Represents a Bezier triangle defined by a triangular array of control points. More...

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

Public Types
using	PointType = Point< T, NDIMS >

using	Barycentric = Point< T, 3 >

using	VectorType = Vector< T, NDIMS >

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

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

using	BoundingBoxType = BoundingBox< T, NDIMS >

using	OrientedBoundingBoxType = OrientedBoundingBox< T, NDIMS >

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

using	EdgesVec = axom::Array< BezierCurveType >

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

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

bool	isRational () const
	Returns true when this triangle is rational. More...

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

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

CoordsVec &	getControlPoints ()
	Returns a reference to the triangle's control points. More...

const CoordsVec &	getControlPoints () const
	Returns a reference to the triangle's control points. More...

WeightsVec &	getWeights ()
	Returns a reference to the triangle's weights. More...

const WeightsVec &	getWeights () const
	Returns a reference to the triangle's weights. More...

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

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

void	setOrder (int ord)
	Sets the order of the Bezier triangle and resizes internal storage. More...

int	getOrder () const
	Returns the polynomial order of the triangle. More...

PointType	getVertex (int vertIdx) const
	Return one vertex from the Bezier triangle. More...

BezierCurveType	getEdge (int edgeIdx) const
	Returns a copy of one of the boundary edges of the Bezier triangle. More...

EdgesVec	getEdges () const
	Returns all three boundary edges of the Bezier triangle. More...

void	restrictToSubtriangle (const Barycentric &Qa, const Barycentric &Qb, const Barycentric &Qc, BezierTriangle &out) const
	Restricts this Bezier triangle to a subtriangle of the parameter domain. More...

void	split (T u0, T v0, BezierTriangle &t0, BezierTriangle &t1, BezierTriangle &t2) const
	Splits a Bezier triangle into three subtriangles by connecting an interior parameter point (u0,v0) to the triangle's three vertices. More...

void	split (int edgeIdx, T s, BezierTriangle &t0, BezierTriangle &t1) const
	Splits a Bezier triangle into two subtriangles by connecting a point on a boundary edge to the opposite vertex. More...

void	uniformSplit (BezierTriangle &t0, BezierTriangle &t1, BezierTriangle &t2, BezierTriangle &t3) const
	Uniform 4-way split at edge midpoints. More...

void	split (T s1, T s2, T s3, BezierTriangle &t1, BezierTriangle &t2, BezierTriangle &t3, BezierTriangle &t4) const
	Splits a Bezier triangle into four subtriangles by inserting one split point on each boundary edge and connecting the split points pairwise. More...

PointType &	operator() (int i, int j)
	Access a control point in the triangular control net. More...

const PointType &	operator() (int i, int j) const
	Access a control point in the triangular control net. More...

const T &	getWeight (int i, int j) const
	Get a specific weight from a rational Bezier triangle. More...

void	setWeight (int i, int j, T weight)
	Set the weight at a specific index for a rational Bezier triangle. More...

PointType	evaluate (T u0, T v0) const
	Evaluates the Bezier triangle at (u0,v0) More...

void	evaluateFirstDerivatives (T u0, T v0, Point< T, NDIMS > &eval, Vector< T, NDIMS > &Du, Vector< T, NDIMS > &Dv) const
	Evaluates first derivatives of the Bezier triangle at (u0,v0) More...

void	evaluateLinearDerivatives (T u0, T v0, Point< T, NDIMS > &eval, Vector< T, NDIMS > &Du, Vector< T, NDIMS > &Dv, Vector< T, NDIMS > &DuDv) const
	Evaluates all linear derivatives of a Bezier triangle at (u, v) More...

void	evaluateSecondDerivatives (T u0, T v0, 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 of a Bezier triangle at (u0, v0) More...

VectorType	du (T u0, T v0) const
	Computes a tangent of a Bezier triangle at (u0, v0) along the u axis. More...

VectorType	dv (T u0, T v0) const
	Computes a tangent of a Bezier triangle at (u0, v0) along the v axis. More...

VectorType	dudu (T u0, T v0) const
	Computes the second derivative of a Bezier triangle at (u0, v0) along the u axis. More...

VectorType	dvdv (T u0, T v0) const
	Computes the second derivative of a Bezier triangle at (u0, v0) along the v axis. More...

VectorType	dudv (T u0, T v0) const
	Computes the mixed second derivative of a Bezier triangle at (u0, v0) More...

VectorType	dvdu (T u0, T v0) const
	Convenience alias for S_vu(u0,v0), which equals S_uv(u0,v0) for polynomial triangles. More...

VectorType	normal (T u0, T v0) const
	Computes the normal vector of a Bezier triangle at (u0, v0) More...

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

Constructors for BezierTriangle
The constructors allow for flexible initialization of BezierTriangle objects from: 1D Axom arrays/views of control points and weights, C-style arrays of control points and weights, a specified polynomial order, rational or polynomial (nonrational) triangles, depending on the presence of weights. The triangle is parametrized over the domain \( u \ge 0, v \ge 0, u+v \le 1 \). Rational triangles are identified by a non-empty weights array, and nonrational triangles by an empty weights array. All weights must be greater than 0 in a rational triangle. For 1D control point/weight arrays, the expected layout corresponds to the indexing used by `operator()(i,j)`: pts[ triIndex(N,i,j) ] <-> (*this)(i,j) for 0<=i<=N and 0<=j<=N-i
	BezierTriangle (axom::ArrayView< const PointType > controlPoints, axom::ArrayView< const T > weights, int ord)
	Constructor from ArrayViews of control points and weights. More...

