BsplineBasis< T > Class Template Reference

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.
	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.
	BsplineBasis (const BsplineBasis< double > &other)
	Conversion constructor.
int	order () const
	The order of this B-spline basis (k in the class description).
int	degree () const
	The degree of the piecewise polynomials comprising this B-spline basis (k - 1 in the class description).
int	num_basis_functions () const
	The number of basis functions in this B-spline basis (n + 1 in the class description).
const std::vector< T > &	knots () const
	The knot vector of this B-spline basis (the vector (t₀, t₁, ..., tₘ)' in the class description).
const T &	initial_parameter_value () const
	The minimum allowable parameter value for B-spline curves using this basis (tₖ₋₁ in the class description).
const T &	final_parameter_value () const
	The maximum allowable parameter value for B-spline curves using this basis (tₙ₊₁ in the class description).
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().
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.
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.
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.
VectorX< T >	EvaluateLinearInControlPoints (const T &parameter_value) const
	Returns the vector, M, such that.
T	EvaluateBasisFunctionI (int i, const T &parameter_value) const
	Returns the value of the i-th basis function evaluated at parameter_value.
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.
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]

template<typename T>

BsplineBasis ( const BsplineBasis< T > & )

default

BsplineBasis() [2/6]

template<typename T>

BsplineBasis ( BsplineBasis< T > && )

default

BsplineBasis() [3/6]

template<typename T>

BsplineBasis

(

)

BsplineBasis() [4/6]

template<typename T>

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]

template<typename T>

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]

template<typename T>

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]

template<typename T>

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]

template<typename T>

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()

template<typename T>

int degree

(

)

const

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

EvaluateBasisFunctionI()

template<typename T>

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

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

EvaluateCurve()

template<typename T>

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.

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()

template<typename T>

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()

template<typename T>

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()

template<typename T>

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()

template<typename T>

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()

template<typename T>

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()

template<typename T>

int num_basis_functions

(

)

const

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

operator!=()

template<typename T>

boolean< T > operator!=

(

const BsplineBasis< T > &

other

)

const

operator=() [1/2]

template<typename T>

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

default

operator=() [2/2]

template<typename T>

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

default

operator==()

template<typename T>

boolean< T > operator==

(

const BsplineBasis< T > &

other

)

const

order()

template<typename T>

int order

(

)

const

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

Serialize()

template<typename T>

template<typename Archive>

void Serialize

(

Archive *

a

)

Passes this object to an Archive.

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

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

• drake/math/bspline_basis.h