_u, 0], ..., pts[ord_u*ord_v] -> nodes[ord_u, ord_v] The patch is parametrized from u=0 to u=1 and v=0 to v=1. Rational patches are identified by a non-empty weights array, and nonrational patches by an empty weights array. All weights must be greater than 0 in a rational patch.
	BezierPatch (axom::ArrayView< const PointType, 2 > controlPoints, axom::ArrayView< const T, 2 > weights, int ord_u, int ord_v)
	Constructor from an ArrayView over the control points and weights. More...

	BezierPatch (axom::ArrayView< PointType, 2 > controlPoints, int ord_u, int ord_v)
	Constructor for polynomial BezierPatch from ArrayView of control points. More...

	BezierPatch (axom::ArrayView< PointType, 2 > controlPoints, axom::ArrayView< T, 2 > weights, int ord_u, int ord_v)
	Constructor for rational BezierPatch from ArrayView of control points. More...

	BezierPatch (int ord_u=-1, int ord_v=-1)
	Constructor for a polynomial (nonrational) Bezier Patch that reserves space for the given order of the surface. More...

	BezierPatch (const PointType *pts, int ord_u, int ord_v)
	Constructor for a Bezier Patch from an array of coordinates. More...

	BezierPatch (PointType pts, T weights, int ord_u, int ord_v)
	Constructor for a rational Bezier Patch from arrays of coordinates and weights. More...

	BezierPatch (const CoordsVec &pts, int ord_u, int ord_v)
	Constructor for a Bezier Patch from a 1D Axom array of coordinates. More...

	BezierPatch (const CoordsVec &pts, const WeightsVec &weights, int ord_u, int ord_v)
	Constructor for a Rational Bezier Patch from 1D Axom arrays of coordinates and weights. More...

	BezierPatch (const CoordsMat &pts, int ord_u, int ord_v)
	Constructor for a Bezier Patch from an Axom array of coordinates. More...

	BezierPatch (const CoordsMat &pts, const WeightsMat &weights, int ord_u, int ord_v)
	Constructor for a rational Bezier Patch from a 2D Axom array of weights and coordinates. More...

Query/modify patch properties (order, rationality, ...)
void	setOrder (int ord_u, int ord_v)
	Sets the order of Bezier patch. More...

int	getOrder_u () const
	Returns the order of the Bezier Patch on the first axis. More...

int	getOrder_v () const
	Returns the order of the Bezier Patch on the second axis. More...

void	clear ()
	Clears the list of control points, make nonrational. More...

bool	isRational () const
	Use array size as flag for rationality. More...

void	makeRational ()
	Make trivially rational. If already rational, do nothing. More...

void	makeNonrational ()
	Make nonrational by shrinking array of weights. More...

Query/modify patch's geometry (control points, weights, bounding box, ...)
PointType &	operator() (int ui, int vi)
	Retrieves the control point at index (idx_p, idx_q) More...

const PointType &	operator() (int ui, int vi) const
	Retrieves the vector of control points at index idx. More...

CoordsMat &	getControlPoints ()
	Returns a reference to the Bezier patch's control points. More...

const CoordsMat &	getControlPoints () const
	Returns a const reference to the Bezier patch's control points. More...

const T &	getWeight (int ui, int vi) const
	Get a specific weight. More...

void	setWeight (int ui, int vi, T weight)
	Set the weight at a specific index. More...

WeightsMat &	getWeights ()
	Returns a reference to the Bezier patch's weights. More...

const WeightsMat &	getWeights () const
	Returns a const reference to the Bezier patch's weights. More...

BoundingBoxType	boundingBox () const
	Returns an axis-aligned bounding box containing the Bezier patch. More...

OrientedBoundingBoxType	orientedBoundingBox () const
	Returns an oriented bounding box containing the Bezier patch. More...

bool	isPlanar (double sq_tol=1e-8, double EPS=1e-8) const
	Predicate to check if the Bezier patch is approximately planar (i.e. flat) More...

bool	isPolygonal (double sq_tol=1e-8, double EPS=1e-8) const
	Predicate to check if the patch can be approximated by a polygon. More...

bool	isBilinear (double sq_tol=1e-8, bool useStrictBilinear=false) const
	Predicate to check if the Bezier patch is approximately bilinear. More...

Query/modify patch parametrization
void	reverseOrientation (int axis)
	Reverses the order of one direction of the Bezier patch's control points and weights. More...

void	reverseOrientation_u ()
	Reverses the order of the Bezier patch's control points and weights on the first axis. More...

void	reverseOrientation_v ()
	Reverses the order of the Bezier patch's control points and weights on the second axis. More...

void	swapAxes ()
	Swap the axes such that s(u, v) becomes s(v, u) More...

Functions to evaluate patch and its derivatives and normals along points or lines
BezierCurveType	isocurve (T uv, int axis) const
	Evaluates a slice Bezier patch for a fixed parameter value of u or v. More...

BezierCurveType	isocurve_u (T u) const
	Returns an isocurve with a fixed value of u. More...

BezierCurveType	isocurve_v (T v) const
	Returns an isocurve with a fixed value of v. More...

PointType	evaluate (T u, T v) const
	Evaluates a Bezier patch at a particular parameter value (u, v) More...

void	evaluateFirstDerivatives (T u, T v, Point< T, NDIMS > &eval, Vector< T, NDIMS > &Du, Vector< T, NDIMS > &Dv) const
	Evaluates all first derivatives Bezier patch at (u, v) More...

void	evaluate_linear_derivatives (T u, T v, Point< T, NDIMS > &eval, Vector< T, NDIMS > &Du, Vector< T, NDIMS > &Dv, Vector< T, NDIMS > &DuDv) const
	Evaluates all linear derivatives Bezier patch at (u, v) More...

void	evaluateSecondDerivatives (T u, T v, Point< T, NDIMS > &eval, Vector< T, NDIMS > &Du, Vector< T, NDIMS > &Dv, Vector< T, NDIMS > &DuDu, Vector< T, NDIMS > &DvDv, Vector< T, NDIMS > &DuDv) const
	Evaluates all second derivatives Bezier patch at (u, v) More...