	BezierTriangle (axom::ArrayView< PointType > controlPoints, axom::ArrayView< T > weights, int ord)
	Constructor from ArrayViews of (non-const) control points and weights. More...

	BezierTriangle (int ord=-1)
	Constructor for a polynomial (nonrational) Bezier Triangle that reserves space. More...

	BezierTriangle (const PointType *pts, int ord)
	Constructor for a polynomial Bezier Triangle from an array of coordinates. More...

	BezierTriangle (const PointType pts, const T weights, int ord)
	Constructor for a rational Bezier Triangle from arrays of coordinates and weights. More...

	BezierTriangle (const CoordsVec &pts, int ord)
	Constructor from an Axom array of control points. More...

	BezierTriangle (const CoordsVec &pts, const WeightsVec &weights, int ord)
	Constructor from Axom arrays of control points and weights. More...

Static Public Member Functions
static constexpr int	triSize (int ord)
	Returns the number of control points for a triangle of order ord. More...

static constexpr size_t	triIndex (int ord, int i, int j)
	Maps triangular indices (i,j) to the linear storage index. More...

static constexpr bool	isValidIndex (int ord, int i, int j)
	Check if a given index is valid in the triangular array. More...

static constexpr PointType	triInterpolate (const PointType &A, const PointType &B, const PointType &C, const Barycentric &Q)
	Do a triangular interpolation from three points and a barycentric coordinate. More...

static constexpr T	triInterpolate (const T &A, const T &B, const T &C, const Barycentric &Q)
	Do a triangular interpolation from three coordinates and a barycentric coordinate. More...

Friends
class	BezierTriangle< T, 1 >

class	BezierTriangle< T, 2 >

class	BezierTriangle< T, 3 >

Detailed Description

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

Represents a Bezier triangle defined by a triangular array of control points.

Template Parameters

T	the coordinate type, e.g., double, float, etc.
NDIMS	The dimension of each control point, e.g. 1 for rational weights

A Bezier triangle of order N has \( (N+1)(N+2)/2 \) control points. It is parametrized over the domain \( u0 \ge 0, v0 \ge 0, u0+v0 \le 1 \).

Control points are indexed using integer coordinates \( (i,j) \) with \( 0 \le i \le N \) and \( 0 \le j \le N-i \) and accessed via operator()(i,j). Internally, the triangular control net is stored in a 1D array and triIndex(N,i,j) maps \( (i,j) \) to that linear storage index.

Rational triangles are represented by an additional set of positive weights. Polynomial (nonrational) Bezier triangles are identified by an empty weights array.

A default-constructed triangle will have order -1, and is "invalid". Arrays of nodes and weights will be empty, and most methods are invalid

Note

This triangle uses permuted barycentric coordinates (u0, v0) for evaluation such that, when getOrder()==1, the parameter values correspond to the triangle vertices:

evaluate(0,0) == (*this)(0,0)
evaluate(0,1) == (*this)(0,1)
evaluate(1,0) == (*this)(1,0)

These are mapped to standard Barycentric coordinates {u,v,w} through (u0, v0) = {1 - u0 - v0, v0, u0}:

Parametric (u0, v0): Barycentric {u,v,w}: (1, 0) {0,0,1} /\ /\ / \ / \ ^ / \ <—> / \ | / \ / \ | / \ / \ v0 /__________\ /__________\ (0, 0) u0 —> (0, 1) {1,0,0} {0,1,0}

Member Typedef Documentation

◆ PointType

template<typename T , int NDIMS>

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

◆ Barycentric

template<typename T , int NDIMS>

using axom::primal::BezierTriangle< T, NDIMS >::Barycentric = Point<T, 3>

◆ VectorType

template<typename T , int NDIMS>

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

◆ CoordsVec

template<typename T , int NDIMS>

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

◆ WeightsVec

template<typename T , int NDIMS>

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

◆ BoundingBoxType

template<typename T , int NDIMS>

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

◆ OrientedBoundingBoxType

template<typename T , int NDIMS>

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

◆ BezierCurveType

template<typename T , int NDIMS>

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

◆ EdgesVec

template<typename T , int NDIMS>

using axom::primal::BezierTriangle< T, NDIMS >::EdgesVec = axom::Array<BezierCurveType>

Constructor & Destructor Documentation

◆ BezierTriangle() [1/7]

template<typename T , int NDIMS>

axom::primal::BezierTriangle< T, NDIMS >::BezierTriangle	(	axom::ArrayView< const PointType >	controlPoints,
		axom::ArrayView< const T >	weights,
		int	ord
	)

inline

Constructor from ArrayViews of control points and weights.

Parameters

[in]	controlPoints	ArrayView of control points (size: (ord+1)(ord+2)/2 or 0)
[in]	weights	ArrayView of weights (size: (ord+1)(ord+2)/2 or 0)
[in]	ord	The triangle's polynomial order

If controlPoints is empty, we still allocate space for the control points.

