A Bezier curve class. More...

#include <bezier.h>

Public Member Functions
	Curve (Eigen::MatrixX2d points)
	Create the Bezier curve.

	Curve (const PointVector &points)
	Create the Bezier curve.

	Curve (const Curve &curve)

	Curve (Curve &&curve) noexcept

Curve &	operator= (const Curve &curve)

Curve &	operator= (Curve &&curve) noexcept

unsigned	order () const
	Get order of the curve (Nth order curve is described with N+1 points)

PointVector	controlPoints () const
	Get a vector of control points.

Point	controlPoint (unsigned idx) const
	Get the control point at index idx.

std::pair< Point, Point >	endPoints () const
	Get first and last control points.

PointVector	polyline () const
	Get a polyline representation of the curve as a vector of points on curve.

PointVector	polyline (double flatness) const
	Get a polyline representation of the curve as a vector of points on curve.

ParamVector	polylineParams () const
	Get curve parameters corresponding to polyline points.

ParamVector	polylineParams (double flatness) const
	Get curve parameters corresponding to polyline points.

double	length () const
	Compute exact arc length using Chebyshev polynomials.

double	length (double t) const
	Compute exact arc length using Chebyshev polynomials.

double	length (double t1, double t2) const
	Compute exact arc length using Chebyshev polynomials.

double	step (double t, double ds) const
	Compute parameter t which is ds distance from given t.

void	reverse ()
	Reverse order of control points.

void	setControlPoint (unsigned idx, const Point &point)
	Set the new coordinates to a control point.

void	raiseOrder ()
	Raise the curve order by 1.

void	lowerOrder ()
	Lower the curve order by 1.

Point	valueAt (double t) const
	Get the point on curve for a given t.

Eigen::MatrixX2d	valueAt (const ParamVector &t_vector) const
	Get the point vector on curve for given parameters.

double	curvatureAt (double t) const
	Get curvature of the curve for a given t.

double	curvatureDerivativeAt (double t) const
	Get curvature derivative of the curve for a given t.

Vector	tangentAt (double t) const
	Get the unit tangent of the curve for a given t.

Vector	normalAt (double t) const
	Get the unit normal of the curve for a given t.

const Curve &	derivative () const
	Get the derivative of a curve.

const Curve &	derivative (unsigned n) const
	Get the nth derivative of a curve.

Vector	derivativeAt (double t) const
	Get value of a derivative for a given t.

Vector	derivativeAt (unsigned n, double t) const
	Get value of an nth derivative for a given t.

ParamVector	roots () const
	Get roots of the curve on both axes.

ParamVector	extrema () const
	Get all extrema of the curve.

BoundingBox	boundingBox () const
	Get the bounding box of curve.

std::vector< Curve >	splitCurve (const ParamVector &t_vector) const
	Split the curve into subcurves at multiple parameters.

std::pair< Curve, Curve >	splitCurve (double t=0.5) const
	Split the curve into two subcurves.

PointVector	intersections (const Curve &curve) const
	Get the points of intersection with another curve.

double	projectPoint (const Point &point) const
	Get the parameter t where curve is closest to given point.

double	distance (const Point &point) const
	Get distance of the point to the curve.

void	applyContinuity (const Curve &curve, const std::vector< double > &beta_coeffs)
	Apply geometric continuity based on another curve.

Static Public Member Functions
static Curve	offsetCurve (const Curve &curve, double offset, unsigned order=0)
	Fit a Bezier approximation of the curve offset by a given distance.

static Curve	joinCurves (const Curve &curve1, const Curve &curve2, unsigned order=0)
	Fit a single Bezier curve through two curves joined end to end.

static Curve	fromPolyline (const PointVector &polyline, unsigned order=0)
	Fit a Bezier curve to an ordered polyline.

Detailed Description

A Bezier curve class.

A class for storing and using any-order Bezier curve. It uses private and static caching for storing often accessed data. Private caching is used for data concerning individual curve, while static caching is used for common data (coefficient matrices)

Constructor & Destructor Documentation

◆ Curve() [1/2]

Curve ( Eigen::MatrixX2d points )

Create the Bezier curve.

Parameters

points Nx2 matrix where each row is one of N control points that define the curve

Warning: Requires at least one control point; undefined behavior otherwise (not enforced)

◆ Curve() [2/2]

Curve ( const PointVector & points )

Create the Bezier curve.

Parameters

points A vector of control points that define the curve

Warning: Requires at least one control point; undefined behavior otherwise (not enforced)

Member Function Documentation

◆ applyContinuity()

void applyContinuity	(	const Curve &	curve,
		const std::vector< double > &	beta_coeffs
	)

Apply geometric continuity based on another curve.

Glues the start of this curve to the end of the given curve.

Parameters

curve	Curve on which calculations are based
beta_coeffs	Beta-constraints used to calculate continuity. Size defines continuity order.