VectorType	du (T u, T v) const
	Computes a tangent of a Bezier patch at a particular parameter value (u, v) along the u axis. More...

VectorType	dudu (T u, T v) const
	Computes the second derivative of a Bezier patch at (u, v) along the u axis. More...

VectorType	dv (T u, T v) const
	Computes a tangent of a Bezier patch at a particular parameter value (u, v) along the v axis. More...

VectorType	dvdv (T u, T v) const
	Computes the second derivative of a Bezier patch at (u, v) along the v axis. More...

VectorType	dudv (T u, T v) const
	Computes the mixed second derivative of a Bezier patch at (u, v) More...

VectorType	dvdu (T u, T v) const
	Computes the mixed second derivative of a Bezier patch at (u, v) More...

VectorType	normal (T u, T v) const
	Computes the normal vector of a Bezier patch at a particular parameter value (u, v) More...

Functions dealing with patch subdivision
void	split (T uv, int axis, BezierPatch &p1, BezierPatch &p2) const
	Splits a Bezier patch into two Bezier patches. More...

void	split_u (T u, BezierPatch &p1, BezierPatch &p2) const
	Split the patch along a fixed value of u. More...

void	split_v (T v, BezierPatch &p1, BezierPatch &p2) const

void	split (T u, T v, BezierPatch &p1, BezierPatch &p2, BezierPatch &p3, BezierPatch &p4) const
	Splits a Bezier patch into four Bezier patches. More...

Friends
bool	operator== (const BezierPatch< T, NDIMS > &lhs, const BezierPatch< T, NDIMS > &rhs)
	Checks equality of two Bezier Patches. More...

bool	operator!= (const BezierPatch< T, NDIMS > &lhs, const BezierPatch< T, NDIMS > &rhs)

Detailed Description

template<typename T, int NDIMS>
class axom::primal::BezierPatch< T, NDIMS >

Represents a 3D Bezier patch defined by a 2D array of control points.

Template Parameters

T	the coordinate type, e.g., double, float, etc.

The order of a Bezier patch with (N+1)(M+1) control points is (N, M). The patch is approximated by the control points, parametrized from u=0 to u=1 and v=0 to v=1.

Contains a 2D array of positive weights to represent a rational Bezier patch. Polynomial (nonrational) Bezier patches are identified by an empty weights array.

Algorithms for Rational Bezier curves derived from Gerald Farin, "Algorithms for rational Bezier curves" Computer-Aided Design, Volume 15, Number 2, 1983,

Member Typedef Documentation

◆ PointType

template<typename T , int NDIMS>

using axom::primal::BezierPatch< T, NDIMS >::PointType = Point<T, NDIMS>

◆ VectorType

template<typename T , int NDIMS>

using axom::primal::BezierPatch< T, NDIMS >::VectorType = Vector<T, NDIMS>

◆ PlaneType

template<typename T , int NDIMS>

using axom::primal::BezierPatch< T, NDIMS >::PlaneType = Plane<T, NDIMS>

◆ CoordsVec

template<typename T , int NDIMS>

using axom::primal::BezierPatch< T, NDIMS >::CoordsVec = axom::Array<PointType, 1>

◆ CoordsMat

template<typename T , int NDIMS>

using axom::primal::BezierPatch< T, NDIMS >::CoordsMat = axom::Array<PointType, 2>

◆ WeightsVec

template<typename T , int NDIMS>

using axom::primal::BezierPatch< T, NDIMS >::WeightsVec = axom::Array<T, 1>

◆ WeightsMat

template<typename T , int NDIMS>

using axom::primal::BezierPatch< T, NDIMS >::WeightsMat = axom::Array<T, 2>

◆ BoundingBoxType

template<typename T , int NDIMS>

using axom::primal::BezierPatch< T, NDIMS >::BoundingBoxType = BoundingBox<T, NDIMS>

◆ OrientedBoundingBoxType

template<typename T , int NDIMS>

using axom::primal::BezierPatch< T, NDIMS >::OrientedBoundingBoxType = OrientedBoundingBox<T, NDIMS>

◆ BezierCurveType

template<typename T , int NDIMS>

using axom::primal::BezierPatch< T, NDIMS >::BezierCurveType = primal::BezierCurve<T, NDIMS>

Constructor & Destructor Documentation

◆ BezierPatch() [1/10]

template<typename T , int NDIMS>

axom::primal::BezierPatch< T, NDIMS >::BezierPatch	(	axom::ArrayView< const PointType, 2 >	controlPoints,
		axom::ArrayView< const T, 2 >	weights,
		int	ord_u,
		int	ord_v
	)

inline

Constructor from an ArrayView over the control points and weights.

Parameters

[in]	controlPoints	ArrayView of control points (size: (ord_u+1, ord_v+1) or (0,0))
[in]	weights	ArrayView of weights (size: (ord_u+1, ord_v+1) or (0,0))
[in]	ord_u	The patch's polynomial order on the first axis
[in]	ord_v	The patch's polynomial order on the second axis

If controlPoints is empty, we still allocate space for (ord_u+1, ord_v+1) control points

Precondition: ord_u and ord_v must either both be at least 0 for a valid patch, or both must be -1 for an empty patch

References axom::ArrayView< T, DIM, SPACE >::data(), axom::ArrayView< T, DIM, SPACE >::empty(), axom::utilities::max(), axom::Array< T, DIM, SPACE, StoragePolicy >::resize(), axom::ArrayBase< T, DIM, ArrayType >::shape(), axom::ArrayView< T, DIM, SPACE >::size(), and SLIC_ASSERT.