Precondition: ord must be greater than or equal to -1

References axom::ArrayView< T, DIM, SPACE >::data(), axom::ArrayView< T, DIM, SPACE >::empty(), axom::Array< T, DIM, SPACE, StoragePolicy >::resize(), axom::ArrayView< T, DIM, SPACE >::size(), SLIC_ASSERT, and axom::primal::BezierTriangle< T, NDIMS >::triSize().

◆ BezierTriangle() [2/7]

template<typename T , int NDIMS>

axom::primal::BezierTriangle< T, NDIMS >::BezierTriangle	(	axom::ArrayView< PointType >	controlPoints,
		axom::ArrayView< T >	weights,
		int	ord
	)

inline

Constructor from ArrayViews of (non-const) control points and weights.

◆ BezierTriangle() [3/7]

template<typename T , int NDIMS>

axom::primal::BezierTriangle< T, NDIMS >::BezierTriangle ( int ord = -1 )

inlineexplicit

Constructor for a polynomial (nonrational) Bezier Triangle that reserves space.

Parameters

[in] ord The triangle's polynomial order

Precondition: ord must be greater than or equal to -1

◆ BezierTriangle() [4/7]

template<typename T , int NDIMS>

axom::primal::BezierTriangle< T, NDIMS >::BezierTriangle	(	const PointType *	pts,
		int	ord
	)

inline

Constructor for a polynomial Bezier Triangle from an array of coordinates.

Parameters

[in]	pts	A 1D C-style array of (ord+1)(ord+2)/2 control points
[in]	ord	The triangle's polynomial order

Precondition: ord is greater than or equal to zero

◆ BezierTriangle() [5/7]

template<typename T , int NDIMS>

axom::primal::BezierTriangle< T, NDIMS >::BezierTriangle	(	const PointType *	pts,
		const T *	weights,
		int	ord
	)

inline

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

Parameters

[in]	pts	A 1D C-style array of (ord+1)(ord+2)/2 control points
[in]	weights	A 1D C-style array of (ord+1)(ord+2)/2 positive weights
[in]	ord	The triangle's polynomial order

Precondition: ord is greater than or equal to zero

If weights is the null pointer, creates a nonrational triangle.

◆ BezierTriangle() [6/7]

template<typename T , int NDIMS>

axom::primal::BezierTriangle< T, NDIMS >::BezierTriangle	(	const CoordsVec &	pts,
		int	ord
	)

inline

Constructor from an Axom array of control points.

Parameters

[in]	pts	A 1D Axom array of (ord+1)(ord+2)/2 control points
[in]	ord	The triangle's polynomial order (>= 0)

◆ BezierTriangle() [7/7]

template<typename T , int NDIMS>

axom::primal::BezierTriangle< T, NDIMS >::BezierTriangle	(	const CoordsVec &	pts,
		const WeightsVec &	weights,
		int	ord
	)

inline

Constructor from Axom arrays of control points and weights.

Parameters

[in]	pts	A 1D Axom array of (ord+1)(ord+2)/2 control points
[in]	weights	A 1D Axom array of (ord+1)(ord+2)/2 positive weights
[in]	ord	The triangle's polynomial order (>= 0)

Member Function Documentation

◆ AXOM_STATIC_ASSERT_MSG() [1/2]

template<typename T , int NDIMS>

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

◆ AXOM_STATIC_ASSERT_MSG() [2/2]

template<typename T , int NDIMS>

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

◆ isRational()

template<typename T , int NDIMS>

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

inline

Returns true when this triangle is rational.

A rational triangle has a weight per control point; polynomial (nonrational) triangles are identified by an empty weight array.

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

◆ makeRational()

template<typename T , int NDIMS>

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

inline

Make trivially rational. If already rational, do nothing.

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

◆ makeNonrational()

template<typename T , int NDIMS>

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

inline

Make nonrational by shrinking array of weights.

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

◆ getControlPoints() [1/2]

template<typename T , int NDIMS>

CoordsVec& axom::primal::BezierTriangle< T, NDIMS >::getControlPoints ( )

inline

Returns a reference to the triangle's control points.

The control net contains triSize(getOrder()) points (or 0 when getOrder()<0).

◆ getControlPoints() [2/2]

template<typename T , int NDIMS>

const CoordsVec& axom::primal::BezierTriangle< T, NDIMS >::getControlPoints ( ) const

inline

Returns a reference to the triangle's control points.

◆ getWeights() [1/2]

template<typename T , int NDIMS>

WeightsVec& axom::primal::BezierTriangle< T, NDIMS >::getWeights ( )

inline

Returns a reference to the triangle's weights.

The weight array is empty for polynomial triangles. For rational triangles it contains triSize(getOrder()) positive weights.

◆ getWeights() [2/2]

template<typename T , int NDIMS>

const WeightsVec& axom::primal::BezierTriangle< T, NDIMS >::getWeights ( ) const

inline

Returns a reference to the triangle's weights.

◆ boundingBox()

template<typename T , int NDIMS>

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

inline

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

Note: The returned box is computed from the control points.

