Detailed Description

template<typename T>
class drake::math::BsplineBasis< T >

Given a set of non-descending breakpoints t₀ ≤ t₁ ≤ ⋅⋅⋅ ≤ tₘ, a B-spline basis of order k is a set of n + 1 (where n = m - k) piecewise polynomials of degree k - 1 defined over those breakpoints.

The elements of this set are called "B-splines". The vector (t₀, t₁, ..., tₘ)' is referred to as the "knot vector" of the basis and its elements are referred to as "knots".

At a breakpoint with multiplicity p (i.e. a breakpoint that appears p times in the knot vector), B-splines are guaranteed to have Cᵏ⁻ᵖ⁻¹ continuity.

A B-spline curve using a B-spline basis B, is a parametric curve mapping parameter values in [tₖ₋₁, tₙ₊₁] to a vector space V. For t ∈ [tₖ₋₁, tₙ₊₁] the value of the curve is given by the linear combination of n + 1 control points, pᵢ ∈ V, with the elements of B evaluated at t.

For more information on B-splines and their uses, see (for example) Patrikalakis et al. [1].

[1] https://web.mit.edu/hyperbook/Patrikalakis-Maekawa-Cho/node15.html

#include <drake/math/bspline_basis.h>

Public Member Functions
	BsplineBasis ()

	BsplineBasis (int order, std::vector< T > knots)
	Constructs a B-spline basis with the specified `order` and `knots`. More...

	BsplineBasis (int order, int num_basis_functions, KnotVectorType type=KnotVectorType::kClampedUniform, const T &initial_parameter_value=0, const T &final_parameter_value=1)
	Constructs a B-spline basis with the specified `order`, `num_basis_functions`, `initial_parameter_value`, `final_parameter_value`, and an auto-generated knot vector of the specified `type`. More...

	BsplineBasis (const BsplineBasis< double > &other)
	Conversion constructor. More...

int	order () const
	The order of this B-spline basis (k in the class description). More...

int	degree () const
	The degree of the piecewise polynomials comprising this B-spline basis (k - 1 in the class description). More...

int	num_basis_functions () const
	The number of basis functions in this B-spline basis (n + 1 in the class description). More...

const std::vector< T > &	knots () const
	The knot vector of this B-spline basis (the vector (t₀, t₁, ..., tₘ)' in the class description). More...

const T &	initial_parameter_value () const
	The minimum allowable parameter value for B-spline curves using this basis (tₖ₋₁ in the class description). More...

const T &	final_parameter_value () const
	The maximum allowable parameter value for B-spline curves using this basis (tₙ₊₁ in the class description). More...

int	FindContainingInterval (const T &parameter_value) const
	For a `parameter_value` = t, the interval that contains it is the pair of knot values [tᵢ, tᵢ₊₁] for the greatest i such that tᵢ ≤ t and tᵢ < final_parameter_value(). More...

std::vector< int >	ComputeActiveBasisFunctionIndices (const std::array< T, 2 > &parameter_interval) const
	Returns the indices of the basis functions which may evaluate to non-zero values for some parameter value in `parameter_interval`; all other basis functions are strictly zero over `parameter_interval`. More...

std::vector< int >	ComputeActiveBasisFunctionIndices (const T &parameter_value) const
	Returns the indices of the basis functions which may evaluate to non-zero values for `parameter_value`; all other basis functions are strictly zero at this point. More...

template<typename T_control_point >
T_control_point	EvaluateCurve (const std::vector< T_control_point > &control_points, const T &parameter_value) const
	Evaluates the B-spline curve defined by `this` and `control_points` at the given `parameter_value`. More...

VectorX< T >	EvaluateLinearInControlPoints (const T &parameter_value) const
	Returns the vector, M, such that. More...

T	EvaluateBasisFunctionI (int i, const T &parameter_value) const
	Returns the value of the `i`-th basis function evaluated at `parameter_value`. More...

boolean< T >	operator== (const BsplineBasis &other) const

boolean< T >	operator!= (const BsplineBasis &other) const

template<typename Archive >
void	Serialize (Archive *a)
	Passes this object to an Archive. More...

Implements CopyConstructible, CopyAssignable, MoveConstructible, MoveAssignable
	BsplineBasis (const BsplineBasis &)=default

BsplineBasis &	operator= (const BsplineBasis &)=default

	BsplineBasis (BsplineBasis &&)=default

BsplineBasis &	operator= (BsplineBasis &&)=default

Constructor & Destructor Documentation

◆ BsplineBasis() [1/6]

BsplineBasis ( const BsplineBasis< T > & )

default

◆ BsplineBasis() [2/6]

BsplineBasis ( BsplineBasis< T > && )

default

◆ BsplineBasis() [3/6]

BsplineBasis ( )

◆ BsplineBasis() [4/6]

BsplineBasis	(	int	order,
		std::vector< T >	knots
	)

Constructs a B-spline basis with the specified order and knots.

Precondition: knots is sorted in non-descending order.

Exceptions

std::exception if knots.size() < 2 * order.

◆ BsplineBasis() [5/6]

BsplineBasis	(	int	order,
		int	num_basis_functions,
		KnotVectorType	type = `KnotVectorType::kClampedUniform`,
		const T &	initial_parameter_value = `0`,
		const T &	final_parameter_value = `1`
	)

Constructs a B-spline basis with the specified order, num_basis_functions, initial_parameter_value, final_parameter_value, and an auto-generated knot vector of the specified type.