◆ BezierPatch() [2/10]

template<typename T , int NDIMS>

axom::primal::BezierPatch< T, NDIMS >::BezierPatch	(	axom::ArrayView< PointType, 2 >	controlPoints,
		int	ord_u,
		int	ord_v
	)

inline

Constructor for polynomial BezierPatch from ArrayView of control points.

◆ BezierPatch() [3/10]

template<typename T , int NDIMS>

axom::primal::BezierPatch< T, NDIMS >::BezierPatch	(	axom::ArrayView< PointType, 2 >	controlPoints,
		axom::ArrayView< T, 2 >	weights,
		int	ord_u,
		int	ord_v
	)

inline

Constructor for rational BezierPatch from ArrayView of control points.

◆ BezierPatch() [4/10]

template<typename T , int NDIMS>

axom::primal::BezierPatch< T, NDIMS >::BezierPatch	(	int	ord_u = `-1`,
		int	ord_v = `-1`
	)

inline

Constructor for a polynomial (nonrational) Bezier Patch that reserves space for the given order of the surface.

Parameters

[in] ord_u,ord_v The patch's polynomial orders

Precondition: ord_u, ord_v greater than or equal to -1.

◆ BezierPatch() [5/10]

template<typename T , int NDIMS>

axom::primal::BezierPatch< T, NDIMS >::BezierPatch	(	const PointType *	pts,
		int	ord_u,
		int	ord_v
	)

inline

Constructor for a Bezier Patch from an array of coordinates.

Parameters

[in]	pts	A 1D C-style array of (ord_u+1)*(ord_v+1) control points
[in]	ord_u,ord_v	The patch's polynomial orders

Precondition: order in both directions is greater than or equal to zero

◆ BezierPatch() [6/10]

template<typename T , int NDIMS>

axom::primal::BezierPatch< T, NDIMS >::BezierPatch	(	PointType *	pts,
		T *	weights,
		int	ord_u,
		int	ord_v
	)

inline

Constructor for a rational Bezier Patch from arrays of coordinates and weights.

Parameters

[in]	pts	A 1D C-style array of (ord_u+1)*(ord_v+1) control points
[in]	weights	A 1D C-style array of (ord_u+1)*(ord_v+1) positive weights
[in]	ord_u,ord_v	The patch's polynomial orders, both greater than or equal to zero

If weights is the null pointer, creates a nonrational curve

◆ BezierPatch() [7/10]

template<typename T , int NDIMS>

axom::primal::BezierPatch< T, NDIMS >::BezierPatch	(	const CoordsVec &	pts,
		int	ord_u,
		int	ord_v
	)

inline

Constructor for a Bezier Patch from a 1D Axom array of coordinates.

Parameters

[in]	pts	A 1D Axom array of (ord_u+1)*(ord_v+1) control points
[in]	ord_u,ord_v	The patch's polynomial orders, both greater than or equal to zero

◆ BezierPatch() [8/10]

template<typename T , int NDIMS>

axom::primal::BezierPatch< T, NDIMS >::BezierPatch	(	const CoordsVec &	pts,
		const WeightsVec &	weights,
		int	ord_u,
		int	ord_v
	)

inline

Constructor for a Rational Bezier Patch from 1D Axom arrays of coordinates and weights.

Parameters

[in]	pts	A 1D Axom array of (ord_u+1)*(ord_v+1) control points
[in]	weights	A 1D Axom array of (ord_u+1)*(ord_v+1) positive weights
[in]	ord_u,ord_v	The patch's polynomial orders, both greater than or equal to zero

◆ BezierPatch() [9/10]

template<typename T , int NDIMS>

axom::primal::BezierPatch< T, NDIMS >::BezierPatch	(	const CoordsMat &	pts,
		int	ord_u,
		int	ord_v
	)

inline

Constructor for a Bezier Patch from an Axom array of coordinates.

Parameters

[in]	pts	A 2D Axom array with (ord_u+1, ord_v+1) control points
[in]	ord_u,ord_v	The patch's polynomial orders, both greater than or equal to zero

◆ BezierPatch() [10/10]

template<typename T , int NDIMS>

axom::primal::BezierPatch< T, NDIMS >::BezierPatch	(	const CoordsMat &	pts,
		const WeightsMat &	weights,
		int	ord_u,
		int	ord_v
	)

inline

Constructor for a rational Bezier Patch from a 2D Axom array of weights and coordinates.

Parameters

[in]	pts	A 2D Axom array with (ord_u+1, ord_v+1) control points
[in]	weights	A 2D Axom array with (ord_u+1, ord_v+1) weights
[in]	ord_u,ord_v	The patch's polynomial orders, both greater than or equal to zero

Member Function Documentation

◆ AXOM_STATIC_ASSERT_MSG() [1/2]

template<typename T , int NDIMS>

axom::primal::BezierPatch< T, NDIMS >::AXOM_STATIC_ASSERT_MSG	(	(NDIMS==1)\|\|(NDIMS==2)\|\|(NDIMS==3)	,
		"A Bezier Patch object may be defined in 1-	,
		2-	,
		or 3-D"
	)

◆ AXOM_STATIC_ASSERT_MSG() [2/2]

template<typename T , int NDIMS>

axom::primal::BezierPatch< T, NDIMS >::AXOM_STATIC_ASSERT_MSG	(	std::is_arithmetic< T >::value	,
		"A Bezier Patch must be defined using an arithmetic type"
	)

◆ setOrder()

template<typename T , int NDIMS>

void axom::primal::BezierPatch< T, NDIMS >::setOrder	(	int	ord_u,
		int	ord_v
	)

inline

Sets the order of Bezier patch.

Parameters

[in] ord_u,ord_v The patch's polynomial orders,

Precondition: ord_u and ord_v must both be -1 or both greater than or equal to zero

Note: Will only resize the arrays, and likely make the patch invalid

References axom::primal::BezierPatch< T, NDIMS >::isRational(), axom::utilities::max(), axom::Array< T, DIM, SPACE, StoragePolicy >::resize(), and SLIC_ASSERT.