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::BezierTriangle< T, NDIMS >::orientedBoundingBox ( ) const

inline

Returns an oriented bounding box containing the Bezier triangle.

Note: The returned box is computed from the control points.

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

◆ setOrder()

template<typename T , int NDIMS>

void axom::primal::BezierTriangle< T, NDIMS >::setOrder ( int ord )

inline

Sets the order of the Bezier triangle and resizes internal storage.

Parameters

[in] ord The polynomial order

Precondition: ord must be greater than or equal to -1

Note: This function only resizes the control point and weight arrays and does not initialize their values. If the triangle is rational (isRational()==true), the weight array is also resized to match the number of control points.

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

◆ getOrder()

template<typename T , int NDIMS>

int axom::primal::BezierTriangle< T, NDIMS >::getOrder ( ) const

inline

Returns the polynomial order of the triangle.

Note: A default constructed triangle has order -1 and no control points.

◆ getVertex()

template<typename T , int NDIMS>

PointType axom::primal::BezierTriangle< T, NDIMS >::getVertex ( int vertIdx ) const

inline

Return one vertex from the Bezier triangle.

Parameters

[in] vertIdx Index of the requested vertex

The vertices are returned in counter-clockwise order with respect to the first control point (*this)(0, 0) == evaluate(0, 0)

Returns: The PointType object at the vertex

References SLIC_ASSERT.

◆ getEdge()

template<typename T , int NDIMS>

BezierCurveType axom::primal::BezierTriangle< T, NDIMS >::getEdge ( int edgeIdx ) const

inline

Returns a copy of one of the boundary edges of the Bezier triangle.

Parameters

[in] edgeIdx Index of the requested edge in [0,2]

The edges are returned in counter-clockwise order with respect to the parameter domain (u,v), with edge 0 across from corner 0 (i.e. evaluate(0,0)):

edgeIdx = 0: u+v = 1 from evaluate(1,0) to evaluate(0,1)
edgeIdx = 1: u = 0 from evaluate(0,1) to evaluate(0,0)
edgeIdx = 2: v = 0 from evaluate(0,0) to evaluate(1,0)

Returns: A copy of the Bezier curve representing the requested edge.

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

◆ getEdges()

template<typename T , int NDIMS>

EdgesVec axom::primal::BezierTriangle< T, NDIMS >::getEdges ( ) const

inline

Returns all three boundary edges of the Bezier triangle.

Returns: An array of three Bezier curves, ordered the same as getEdge(int).

References axom::primal::BezierTriangle< T, NDIMS >::getEdge(), axom::Array< T, DIM, SPACE, StoragePolicy >::push_back(), axom::Array< T, DIM, SPACE, StoragePolicy >::reserve(), and SLIC_ASSERT.

◆ restrictToSubtriangle()

template<typename T , int NDIMS>

void axom::primal::BezierTriangle< T, NDIMS >::restrictToSubtriangle	(	const Barycentric &	Qa,
		const Barycentric &	Qb,
		const Barycentric &	Qc,
		BezierTriangle< T, NDIMS > &	out
	)		const

inline

Restricts this Bezier triangle to a subtriangle of the parameter domain.

See overload returning a BezierTriangle for the vertex mapping convention.

Parameters

[in]	Qa	Barycentric coordinates of the first subtriangle vertex `(u,v,w)`
[in]	Qb	Barycentric coordinates of the second subtriangle vertex `(u,v,w)`
[in]	Qc	Barycentric coordinates of the third subtriangle vertex `(u,v,w)`
[out]	out	Output restricted Bezier triangle

Precondition: getOrder() >= 0

Note: The barycentric inputs Qa, Qb, Qc are standard Barycentric coordiantes related to parameter convention through (u0 = Qc, v0 = Qb)

References axom::primal::BezierTriangle< T, NDIMS >::isRational(), axom::primal::BezierTriangle< T, NDIMS >::makeNonrational(), axom::primal::BezierTriangle< T, NDIMS >::restrictToSubtriangle(), axom::primal::BezierTriangle< T, NDIMS >::setOrder(), SLIC_ASSERT, axom::Array< T, DIM, SPACE, StoragePolicy >::swap(), axom::primal::BezierTriangle< T, NDIMS >::triIndex(), axom::primal::BezierTriangle< T, NDIMS >::triInterpolate(), and axom::primal::BezierTriangle< T, NDIMS >::triSize().

◆ split() [1/3]

template<typename T , int NDIMS>

void axom::primal::BezierTriangle< T, NDIMS >::split	(	T	u0,
		T	v0,
		BezierTriangle< T, NDIMS > &	t0,
		BezierTriangle< T, NDIMS > &	t1,
		BezierTriangle< T, NDIMS > &	t2
	)		const

inline

Splits a Bezier triangle into three subtriangles by connecting an interior parameter point (u0,v0) to the triangle's three vertices.