Note: Raises the order of this curve if it has too few control points for the requested continuity

Warning: Resets cached data

◆ boundingBox()

BoundingBox boundingBox ( ) const

Get the bounding box of curve.

Returns: Bounding box

◆ controlPoint()

Point controlPoint ( unsigned idx ) const

Get the control point at index idx.

Parameters

idx	Index of chosen control point

Returns: The control point at index idx

Note: idx must be in range [0, order()]

◆ controlPoints()

PointVector controlPoints ( ) const

Get a vector of control points.

Returns: A vector of control points

◆ curvatureAt()

double curvatureAt ( double t ) const

Get curvature of the curve for a given t.

Parameters

t	Curve parameter

Returns: Curvature of a curve for a given t

◆ curvatureDerivativeAt()

double curvatureDerivativeAt ( double t ) const

Get curvature derivative of the curve for a given t.

Parameters

t	Curve parameter

Returns: Curvature derivative of a curve for a given t

◆ derivative() [1/2]

const Curve & derivative ( ) const

Get the derivative of a curve.

Returns: Derivative curve

Warning: The returned reference lives in this curve's cache; it is invalidated by any mutation, assignment or move of this curve.

◆ derivative() [2/2]

const Curve & derivative ( unsigned n ) const

Get the nth derivative of a curve.

Parameters

n	Desired number of derivative

Returns: Derivative curve

Warning: The returned reference lives in this curve's cache; it is invalidated by any mutation, assignment or move of this curve.

◆ derivativeAt() [1/2]

Vector derivativeAt ( double t ) const

Get value of a derivative for a given t.

Parameters

t	Curve parameter

Returns: Curve derivative at t

◆ derivativeAt() [2/2]

Vector derivativeAt	(	unsigned	n,
		double	t
	)		const

Get value of an nth derivative for a given t.

Parameters

n	Desired number of derivative
t	Curve parameter

Returns: nth curve derivative at t

◆ distance()

double distance ( const Point & point ) const

Get distance of the point to the curve.

Parameters

point Point to project on curve

Returns: Distance to the curve

◆ endPoints()

std::pair< Point, Point > endPoints ( ) const

Get first and last control points.

Returns: A pair of end points

◆ extrema()

ParamVector extrema ( ) const

Get all extrema of the curve.

Returns: A vector of parameters where extrema are

◆ fromPolyline()

static Curve fromPolyline	(	const PointVector &	polyline,
		unsigned	order = `0`
	)

static

Fit a Bezier curve to an ordered polyline.

Parameters

polyline	Polyline vertices to approximate
order	Order of the fitted curve; 0 selects it automatically

Returns: Fitted curve

Exceptions

std::logic_error if the polyline has fewer than two points

◆ intersections()

PointVector intersections ( const Curve & curve ) const

Get the points of intersection with another curve.

Parameters

curve Curve to intersect with

Returns: A vector of points of intersection between curves

Note: Transversal crossings and endpoint/vertex contacts are reported; tangential (non-crossing) touches are not.

◆ joinCurves()

static Curve joinCurves	(	const Curve &	curve1,
		const Curve &	curve2,
		unsigned	order = `0`
	)

static

Fit a single Bezier curve through two curves joined end to end.

Parameters

curve1	First curve
curve2	Second curve
order	Order of the resulting curve; 0 selects it automatically

Returns: Joined curve

Note: Curves need not be contiguous; a gap between them is bridged by the fit.

◆ length() [1/3]

double length ( ) const

Compute exact arc length using Chebyshev polynomials.

Returns: Arc length

◆ length() [2/3]

double length ( double t ) const

Compute exact arc length using Chebyshev polynomials.

Parameters

t	Curve parameter to which length is computed

Returns: Arc length from start to parameter t

Exceptions

std::logic_error if t is outside [0.0, 1.0]

◆ length() [3/3]

double length	(	double	t1,
		double	t2
	)		const

Compute exact arc length using Chebyshev polynomials.

Parameters

t1	Curve parameter from which length is computed
t2	Curve parameter to which length is computed

Returns: Arc length between parameters t1 and t2

Exceptions

std::logic_error if t1 or t2 is outside [0.0, 1.0]

◆ lowerOrder()

void lowerOrder ( )

Lower the curve order by 1.

If current shape cannot be described by lower order, it will be best approximation

Warning: Resets cached data

Exceptions

std::logic_error if the curve is already 1st order

◆ normalAt()

Vector normalAt ( double t ) const

Get the unit normal of the curve for a given t.

Parameters

t	Curve parameter

Returns: Unit normal of a curve for given t

Note: See tangentAt regarding behaviour at a cusp.

◆ offsetCurve()

static Curve offsetCurve	(	const Curve &	curve,
		double	offset,
		unsigned	order = `0`
	)

static

Fit a Bezier approximation of the curve offset by a given distance.