◆ getOrder_u()

template<typename T , int NDIMS>

int axom::primal::BezierPatch< T, NDIMS >::getOrder_u ( ) const

inline

Returns the order of the Bezier Patch on the first axis.

References axom::ArrayBase< T, DIM, ArrayType >::shape().

◆ getOrder_v()

template<typename T , int NDIMS>

int axom::primal::BezierPatch< T, NDIMS >::getOrder_v ( ) const

inline

Returns the order of the Bezier Patch on the second axis.

References axom::ArrayBase< T, DIM, ArrayType >::shape().

◆ clear()

template<typename T , int NDIMS>

void axom::primal::BezierPatch< T, NDIMS >::clear ( )

inline

Clears the list of control points, make nonrational.

References axom::Array< T, DIM, SPACE, StoragePolicy >::clear(), and axom::primal::BezierPatch< T, NDIMS >::makeNonrational().

◆ isRational()

template<typename T , int NDIMS>

bool axom::primal::BezierPatch< T, NDIMS >::isRational ( ) const

inline

Use array size as flag for rationality.

References axom::Array< T, DIM, SPACE, StoragePolicy >::empty().

◆ makeRational()

template<typename T , int NDIMS>

void axom::primal::BezierPatch< T, NDIMS >::makeRational ( )

inline

Make trivially rational. If already rational, do nothing.

References axom::Array< T, DIM, SPACE, StoragePolicy >::fill(), axom::primal::BezierPatch< T, NDIMS >::getOrder_u(), axom::primal::BezierPatch< T, NDIMS >::getOrder_v(), axom::primal::BezierPatch< T, NDIMS >::isRational(), and axom::Array< T, DIM, SPACE, StoragePolicy >::resize().

◆ makeNonrational()

template<typename T , int NDIMS>

void axom::primal::BezierPatch< T, NDIMS >::makeNonrational ( )

inline

Make nonrational by shrinking array of weights.

References axom::Array< T, DIM, SPACE, StoragePolicy >::clear().

◆ operator()() [1/2]

template<typename T , int NDIMS>

PointType& axom::primal::BezierPatch< T, NDIMS >::operator()	(	int	ui,
		int	vi
	)

inline

Retrieves the control point at index (idx_p, idx_q)

◆ operator()() [2/2]

template<typename T , int NDIMS>

const PointType& axom::primal::BezierPatch< T, NDIMS >::operator()	(	int	ui,
		int	vi
	)		const

inline

Retrieves the vector of control points at index idx.

◆ getControlPoints() [1/2]

template<typename T , int NDIMS>

CoordsMat& axom::primal::BezierPatch< T, NDIMS >::getControlPoints ( )

inline

Returns a reference to the Bezier patch's control points.

◆ getControlPoints() [2/2]

template<typename T , int NDIMS>

const CoordsMat& axom::primal::BezierPatch< T, NDIMS >::getControlPoints ( ) const

inline

Returns a const reference to the Bezier patch's control points.

◆ getWeight()

template<typename T , int NDIMS>

const T& axom::primal::BezierPatch< T, NDIMS >::getWeight	(	int	ui,
		int	vi
	)		const

inline

Get a specific weight.

Parameters

[in]	ui	The index of the weight on the first axis
[in]	vi	The index of the weight on the second axis

Precondition: Requires that the surface be rational

References axom::primal::BezierPatch< T, NDIMS >::isRational(), and SLIC_ASSERT.

◆ setWeight()

template<typename T , int NDIMS>

void axom::primal::BezierPatch< T, NDIMS >::setWeight	(	int	ui,
		int	vi,
		T	weight
	)

inline

Set the weight at a specific index.

Parameters

[in]	ui	The index of the weight in on the first axis
[in]	vi	The index of the weight in on the second axis
[in]	weight	The updated value of the weight

Precondition: Requires that the surface be rational; Requires that the weight be positive

References axom::primal::BezierPatch< T, NDIMS >::isRational(), and SLIC_ASSERT.

◆ getWeights() [1/2]

template<typename T , int NDIMS>

WeightsMat& axom::primal::BezierPatch< T, NDIMS >::getWeights ( )

inline

Returns a reference to the Bezier patch's weights.

◆ getWeights() [2/2]

template<typename T , int NDIMS>

const WeightsMat& axom::primal::BezierPatch< T, NDIMS >::getWeights ( ) const

inline

Returns a const reference to the Bezier patch's weights.

◆ boundingBox()

template<typename T , int NDIMS>

BoundingBoxType axom::primal::BezierPatch< T, NDIMS >::boundingBox ( ) const

inline

Returns an axis-aligned bounding box containing the Bezier patch.

References axom::Array< T, DIM, SPACE, StoragePolicy >::data(), and axom::Array< T, DIM, SPACE, StoragePolicy >::size().

◆ orientedBoundingBox()

template<typename T , int NDIMS>

OrientedBoundingBoxType axom::primal::BezierPatch< T, NDIMS >::orientedBoundingBox ( ) const

inline

Returns an oriented bounding box containing the Bezier patch.

References axom::Array< T, DIM, SPACE, StoragePolicy >::data(), and axom::Array< T, DIM, SPACE, StoragePolicy >::size().

◆ isPlanar()

template<typename T , int NDIMS>

bool axom::primal::BezierPatch< T, NDIMS >::isPlanar	(	double	sq_tol = `1e-8`,
		double	EPS = `1e-8`
	)		const

inline

Predicate to check if the Bezier patch is approximately planar (i.e. flat)

This function checks if all control points of the BezierPatch are approximately on the plane defined by its four corners

Parameters

[in]	sq_tol	Threshold for sum of squared distances
[in]	EPS	Threshold for nearness to zero

Returns: True if the object is planar up to tolerance sq_tol

References axom::primal::Vector< T, NDIMS >::cross_product(), axom::primal::Vector< T, NDIMS >::dot(), axom::primal::BezierPatch< T, NDIMS >::getOrder_u(), axom::primal::BezierPatch< T, NDIMS >::getOrder_v(), axom::utilities::isNearlyEqual(), axom::primal::Vector< T, NDIMS >::norm(), axom::primal::Vector< T, NDIMS >::scalar_triple_product(), and axom::primal::Vector< T, NDIMS >::unitVector().