Parameters

[in]	u0	Parameter value along the u axis for the split point
[in]	v0	Parameter value along the v axis for the split point
[out]	t0	Subtriangle over the parameter triangle with vertices `(0,1)`, `(1,0)`, and `(u0,v0)` (preserves edge 0)
[out]	t1	Subtriangle over the parameter triangle with vertices `(1,0)`, `(0,0)`, and `(u0,v0)` (preserves edge 1)
[out]	t2	Subtriangle over the parameter triangle with vertices `(0,0)`, `(0,1)`, and `(u0,v0)` (preserves edge 2)

Precondition

u0 > 0, v0 > 0, and u0 + v0 < 1

       A
      /|\
     / | \
    /t2|t1\
   /  /Q\  \
  / /  t0 \ \
 //_________\\
B             C

Returns: t0 = Tri(B,C,Q), t1 = Tri(C,A,Q), t2 = Tri(A,B,Q)

References axom::primal::BezierTriangle< T, NDIMS >::isRational(), axom::primal::BezierTriangle< T, NDIMS >::makeNonrational(), axom::Array< T, DIM, SPACE, StoragePolicy >::resize(), axom::primal::BezierTriangle< T, NDIMS >::setOrder(), SLIC_ASSERT, axom::primal::BezierTriangle< T, NDIMS >::split(), axom::primal::BezierTriangle< T, NDIMS >::triIndex(), axom::primal::BezierTriangle< T, NDIMS >::triInterpolate(), and axom::primal::BezierTriangle< T, NDIMS >::triSize().

◆ split() [2/3]

template<typename T , int NDIMS>

void axom::primal::BezierTriangle< T, NDIMS >::split	(	int	edgeIdx,
		T	s,
		BezierTriangle< T, NDIMS > &	t0,
		BezierTriangle< T, NDIMS > &	t1
	)		const

inline

Splits a Bezier triangle into two subtriangles by connecting a point on a boundary edge to the opposite vertex.

Parameters

[in]	edgeIdx	Index of the boundary edge to split (same convention as `getEdge(int)`)
[in]	s	Parameter in (0,1) locating the split point along the chosen edge
[out]	t0	First output subtriangle
[out]	t1	Second output subtriangle

Precondition: edgeIdx is 0, 1, or 2; s is in (0,1)

Taking P0 as the vertex opposite edge edgeIdx:

       P0
      /|\
     / | \
    /  |  \
   /   |   \
  / t0 | t1 \
 /_____|_____\

P1 Q P2 s=0 s s=1

Returns: t0 = Tri( P0, P1, Q ), t1 = Tri( P2, P0, Q )

References axom::primal::BezierTriangle< T, NDIMS >::isRational(), axom::Array< T, DIM, SPACE, StoragePolicy >::resize(), SLIC_ASSERT, axom::primal::BezierTriangle< T, NDIMS >::split(), axom::primal::BezierTriangle< T, NDIMS >::triIndex(), axom::primal::BezierTriangle< T, NDIMS >::triInterpolate(), and axom::primal::BezierTriangle< T, NDIMS >::triSize().

◆ uniformSplit()

template<typename T , int NDIMS>

void axom::primal::BezierTriangle< T, NDIMS >::uniformSplit	(	BezierTriangle< T, NDIMS > &	t0,
		BezierTriangle< T, NDIMS > &	t1,
		BezierTriangle< T, NDIMS > &	t2,
		BezierTriangle< T, NDIMS > &	t3
	)		const

inline

Uniform 4-way split at edge midpoints.

Parameters

[out]	t0	Subtriangle near vertex `(0,0)`
[out]	t1	Subtriangle near vertex `(0,1)`
[out]	t2	Subtriangle near vertex `(1,0)`
[out]	t3	Central subtriangle

This is equivalent to split(0.5, 0.5, 0.5, ...), but with optimizations to reduce redundant computations by sharing intermediate control nets. We also separate the implementation based on triangle rationality to improve performance

      C
      /\
     /t2\
 P1 /____\ P0
   /\ t3 /\
  /t0\  /t1\
 /____\/____\
A     P2      B

Returns: t0 = Tri( A, P2, P1 ), t1 = Tri( B, P0, P2 ), t2 = Tri( C, P1, P0 ), t3 = Tri( P1, P0, P2 )

References axom::primal::BezierTriangle< T, NDIMS >::isRational(), axom::primal::BezierTriangle< T, NDIMS >::makeNonrational(), axom::primal::BezierTriangle< T, NDIMS >::makeRational(), axom::primal::BezierTriangle< T, NDIMS >::setOrder(), SLIC_ASSERT, and axom::primal::BezierTriangle< T, NDIMS >::triIndex().

◆ split() [3/3]

template<typename T , int NDIMS>

void axom::primal::BezierTriangle< T, NDIMS >::split	(	T	s1,
		T	s2,
		T	s3,
		BezierTriangle< T, NDIMS > &	t1,
		BezierTriangle< T, NDIMS > &	t2,
		BezierTriangle< T, NDIMS > &	t3,
		BezierTriangle< T, NDIMS > &	t4
	)		const

inline

Splits a Bezier triangle into four subtriangles by inserting one split point on each boundary edge and connecting the split points pairwise.