Exceptions

std::exception if num_basis_functions < order

Precondition: initial_parameter_value ≤ final_parameter_value

◆ BsplineBasis() [6/6]

BsplineBasis ( const BsplineBasis< double > & other )

explicit

Conversion constructor.

Constructs an instance of BsplineBasis<T> from a double-valued basis.

Member Function Documentation

◆ ComputeActiveBasisFunctionIndices() [1/2]

std::vector<int> ComputeActiveBasisFunctionIndices ( const std::array< T, 2 > & parameter_interval ) const

Returns the indices of the basis functions which may evaluate to non-zero values for some parameter value in parameter_interval; all other basis functions are strictly zero over parameter_interval.

Precondition: parameter_interval[0] ≤ parameter_interval[1]; parameter_interval[0] ≥ initial_parameter_value(); parameter_interval[1] ≤ final_parameter_value()

◆ ComputeActiveBasisFunctionIndices() [2/2]

std::vector<int> ComputeActiveBasisFunctionIndices ( const T & parameter_value ) const

Returns the indices of the basis functions which may evaluate to non-zero values for parameter_value; all other basis functions are strictly zero at this point.

Precondition: parameter_value ≥ initial_parameter_value(); parameter_value ≤ final_parameter_value()

◆ degree()

int degree ( ) const

The degree of the piecewise polynomials comprising this B-spline basis (k - 1 in the class description).

◆ EvaluateBasisFunctionI()

T EvaluateBasisFunctionI	(	int	i,
		const T &	parameter_value
	)		const

Returns the value of the i-th basis function evaluated at parameter_value.

◆ EvaluateCurve()

T_control_point EvaluateCurve	(	const std::vector< T_control_point > &	control_points,
		const T &	parameter_value
	)		const

Evaluates the B-spline curve defined by this and control_points at the given parameter_value.

Parameters

control_points	Control points of the B-spline curve.
parameter_value	Parameter value at which to evaluate the B-spline curve defined by `this` and `control_points`.

Precondition: control_points.size() == num_basis_functions(); parameter_value ≥ initial_parameter_value(); parameter_value ≤ final_parameter_value()

◆ EvaluateLinearInControlPoints()

VectorX<T> EvaluateLinearInControlPoints ( const T & parameter_value ) const

Returns the vector, M, such that.

EvaluateCurve(control_points, parameter_value) = control_points * M

where T_control_points==VectorX<T> (so control_points is a matrix). This is useful for writing linear constraints on the control points.

Precondition: parameter_value ≥ initial_parameter_value(); parameter_value ≤ final_parameter_value()

◆ final_parameter_value()

const T& final_parameter_value ( ) const

The maximum allowable parameter value for B-spline curves using this basis (tₙ₊₁ in the class description).

◆ FindContainingInterval()

int FindContainingInterval ( const T & parameter_value ) const

For a parameter_value = t, the interval that contains it is the pair of knot values [tᵢ, tᵢ₊₁] for the greatest i such that tᵢ ≤ t and tᵢ < final_parameter_value().

This function returns that value of i.

Precondition: parameter_value ≥ initial_parameter_value(); parameter_value ≤ final_parameter_value()

◆ initial_parameter_value()

const T& initial_parameter_value ( ) const

The minimum allowable parameter value for B-spline curves using this basis (tₖ₋₁ in the class description).

◆ knots()

const std::vector<T>& knots ( ) const

The knot vector of this B-spline basis (the vector (t₀, t₁, ..., tₘ)' in the class description).

◆ num_basis_functions()

int num_basis_functions ( ) const

The number of basis functions in this B-spline basis (n + 1 in the class description).

◆ operator!=()

boolean<T> operator!= ( const BsplineBasis< T > & other ) const

◆ operator=() [1/2]

BsplineBasis& operator= ( BsplineBasis< T > && )

default

◆ operator=() [2/2]

BsplineBasis& operator= ( const BsplineBasis< T > & )

default

◆ operator==()

boolean<T> operator== ( const BsplineBasis< T > & other ) const

◆ order()

int order ( ) const

The order of this B-spline basis (k in the class description).

◆ Serialize()

void Serialize ( Archive * a )

Passes this object to an Archive.

Refer to YAML Serialization for background. This method is only available when T = double.

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

drake/math/bspline_basis.h

Detailed Description

template<typename T> class drake::math::BsplineBasis< T >

Public Member Functions

Constructor & Destructor Documentation

◆ BsplineBasis() [1/6]

◆ BsplineBasis() [2/6]

◆ BsplineBasis() [3/6]

◆ BsplineBasis() [4/6]

◆ BsplineBasis() [5/6]

◆ BsplineBasis() [6/6]

Member Function Documentation

◆ ComputeActiveBasisFunctionIndices() [1/2]

◆ ComputeActiveBasisFunctionIndices() [2/2]

◆ degree()

◆ EvaluateBasisFunctionI()

◆ EvaluateCurve()

◆ EvaluateLinearInControlPoints()

◆ final_parameter_value()

◆ FindContainingInterval()

◆ initial_parameter_value()

◆ knots()

◆ num_basis_functions()

◆ operator!=()

◆ operator=() [1/2]

◆ operator=() [2/2]

◆ operator==()

◆ order()

◆ Serialize()

template<typename T>
class drake::math::BsplineBasis< T >