◆ isPolygonal()

template<typename T , int NDIMS>

bool axom::primal::BezierPatch< T, NDIMS >::isPolygonal	(	double	sq_tol = `1e-8`,
		double	EPS = `1e-8`
	)		const

inline

Predicate to check if the patch can be approximated by a polygon.

This function checks if a BezierPatch lies in a plane and that the edges are linear up to tolerance sq_tol

Parameters

[in]	tol	Threshold for sum of squared distances
[in]	EPS	Threshold for nearness to zero

Returns: True if the patch is a planar polygon, up to the desired tolerances

References axom::primal::BezierPatch< T, NDIMS >::getOrder_u(), axom::primal::BezierPatch< T, NDIMS >::getOrder_v(), axom::primal::BezierPatch< T, NDIMS >::isocurve_u(), axom::primal::BezierPatch< T, NDIMS >::isocurve_v(), and axom::primal::BezierPatch< T, NDIMS >::isPlanar().

◆ isBilinear()

template<typename T , int NDIMS>

bool axom::primal::BezierPatch< T, NDIMS >::isBilinear	(	double	sq_tol = `1e-8`,
		bool	useStrictBilinear = `false`
	)		const

inline

Predicate to check if the Bezier patch is approximately bilinear.

This function checks if the patch is (nearly) bilinear. A necessary condition for a geometrically bilinear patch is that each line of control points in the net is approximately linear. A necessary condition for a parametrically bilinear patch is that the control points are coincident with the surface of the bilinear patch defined by its corners evaluated at uniform parameter values, i.e. the control points are also equally spaced on the net.

Parameters

[in]	sq_tol	Threshold for absolute squared distances
[in]	useStrictBilinear	If true, require the patch be parametrically bilinear

Returns: True if patch is bilinear up to tolerance sq_tol

References axom::primal::BezierPatch< T, NDIMS >::getOrder_u(), axom::primal::BezierPatch< T, NDIMS >::getOrder_v(), axom::utilities::lerp(), and axom::primal::squared_distance().

◆ reverseOrientation()

template<typename T , int NDIMS>

void axom::primal::BezierPatch< T, NDIMS >::reverseOrientation ( int axis )

inline

Reverses the order of one direction of the Bezier patch's control points and weights.

Parameters

[in] axis orientation of patch. 0 to reverse in u, 1 for reverse in v

References axom::primal::BezierPatch< T, NDIMS >::reverseOrientation_u(), and axom::primal::BezierPatch< T, NDIMS >::reverseOrientation_v().

◆ reverseOrientation_u()

template<typename T , int NDIMS>

void axom::primal::BezierPatch< T, NDIMS >::reverseOrientation_u ( )

inline

Reverses the order of the Bezier patch's control points and weights on the first axis.

References axom::primal::BezierPatch< T, NDIMS >::getOrder_u(), axom::primal::BezierPatch< T, NDIMS >::getOrder_v(), axom::primal::BezierPatch< T, NDIMS >::isRational(), and axom::utilities::swap().

◆ reverseOrientation_v()

template<typename T , int NDIMS>

void axom::primal::BezierPatch< T, NDIMS >::reverseOrientation_v ( )

inline

Reverses the order of the Bezier patch's control points and weights on the second axis.

References axom::primal::BezierPatch< T, NDIMS >::getOrder_u(), axom::primal::BezierPatch< T, NDIMS >::getOrder_v(), axom::primal::BezierPatch< T, NDIMS >::isRational(), and axom::utilities::swap().

◆ swapAxes()

template<typename T , int NDIMS>

void axom::primal::BezierPatch< T, NDIMS >::swapAxes ( )

inline

Swap the axes such that s(u, v) becomes s(v, u)

References axom::primal::BezierPatch< T, NDIMS >::getOrder_u(), axom::primal::BezierPatch< T, NDIMS >::getOrder_v(), and axom::primal::BezierPatch< T, NDIMS >::isRational().

◆ isocurve()

template<typename T , int NDIMS>

BezierCurveType axom::primal::BezierPatch< T, NDIMS >::isocurve	(	T	uv,
		int	axis
	)		const

inline

Evaluates a slice Bezier patch for a fixed parameter value of u or v.

Parameters

[in]	u	parameter value at which to evaluate the first axis
[in]	v	parameter value at which to evaluate the second axis
[in]	axis	orientation of curve. 0 for fixed u, 1 for fixed v

Returns: p the value of the Bezier patch at (u, v)

Note: We typically evaluate the patch at u or v between 0 and 1

References axom::primal::BezierPatch< T, NDIMS >::isocurve_u(), axom::primal::BezierPatch< T, NDIMS >::isocurve_v(), and SLIC_ASSERT.

◆ isocurve_u()

template<typename T , int NDIMS>

BezierCurveType axom::primal::BezierPatch< T, NDIMS >::isocurve_u ( T u ) const

inline

Returns an isocurve with a fixed value of u.

Parameters

[in] u Parameter value fixed in the isocurve

Returns: c The isocurve C(v) = S(u, v) for fixed u

References axom::utilities::annotations::end(), axom::primal::BezierPatch< T, NDIMS >::getOrder_u(), axom::primal::BezierPatch< T, NDIMS >::getOrder_v(), axom::primal::BezierPatch< T, NDIMS >::isocurve_u(), axom::primal::BezierPatch< T, NDIMS >::isRational(), axom::utilities::lerp(), axom::primal::BezierCurve< T, NDIMS >::makeRational(), and axom::primal::BezierCurve< T, NDIMS >::setWeight().