Parameters

[in]	s1	Parameter in (0,1) locating the split point on edge 0 (same convention as `getEdge(0)`)
[in]	s2	Parameter in (0,1) locating the split point on edge 1 (same convention as `getEdge(1)`)
[in]	s3	Parameter in (0,1) locating the split point on edge 2 (same convention as `getEdge(2)`)
[out]	t1	Subtriangle near vertex `(0,0)` with vertices `(0,0)`, point on edge 2, point on edge 1
[out]	t2	Subtriangle near vertex `(0,1)` with vertices `(0,1)`, point on edge 0, point on edge 2
[out]	t3	Subtriangle near vertex `(1,0)` with vertices `(1,0)`, point on edge 1, point on edge 0
[out]	t4	Central subtriangle with vertices point on edge 1, point on edge 0, point on edge 2

Precondition: s1, s2, s3 are all in (0,1) C /\ /t3\ P1 /____\ P0 /\ t4 /\ /t1\ /t2\ /____\/____\ A P2 B

Returns: t1 = Tri( A, P2, P1 ), t2 = Tri( B, P0, P2 ), t3 = Tri( C, P1, P0 ), t4 = Tri( P1, P0, P2 )

References axom::primal::BezierTriangle< T, NDIMS >::restrictToSubtriangle(), and SLIC_ASSERT.

◆ operator()() [1/2]

template<typename T , int NDIMS>

PointType& axom::primal::BezierTriangle< T, NDIMS >::operator()	(	int	i,
		int	j
	)

inline

Access a control point in the triangular control net.

Parameters

[in]	i	The first index (0 <= i <= getOrder())
[in]	j	The second index (0 <= j <= getOrder()-i)

Precondition: i and j are in range and i+j <= getOrder()

References axom::primal::BezierTriangle< T, NDIMS >::isValidIndex(), SLIC_ASSERT, and axom::primal::BezierTriangle< T, NDIMS >::triIndex().

◆ operator()() [2/2]

template<typename T , int NDIMS>

const PointType& axom::primal::BezierTriangle< T, NDIMS >::operator()	(	int	i,
		int	j
	)		const

inline

Access a control point in the triangular control net.

References axom::primal::BezierTriangle< T, NDIMS >::isValidIndex(), SLIC_ASSERT, and axom::primal::BezierTriangle< T, NDIMS >::triIndex().

◆ getWeight()

template<typename T , int NDIMS>

const T& axom::primal::BezierTriangle< T, NDIMS >::getWeight	(	int	i,
		int	j
	)		const

inline

Get a specific weight from a rational Bezier triangle.

Parameters

[in]	i	First control net index
[in]	j	Second control net index

Precondition: Requires that the triangle be rational

References axom::primal::BezierTriangle< T, NDIMS >::isRational(), axom::primal::BezierTriangle< T, NDIMS >::isValidIndex(), SLIC_ASSERT, and axom::primal::BezierTriangle< T, NDIMS >::triIndex().

◆ setWeight()

template<typename T , int NDIMS>

void axom::primal::BezierTriangle< T, NDIMS >::setWeight	(	int	i,
		int	j,
		T	weight
	)

inline

Set the weight at a specific index for a rational Bezier triangle.

Parameters

[in]	i	First control net index
[in]	j	Second control net index
[in]	weight	The updated value of the weight

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

References axom::primal::BezierTriangle< T, NDIMS >::isRational(), axom::primal::BezierTriangle< T, NDIMS >::isValidIndex(), SLIC_ASSERT, and axom::primal::BezierTriangle< T, NDIMS >::triIndex().

◆ evaluate()

template<typename T , int NDIMS>

PointType axom::primal::BezierTriangle< T, NDIMS >::evaluate	(	T	u0,
		T	v0
	)		const

inline

Evaluates the Bezier triangle at (u0,v0)

Parameters

[in]	u0	Parameter value along the u axis
[in]	v0	Parameter value along the v axis

Note

Evaluation uses permuted barycentric coordinates such that parameter values (u0, v0) correspond to the triangle vertices:

evaluate(0,0) == (*this)(0,0)
evaluate(0,1) == (*this)(0,order)
evaluate(1,0) == (*this)(order,0)

Warning: Will automatically extrapolate if (u0, v0) are outside the triangular domain (u0 >= 0, v0 >= 0, and u0+v0 <= 1)

Returns: Point value S(u0,v0)

References axom::utilities::annotations::end(), axom::primal::BezierTriangle< T, NDIMS >::evaluate(), axom::primal::BezierTriangle< T, NDIMS >::isRational(), axom::Array< T, DIM, SPACE, StoragePolicy >::size(), SLIC_ASSERT, axom::primal::BezierTriangle< T, NDIMS >::triIndex(), and axom::primal::BezierTriangle< T, NDIMS >::triInterpolate().

◆ evaluateFirstDerivatives()

template<typename T , int NDIMS>

void axom::primal::BezierTriangle< T, NDIMS >::evaluateFirstDerivatives	(	T	u0,
		T	v0,
		Point< T, NDIMS > &	eval,
		Vector< T, NDIMS > &	Du,
		Vector< T, NDIMS > &	Dv
	)		const

inline

Evaluates first derivatives of the Bezier triangle at (u0,v0)

Parameters

[in]	u0	Parameter value along the u axis
[in]	v0	Parameter value along the v axis
[out]	eval	Point value S(u,v)
[out]	Du	First derivative S_u(u,v)
[out]	Dv	First derivative S_v(u,v)

Warning: Will automatically extrapolate if (u0, v0) are outside the triangular domain (u0 >= 0, v0 >= 0, and u0+v0 <= 1)

References axom::utilities::annotations::end(), axom::primal::BezierTriangle< T, NDIMS >::evaluateFirstDerivatives(), axom::primal::BezierTriangle< T, NDIMS >::isRational(), axom::Array< T, DIM, SPACE, StoragePolicy >::size(), SLIC_ASSERT, axom::primal::BezierTriangle< T, NDIMS >::triIndex(), and axom::primal::BezierTriangle< T, NDIMS >::triInterpolate().