Parameters

curve	Source curve
offset	Offset distance (positive to the left of the curve's direction, negative to the right)
order	Order of the resulting curve; 0 selects it automatically

Returns: Offset curve

◆ order()

unsigned order ( ) const

Get order of the curve (Nth order curve is described with N+1 points)

Returns: Order of curve

◆ polyline() [1/2]

PointVector polyline ( ) const

Get a polyline representation of the curve as a vector of points on curve.

Returns: A vector of polyline vertices

Note: Default flatness parameter is calculated as 0.1% of bounding box diagonal

◆ polyline() [2/2]

PointVector polyline ( double flatness ) const

Get a polyline representation of the curve as a vector of points on curve.

Parameters

flatness Error tolerance of approximation

Returns: A vector of polyline vertices

◆ polylineParams() [1/2]

ParamVector polylineParams ( ) const

Get curve parameters corresponding to polyline points.

Returns: A vector of curve parameters for each polyline vertex

Note: Default flatness parameter is calculated as 0.1% of bounding box diagonal

◆ polylineParams() [2/2]

ParamVector polylineParams ( double flatness ) const

Get curve parameters corresponding to polyline points.

Parameters

flatness Error tolerance of approximation

Returns: A vector of curve parameters for each polyline vertex

◆ projectPoint()

double projectPoint ( const Point & point ) const

Get the parameter t where curve is closest to given point.

Parameters

point Point to project on curve

Returns: double t

◆ raiseOrder()

void raiseOrder ( )

Raise the curve order by 1.

Curve will always retain its shape

Warning: Resets cached data

◆ reverse()

void reverse ( )

Reverse order of control points.

Warning: Resets cached data

◆ roots()

ParamVector roots ( ) const

Get roots of the curve on both axes.

Returns: A vector of parameters where curve passes through axes

◆ setControlPoint()

void setControlPoint	(	unsigned	idx,
		const Point &	point
	)

Set the new coordinates to a control point.

Parameters

idx	Index of chosen control point
point	New control point

Note: idx must be in range [0, order()]

Warning: Resets cached data

◆ splitCurve() [1/2]

std::vector< Curve > splitCurve ( const ParamVector & t_vector ) const

Split the curve into subcurves at multiple parameters.

Parameters

t_vector Vector of curve parameters at which to split the curve

Returns: A vector of subcurves

◆ splitCurve() [2/2]

std::pair< Curve, Curve > splitCurve ( double t = 0.5 ) const

Split the curve into two subcurves.

Parameters

t	Curve parameter at which to split the curve

Returns: Pair of two subcurves

◆ step()

double step	(	double	t,
		double	ds
	)		const

Compute parameter t which is ds distance from given t.

Parameters

t	Curve parameter
ds	Distance to iterate

Returns: New parameter t

Note: Result saturates at t = 0.0 / t = 1.0 if ds walks off the curve

Exceptions

std::logic_error if t is outside [0.0, 1.0]

◆ tangentAt()

Vector tangentAt ( double t ) const

Get the unit tangent of the curve for a given t.

Parameters

t	Curve parameter

Returns: Unit tangent of a curve for a given t

Note: At a cusp (zero velocity) the direction is taken from the first non-vanishing derivative; it is the zero vector only if the curve is a point.

◆ valueAt() [1/2]

Eigen::MatrixX2d valueAt ( const ParamVector & t_vector ) const

Get the point vector on curve for given parameters.

Parameters

t_vector Curve parameters

Returns: Matrix of points on a curve for given parameters

◆ valueAt() [2/2]

Point valueAt ( double t ) const

Get the point on curve for a given t.

Parameters

t	Curve parameter

Returns: Point on a curve for a given t

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

include/Bezier/bezier.h

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ Curve() [1/2]

◆ Curve() [2/2]

Member Function Documentation

◆ applyContinuity()

◆ boundingBox()

◆ controlPoint()

◆ controlPoints()

◆ curvatureAt()

◆ curvatureDerivativeAt()

◆ derivative() [1/2]

◆ derivative() [2/2]

◆ derivativeAt() [1/2]

◆ derivativeAt() [2/2]

◆ distance()

◆ endPoints()

◆ extrema()

◆ fromPolyline()

◆ intersections()

◆ joinCurves()

◆ length() [1/3]

◆ length() [2/3]

◆ length() [3/3]

◆ lowerOrder()

◆ normalAt()

◆ offsetCurve()

◆ order()

◆ polyline() [1/2]

◆ polyline() [2/2]

◆ polylineParams() [1/2]

◆ polylineParams() [2/2]

◆ projectPoint()

◆ raiseOrder()

◆ reverse()

◆ roots()

◆ setControlPoint()

◆ splitCurve() [1/2]

◆ splitCurve() [2/2]

◆ step()

◆ tangentAt()

◆ valueAt() [1/2]

◆ valueAt() [2/2]