◆ isocurve_v()

template<typename T , int NDIMS>

BezierCurveType axom::primal::BezierPatch< T, NDIMS >::isocurve_v ( T v ) const

inline

Returns an isocurve with a fixed value of v.

Parameters

[in] v Parameter value fixed in the isocurve

Returns: c The isocurve C(u) = S(u, v) for fixed v

References axom::utilities::annotations::end(), axom::primal::BezierPatch< T, NDIMS >::getOrder_u(), axom::primal::BezierPatch< T, NDIMS >::getOrder_v(), axom::primal::BezierPatch< T, NDIMS >::isocurve_v(), axom::primal::BezierPatch< T, NDIMS >::isRational(), axom::utilities::lerp(), axom::primal::BezierCurve< T, NDIMS >::makeRational(), and axom::primal::BezierCurve< T, NDIMS >::setWeight().

◆ evaluate()

template<typename T , int NDIMS>

PointType axom::primal::BezierPatch< T, NDIMS >::evaluate	(	T	u,
		T	v
	)		const

inline

Evaluates a Bezier patch at a particular parameter value (u, v)

Parameters

[in]	u	parameter value at which to evaluate on the first axis
[in]	v	parameter value at which to evaluate on the second axis

Returns: p the value of the Bezier patch at (u, v)

Note: We typically evaluate the patch at u and v between 0 and 1

References axom::primal::BezierCurve< T, NDIMS >::evaluate(), axom::primal::BezierPatch< T, NDIMS >::getOrder_u(), axom::primal::BezierPatch< T, NDIMS >::getOrder_v(), axom::primal::BezierPatch< T, NDIMS >::isocurve_u(), and axom::primal::BezierPatch< T, NDIMS >::isocurve_v().

◆ evaluateFirstDerivatives()

template<typename T , int NDIMS>

void axom::primal::BezierPatch< T, NDIMS >::evaluateFirstDerivatives	(	T	u,
		T	v,
		Point< T, NDIMS > &	eval,
		Vector< T, NDIMS > &	Du,
		Vector< T, NDIMS > &	Dv
	)		const

inline

Evaluates all first derivatives Bezier patch at (u, v)

Parameters

[in]	u	Parameter value at which to evaluate on the first axis
[in]	v	Parameter value at which to evaluate on the second axis
[out]	eval	The point value of the Bezier patch at (u, v)
[out]	Du	The vector value of S_u(u, v)
[out]	Dv	The vector value of S_v(u, v)

Note: We typically evaluate the patch at u and v between 0 and 1

References axom::primal::BezierPatch< T, NDIMS >::evaluateSecondDerivatives(), axom::primal::BezierPatch< T, NDIMS >::getOrder_u(), axom::primal::BezierPatch< T, NDIMS >::getOrder_v(), axom::primal::BezierPatch< T, NDIMS >::isRational(), and axom::utilities::lerp().

◆ evaluate_linear_derivatives()

template<typename T , int NDIMS>

void axom::primal::BezierPatch< T, NDIMS >::evaluate_linear_derivatives	(	T	u,
		T	v,
		Point< T, NDIMS > &	eval,
		Vector< T, NDIMS > &	Du,
		Vector< T, NDIMS > &	Dv,
		Vector< T, NDIMS > &	DuDv
	)		const

inline

Evaluates all linear derivatives Bezier patch at (u, v)

Parameters

[in]	u	Parameter value at which to evaluate on the first axis
[in]	v	Parameter value at which to evaluate on the second axis
[out]	eval	The point value of the Bezier patch at (u, v)
[out]	Du	The vector value of S_u(u, v)
[out]	Dv	The vector value of S_v(u, v)
[out]	DuDv	The vector value of S_uv(u, v) == S_vu(u, v)

Note: We typically evaluate the patch at u and v between 0 and 1

References axom::primal::BezierPatch< T, NDIMS >::evaluateSecondDerivatives(), axom::primal::BezierPatch< T, NDIMS >::getOrder_u(), axom::primal::BezierPatch< T, NDIMS >::getOrder_v(), axom::primal::BezierPatch< T, NDIMS >::isRational(), and axom::utilities::lerp().

◆ evaluateSecondDerivatives()

template<typename T , int NDIMS>

void axom::primal::BezierPatch< T, NDIMS >::evaluateSecondDerivatives	(	T	u,
		T	v,
		Point< T, NDIMS > &	eval,
		Vector< T, NDIMS > &	Du,
		Vector< T, NDIMS > &	Dv,
		Vector< T, NDIMS > &	DuDu,
		Vector< T, NDIMS > &	DvDv,
		Vector< T, NDIMS > &	DuDv
	)		const

inline

Evaluates all second derivatives Bezier patch at (u, v)

Parameters

[in]	u	Parameter value at which to evaluate on the first axis
[in]	v	Parameter value at which to evaluate on the second axis
[out]	eval	The point value of the Bezier patch at (u, v)
[out]	Du	The vector value of S_u(u, v)
[out]	Dv	The vector value of S_v(u, v)
[out]	DuDu	The vector value of S_uu(u, v)
[out]	DvDv	The vector value of S_vv(u, v)
[out]	DuDv	The vector value of S_uv(u, v) == S_vu(u, v)

Note: We typically evaluate the patch at u and v between 0 and 1

References axom::primal::BezierPatch< T, NDIMS >::evaluateSecondDerivatives(), axom::primal::BezierPatch< T, NDIMS >::getOrder_u(), axom::primal::BezierPatch< T, NDIMS >::getOrder_v(), axom::primal::BezierPatch< T, NDIMS >::isRational(), and axom::utilities::lerp().

◆ du()

template<typename T , int NDIMS>

VectorType axom::primal::BezierPatch< T, NDIMS >::du	(	T	u,
		T	v
	)		const

inline

Computes a tangent of a Bezier patch at a particular parameter value (u, v) along the u axis.