◆ evaluateLinearDerivatives()

template<typename T , int NDIMS>

void axom::primal::BezierTriangle< T, NDIMS >::evaluateLinearDerivatives	(	T	u0,
		T	v0,
		Point< T, NDIMS > &	eval,
		Vector< T, NDIMS > &	Du,
		Vector< T, NDIMS > &	Dv,
		Vector< T, NDIMS > &	DuDv
	)		const

inline

Evaluates all linear derivatives of a Bezier triangle at (u, v)

Parameters

[in]	u0	Parameter value at which to evaluate along the u axis
[in]	v0	Parameter value at which to evaluate along the v axis
[out]	eval	The point value of the Bezier triangle 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)

Warning: Will automatically extrapolate if (u0, v0) are outside the triangular domain (u0 >= 0, v0 >= 0, and u0+v0 <= 1)

References axom::utilities::annotations::end(), axom::primal::BezierTriangle< T, NDIMS >::evaluateFirstDerivatives(), axom::primal::BezierTriangle< T, NDIMS >::evaluateLinearDerivatives(), axom::primal::BezierTriangle< T, NDIMS >::isRational(), axom::Array< T, DIM, SPACE, StoragePolicy >::size(), SLIC_ASSERT, axom::primal::BezierTriangle< T, NDIMS >::triIndex(), and axom::primal::BezierTriangle< T, NDIMS >::triInterpolate().

◆ evaluateSecondDerivatives()

template<typename T , int NDIMS>

void axom::primal::BezierTriangle< T, NDIMS >::evaluateSecondDerivatives	(	T	u0,
		T	v0,
		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 of a Bezier triangle at (u0, v0)

Parameters

[in]	u0	Parameter value at which to evaluate along the u axis
[in]	v0	Parameter value at which to evaluate along the v axis
[out]	eval	The point value of the Bezier triangle 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)

Warning: Will automatically extrapolate if (u0, v0) are outside the triangular domain (u0 >= 0, v0 >= 0, and u0+v0 <= 1)

References axom::utilities::annotations::end(), axom::primal::BezierTriangle< T, NDIMS >::evaluateFirstDerivatives(), axom::primal::BezierTriangle< T, NDIMS >::evaluateSecondDerivatives(), axom::primal::BezierTriangle< T, NDIMS >::isRational(), axom::Array< T, DIM, SPACE, StoragePolicy >::size(), SLIC_ASSERT, axom::primal::BezierTriangle< T, NDIMS >::triIndex(), and axom::primal::BezierTriangle< T, NDIMS >::triInterpolate().

◆ du()

template<typename T , int NDIMS>

VectorType axom::primal::BezierTriangle< T, NDIMS >::du	(	T	u0,
		T	v0
	)		const

inline

Computes a tangent of a Bezier triangle at (u0, v0) along the u axis.

References axom::primal::BezierTriangle< T, NDIMS >::evaluateFirstDerivatives().

◆ dv()

template<typename T , int NDIMS>

VectorType axom::primal::BezierTriangle< T, NDIMS >::dv	(	T	u0,
		T	v0
	)		const

inline

Computes a tangent of a Bezier triangle at (u0, v0) along the v axis.

References axom::primal::BezierTriangle< T, NDIMS >::evaluateFirstDerivatives().

◆ dudu()

template<typename T , int NDIMS>

VectorType axom::primal::BezierTriangle< T, NDIMS >::dudu	(	T	u0,
		T	v0
	)		const

inline

Computes the second derivative of a Bezier triangle at (u0, v0) along the u axis.

References axom::primal::BezierTriangle< T, NDIMS >::evaluateSecondDerivatives().

◆ dvdv()

template<typename T , int NDIMS>

VectorType axom::primal::BezierTriangle< T, NDIMS >::dvdv	(	T	u0,
		T	v0
	)		const

inline

Computes the second derivative of a Bezier triangle at (u0, v0) along the v axis.

References axom::primal::BezierTriangle< T, NDIMS >::evaluateSecondDerivatives().

◆ dudv()

template<typename T , int NDIMS>

VectorType axom::primal::BezierTriangle< T, NDIMS >::dudv	(	T	u0,
		T	v0
	)		const

inline

Computes the mixed second derivative of a Bezier triangle at (u0, v0)

References axom::primal::BezierTriangle< T, NDIMS >::evaluateSecondDerivatives().

◆ dvdu()

template<typename T , int NDIMS>

VectorType axom::primal::BezierTriangle< T, NDIMS >::dvdu	(	T	u0,
		T	v0
	)		const

inline

Convenience alias for S_vu(u0,v0), which equals S_uv(u0,v0) for polynomial triangles.