Parameters

[in]	u	parameter value at which to evaluate on the first axis
[in]	v	parameter value at which to evaluate on the second axis

Returns: vec a tangent vector in u of the Bezier patch at (u, v)

Note: We typically find the tangent of the patch at u and v between 0 and 1

References axom::primal::BezierCurve< T, NDIMS >::dt(), and axom::primal::BezierPatch< T, NDIMS >::isocurve_v().

◆ dudu()

template<typename T , int NDIMS>

VectorType axom::primal::BezierPatch< T, NDIMS >::dudu	(	T	u,
		T	v
	)		const

inline

Computes the second derivative of a Bezier patch at (u, v) along the u axis.

Parameters

[in]	u	Parameter value at which to evaluate on the first axis
[in]	v	Parameter value at which to evaluate on the second axis

Returns: vec The vector value of S_uu(u, v)

Note: We typically find the tangent of the patch at u and v between 0 and 1

References axom::primal::BezierCurve< T, NDIMS >::dtdt(), and axom::primal::BezierPatch< T, NDIMS >::isocurve_v().

◆ dv()

template<typename T , int NDIMS>

VectorType axom::primal::BezierPatch< T, NDIMS >::dv	(	T	u,
		T	v
	)		const

inline

Computes a tangent of a Bezier patch at a particular parameter value (u, v) along the v axis.

Parameters

[in]	u	parameter value at which to evaluate on the first axis
[in]	v	parameter value at which to evaluate on the second axis

Returns: vec a tangent vector in v of the Bezier patch at (u, v)

Note: We typically find the tangent of the patch at u and v between 0 and 1

References axom::primal::BezierCurve< T, NDIMS >::dt(), and axom::primal::BezierPatch< T, NDIMS >::isocurve_u().

◆ dvdv()

template<typename T , int NDIMS>

VectorType axom::primal::BezierPatch< T, NDIMS >::dvdv	(	T	u,
		T	v
	)		const

inline

Computes the second derivative of a Bezier patch at (u, v) along the v axis.

Parameters

[in]	u	Parameter value at which to evaluate on the first axis
[in]	v	Parameter value at which to evaluate on the second axis

Returns: vec The vector value of S_vv(u, v)

Note: We typically find the tangent of the patch at u and v between 0 and 1

References axom::primal::BezierCurve< T, NDIMS >::dtdt(), and axom::primal::BezierPatch< T, NDIMS >::isocurve_u().

◆ dudv()

template<typename T , int NDIMS>

VectorType axom::primal::BezierPatch< T, NDIMS >::dudv	(	T	u,
		T	v
	)		const

inline

Computes the mixed second derivative of a Bezier patch at (u, v)

Parameters

[in]	u	Parameter value at which to evaluate on the first axis
[in]	v	Parameter value at which to evaluate on the second axis

Returns: vec The vector value of S_uv(u, v) == S_vu(u, v)

Note: We typically find the tangent of the patch at u and v between 0 and 1

References axom::primal::BezierPatch< T, NDIMS >::evaluate_linear_derivatives(), axom::primal::BezierPatch< T, NDIMS >::getOrder_u(), axom::primal::BezierPatch< T, NDIMS >::getOrder_v(), axom::primal::BezierPatch< T, NDIMS >::isRational(), and axom::utilities::lerp().

◆ dvdu()

template<typename T , int NDIMS>

VectorType axom::primal::BezierPatch< T, NDIMS >::dvdu	(	T	u,
		T	v
	)		const

inline

Computes the mixed second derivative of a Bezier patch at (u, v)

Parameters

[in]	u	Parameter value at which to evaluate on the first axis
[in]	v	Parameter value at which to evaluate on the second axis

Returns: vec The vector value of S_uv(u, v) == S_vu(u, v)

Note: We typically find the tangent of the patch at u and v between 0 and 1

References axom::primal::BezierPatch< T, NDIMS >::dudv().

◆ normal()

template<typename T , int NDIMS>

VectorType axom::primal::BezierPatch< T, NDIMS >::normal	(	T	u,
		T	v
	)		const

inline

Computes the normal vector of a Bezier patch at a particular parameter value (u, v)

Parameters

[in]	u	parameter value at which to evaluate on the first axis
[in]	v	parameter value at which to evaluate on the second axis

Returns: vec the normal vector of the Bezier patch at (u, v)

Note: We typically find the normal of the patch at u and v between 0 and 1

References axom::primal::Vector< T, NDIMS >::cross_product(), and axom::primal::BezierPatch< T, NDIMS >::evaluateFirstDerivatives().

◆ split() [1/2]

template<typename T , int NDIMS>

void axom::primal::BezierPatch< T, NDIMS >::split	(	T	uv,
		int	axis,
		BezierPatch< T, NDIMS > &	p1,
		BezierPatch< T, NDIMS > &	p2
	)		const

inline

Splits a Bezier patch into two Bezier patches.

Parameters

[in]	uv	parameter value between 0 and 1 at which to bisect the patch
[in]	axis	orientation of split. 0 for fixed u, 1 for fixed v
[out]	p1	First output Bezier patch
[out]	p2	Second output Bezier patch

Precondition: Parameter uv must be between 0 and 1

References SLIC_ASSERT, axom::primal::BezierPatch< T, NDIMS >::split_u(), and axom::primal::BezierPatch< T, NDIMS >::split_v().

◆ split_u()

template<typename T , int NDIMS>

void axom::primal::BezierPatch< T, NDIMS >::split_u	(	T	u,
		BezierPatch< T, NDIMS > &	p1,
		BezierPatch< T, NDIMS > &	p2
	)		const

inline

Split the patch along a fixed value of u.

References axom::utilities::annotations::end(), axom::primal::BezierPatch< T, NDIMS >::getOrder_u(), axom::primal::BezierPatch< T, NDIMS >::getOrder_v(), axom::primal::BezierPatch< T, NDIMS >::getWeight(), axom::primal::BezierPatch< T, NDIMS >::isRational(), axom::utilities::lerp(), axom::primal::BezierPatch< T, NDIMS >::makeRational(), axom::primal::BezierPatch< T, NDIMS >::setOrder(), and axom::primal::BezierPatch< T, NDIMS >::setWeight().

◆ split_v()

template<typename T , int NDIMS>

void axom::primal::BezierPatch< T, NDIMS >::split_v	(	T	v,
		BezierPatch< T, NDIMS > &	p1,
		BezierPatch< T, NDIMS > &	p2
	)		const

inline

References axom::utilities::annotations::end(), axom::primal::BezierPatch< T, NDIMS >::getOrder_u(), axom::primal::BezierPatch< T, NDIMS >::getOrder_v(), axom::primal::BezierPatch< T, NDIMS >::getWeight(), axom::primal::BezierPatch< T, NDIMS >::isRational(), axom::utilities::lerp(), axom::primal::BezierPatch< T, NDIMS >::makeRational(), axom::primal::BezierPatch< T, NDIMS >::setOrder(), and axom::primal::BezierPatch< T, NDIMS >::setWeight().

◆ split() [2/2]

template<typename T , int NDIMS>

void axom::primal::BezierPatch< T, NDIMS >::split	(	T	u,
		T	v,
		BezierPatch< T, NDIMS > &	p1,
		BezierPatch< T, NDIMS > &	p2,
		BezierPatch< T, NDIMS > &	p3,
		BezierPatch< T, NDIMS > &	p4
	)		const

inline

Splits a Bezier patch into four Bezier patches.

Parameters

[in]	u	parameter value between 0 and 1 at which to bisect on the first axis
[in]	v	parameter value between 0 and 1 at which to bisect on the second axis
[out]	p1	First output Bezier patch
[out]	p2	Second output Bezier patch
[out]	p3	Third output Bezier patch
[out]	p4	Fourth output Bezier patch

v = 1

| | | | p3 | p4 | | | | -----—(u,v)------— | | | | p1 | p2 | | | | -------------------— u = 1

Precondition: Parameter u and v must be between 0 and 1

References axom::primal::BezierPatch< T, NDIMS >::split_u(), and axom::primal::BezierPatch< T, NDIMS >::split_v().

◆ print()

template<typename T , int NDIMS>

std::ostream& axom::primal::BezierPatch< T, NDIMS >::print ( std::ostream & os ) const

inline

Simple formatted print of a Bezier Patch instance.

Parameters

os	The output stream to write to

Returns: A reference to the modified ostream

References axom::primal::BezierPatch< T, NDIMS >::getOrder_u(), axom::primal::BezierPatch< T, NDIMS >::getOrder_v(), and axom::primal::BezierPatch< T, NDIMS >::isRational().

Friends And Related Function Documentation

◆ operator==

template<typename T , int NDIMS>

bool operator==	(	const BezierPatch< T, NDIMS > &	lhs,
		const BezierPatch< T, NDIMS > &	rhs
	)

friend

Checks equality of two Bezier Patches.

◆ operator!=

template<typename T , int NDIMS>

bool operator!=	(	const BezierPatch< T, NDIMS > &	lhs,
		const BezierPatch< T, NDIMS > &	rhs
	)

friend

The documentation for this class was generated from the following file:

/home/docs/checkouts/readthedocs.org/user_builds/axom/checkouts/main/src/axom/primal/geometry/BezierPatch.hpp

Public Types

Public Member Functions

Friends

Detailed Description

template<typename T, int NDIMS> class axom::primal::BezierPatch< T, NDIMS >

Member Typedef Documentation

◆ PointType

◆ VectorType

◆ PlaneType

◆ CoordsVec

◆ CoordsMat

◆ WeightsVec

◆ WeightsMat

◆ BoundingBoxType

◆ OrientedBoundingBoxType

◆ BezierCurveType

Constructor & Destructor Documentation

◆ BezierPatch() [1/10]

◆ BezierPatch() [2/10]

◆ BezierPatch() [3/10]

◆ BezierPatch() [4/10]

◆ BezierPatch() [5/10]

◆ BezierPatch() [6/10]

◆ BezierPatch() [7/10]

◆ BezierPatch() [8/10]

◆ BezierPatch() [9/10]

◆ BezierPatch() [10/10]

Member Function Documentation

◆ AXOM_STATIC_ASSERT_MSG() [1/2]

◆ AXOM_STATIC_ASSERT_MSG() [2/2]

◆ setOrder()

◆ getOrder_u()

◆ getOrder_v()

◆ clear()

◆ isRational()

◆ makeRational()

◆ makeNonrational()

◆ operator()() [1/2]

◆ operator()() [2/2]

◆ getControlPoints() [1/2]

◆ getControlPoints() [2/2]

◆ getWeight()

◆ setWeight()

◆ getWeights() [1/2]

◆ getWeights() [2/2]

◆ boundingBox()

◆ orientedBoundingBox()

◆ isPlanar()

◆ isPolygonal()

◆ isBilinear()

◆ reverseOrientation()

◆ reverseOrientation_u()

◆ reverseOrientation_v()

◆ swapAxes()

◆ isocurve()

◆ isocurve_u()

◆ isocurve_v()

◆ evaluate()

◆ evaluateFirstDerivatives()

◆ evaluate_linear_derivatives()

◆ evaluateSecondDerivatives()

◆ du()

◆ dudu()

◆ dv()

◆ dvdv()

◆ dudv()

◆ dvdu()

◆ normal()

◆ split() [1/2]

◆ split_u()

◆ split_v()

◆ split() [2/2]

v = 1

◆ print()

Friends And Related Function Documentation

◆ operator==

◆ operator!=

template<typename T, int NDIMS>
class axom::primal::BezierPatch< T, NDIMS >