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

◆ normal()

template<typename T , int NDIMS>

VectorType axom::primal::BezierTriangle< T, NDIMS >::normal	(	T	u0,
		T	v0
	)		const

inline

Computes the normal vector of a Bezier triangle at (u0, v0)

Note: Only meaningful for NDIMS==3.

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

◆ print()

template<typename T , int NDIMS>

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

inline

Simple formatted print of a Bezier Triangle instance.

Parameters

os	The output stream to write to

Returns: A reference to the modified ostream

References axom::primal::BezierTriangle< T, NDIMS >::isRational(), and axom::primal::BezierTriangle< T, NDIMS >::triIndex().

◆ triSize()

template<typename T , int NDIMS>

static constexpr int axom::primal::BezierTriangle< T, NDIMS >::triSize ( int ord )

inlinestaticconstexpr

Returns the number of control points for a triangle of order ord.

Parameters

[in] ord Triangle order

◆ triIndex()

template<typename T , int NDIMS>

static constexpr size_t axom::primal::BezierTriangle< T, NDIMS >::triIndex	(	int	ord,
		int	i,
		int	j
	)

inlinestaticconstexpr

Maps triangular indices (i,j) to the linear storage index.

Parameters

[in]	ord	Triangle order
[in]	i	First control net index
[in]	j	Second control net index

Precondition: ord >= 0, i >= 0, j >= 0, and i+j <= ord

◆ isValidIndex()

template<typename T , int NDIMS>

static constexpr bool axom::primal::BezierTriangle< T, NDIMS >::isValidIndex	(	int	ord,
		int	i,
		int	j
	)

inlinestaticconstexpr

Check if a given index is valid in the triangular array.

◆ triInterpolate() [1/2]

template<typename T , int NDIMS>

static constexpr PointType axom::primal::BezierTriangle< T, NDIMS >::triInterpolate	(	const PointType &	A,
		const PointType &	B,
		const PointType &	C,
		const Barycentric &	Q
	)

inlinestaticconstexpr

Do a triangular interpolation from three points and a barycentric coordinate.

References axom::primal::Point< T, NDIMS >::array().

◆ triInterpolate() [2/2]

template<typename T , int NDIMS>

static constexpr T axom::primal::BezierTriangle< T, NDIMS >::triInterpolate	(	const T &	A,
		const T &	B,
		const T &	C,
		const Barycentric &	Q
	)

inlinestaticconstexpr

Do a triangular interpolation from three coordinates and a barycentric coordinate.

Friends And Related Function Documentation

◆ BezierTriangle< T, 1 >

template<typename T , int NDIMS>

friend class BezierTriangle< T, 1 >

friend

◆ BezierTriangle< T, 2 >

template<typename T , int NDIMS>

friend class BezierTriangle< T, 2 >

friend

◆ BezierTriangle< T, 3 >

template<typename T , int NDIMS>

friend class BezierTriangle< T, 3 >

friend

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

/home/docs/checkouts/readthedocs.org/user_builds/axom/checkouts/develop/src/axom/primal/geometry/BezierTriangle.hpp

Public Types

Public Member Functions

Static Public Member Functions

Friends

Detailed Description

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

Member Typedef Documentation

◆ PointType

◆ Barycentric

◆ VectorType

◆ CoordsVec

◆ WeightsVec

◆ BoundingBoxType

◆ OrientedBoundingBoxType

◆ BezierCurveType

◆ EdgesVec

Constructor & Destructor Documentation

◆ BezierTriangle() [1/7]

◆ BezierTriangle() [2/7]

◆ BezierTriangle() [3/7]

◆ BezierTriangle() [4/7]

◆ BezierTriangle() [5/7]

◆ BezierTriangle() [6/7]

◆ BezierTriangle() [7/7]

Member Function Documentation

◆ AXOM_STATIC_ASSERT_MSG() [1/2]

◆ AXOM_STATIC_ASSERT_MSG() [2/2]

◆ isRational()

◆ makeRational()

◆ makeNonrational()

◆ getControlPoints() [1/2]

◆ getControlPoints() [2/2]

◆ getWeights() [1/2]

◆ getWeights() [2/2]

◆ boundingBox()

◆ orientedBoundingBox()

◆ setOrder()

◆ getOrder()

◆ getVertex()

◆ getEdge()

◆ getEdges()

◆ restrictToSubtriangle()

◆ split() [1/3]

◆ split() [2/3]

◆ uniformSplit()

◆ split() [3/3]

◆ operator()() [1/2]

◆ operator()() [2/2]

◆ getWeight()

◆ setWeight()

◆ evaluate()

◆ evaluateFirstDerivatives()

◆ evaluateLinearDerivatives()

◆ evaluateSecondDerivatives()

◆ du()

◆ dv()

◆ dudu()

◆ dvdv()

◆ dudv()

◆ dvdu()

◆ normal()

◆ print()

◆ triSize()

◆ triIndex()

◆ isValidIndex()

◆ triInterpolate() [1/2]

◆ triInterpolate() [2/2]

Friends And Related Function Documentation

◆ BezierTriangle< T, 1 >

◆ BezierTriangle< T, 2 >

◆ BezierTriangle< T, 3 >

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