Inkscape: Geom::BezierCurve Class Reference

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Two-dimensional Bezier curve of arbitrary order. More...

#include <bezier-curve.h>

Inheritance diagram for Geom::BezierCurve:

Public Member Functions
	BezierCurve (D2< Bezier > const &b)

Point	initialPoint () const override
	Retrieve the start of the curve.

Point	finalPoint () const override
	Retrieve the end of the curve.

bool	isDegenerate () const override
	Check whether the curve has exactly zero length.

bool	isLineSegment () const override
	Return false if there are at least 3 distinct control points, true otherwise.

void	setInitial (Point const &v) override
	Change the starting point of the curve.

void	setFinal (Point const &v) override
	Change the ending point of the curve.

Rect	boundsFast () const override
	Quickly compute the curve's approximate bounding box.

Rect	boundsExact () const override
	Compute the curve's exact bounding box.

void	expandToTransformed (Rect &bbox, Affine const &transform) const override
	Expand the given rectangle to include the transformed curve, assuming it already contains its initial point.

OptRect	boundsLocal (OptInterval const &i, unsigned deg) const override

Curve *	duplicate () const override
	Create an exact copy of this curve.

Curve *	portion (Coord f, Coord t) const override
	Create a curve that corresponds to a part of this curve.

Curve *	reverse () const override
	Create a reversed version of this curve.

void	operator*= (Translate const &tr) override

void	operator*= (Scale const &s) override

void	operator*= (Affine const &m) override

Curve *	derivative () const override
	Create a derivative of this curve.

int	degreesOfFreedom () const override
	Return the number of independent parameters required to represent all variations of this curve.

std::vector< Coord >	roots (Coord v, Dim2 d) const override
	Computes time values at which the curve intersects an axis-aligned line.

Coord	nearestTime (Point const &p, Coord from=0, Coord to=1) const override
	Compute a time value at which the curve comes closest to a specified point.

Coord	length (Coord tolerance) const override
	Compute the arc length of this curve.

std::vector< CurveIntersection >	intersect (Curve const &other, Coord eps=EPSILON) const override
	Compute intersections with another curve.

Point	pointAt (Coord t) const override
	Evaluate the curve at a specified time value.

std::vector< Point >	pointAndDerivatives (Coord t, unsigned n) const override
	Evaluate the curve and its derivatives.

Coord	valueAt (Coord t, Dim2 d) const override
	Evaluate one of the coordinates at the specified time value.

D2< SBasis >	toSBasis () const override
	Convert the curve to a symmetric power basis polynomial.

bool	isNear (Curve const &c, Coord precision) const override
	Test whether two curves are approximately the same.

void	feed (PathSink &sink, bool) const override
	Feed the curve to a PathSink.

std::vector< Coord >	timesWithRadiusOfCurvature (double radius) const
	Computes the times where the radius of curvature of the bezier curve equals the given radius.

virtual void	operator*= (Translate const &tr)

virtual void	operator*= (Scale const &s)

virtual void	operator*= (Rotate const &r)

virtual void	operator*= (HShear const &hs)

virtual void	operator*= (VShear const &vs)

virtual void	operator*= (Zoom const &z)

virtual void	operator*= (Affine const &m)=0

Access and modify control points
unsigned	order () const
	Get the order of the Bezier curve.

unsigned	size () const
	Get the number of control points.

Point	controlPoint (unsigned ix) const
	Access control points of the curve.

Point	operator[] (unsigned ix) const

std::vector< Point >	controlPoints () const
	Get the control points.

D2< Bezier > const &	fragment () const

void	setPoint (unsigned ix, Point const &v)
	Modify a control point.

virtual void	setPoints (std::vector< Point > const &ps)
	Set new control points.

Public Member Functions inherited from Geom::Curve
virtual	~Curve ()

virtual Interval	timeRange () const
	Get the interval of allowed time values.

virtual Point	operator() (Coord t) const
	Evaluate the function at the specified time value.

OptRect	boundsLocal (OptInterval const &a) const
	Compute the bounding box of a part of the curve.

void	transform (Affine const &m)
	Transform this curve by an affine transformation.

virtual Curve *	transformed (Affine const &m) const
	Create a curve transformed by an affine transformation.

Curve *	portion (Interval const &i) const
	A version of that accepts an Interval.

Coord	nearestTime (Point const &p, Interval const &i) const
	A version that takes an Interval.

virtual std::vector< Coord >	allNearestTimes (Point const &p, Coord from=0, Coord to=1) const
	Compute time values at which the curve comes closest to a specified point.

std::vector< Coord >	allNearestTimes (Point const &p, Interval const &i)
	A version that takes an Interval.

virtual int	winding (Point const &p) const
	Compute the partial winding number of this curve.

virtual std::vector< CurveIntersection >	intersectSelf (Coord eps=EPSILON) const
	Compute intersections of this curve with itself.

virtual Point	unitTangentAt (Coord t, unsigned n=3) const
	Compute a vector tangent to the curve.

bool	operator== (Curve const &c) const
	Test equality of two curves.

Static Public Member Functions
Construct a Bezier curve with runtime-determined order.
static BezierCurve *	create (std::vector< Point > const &pts)
	Construct a curve from a vector of control points.

Protected Member Functions
	BezierCurve ()

	BezierCurve (Bezier const &x, Bezier const &y)

	BezierCurve (std::vector< Point > const &pts)

bool	_equalTo (Curve const &c) const override

Protected Attributes
D2< Bezier >	inner

Related Symbols
(Note that these are not member symbols.)
Coord	bezier_length (std::vector< Point > const &points, Coord tolerance)
	Compute the length of a bezier curve given by a vector of its control points.

Detailed Description

Two-dimensional Bezier curve of arbitrary order.

Bezier curves are an expansion of the concept of linear interpolation to n points. Linear segments in 2Geom are in fact Bezier curves of order 1.

Let \(\mathbf{B}_{\mathbf{p}_0\mathbf{p}_1\ldots\mathbf{p}_n}\) denote a Bezier curve of order \(n\) defined by the points \(\mathbf{p}_0, \mathbf{p}_1, \ldots, \mathbf{p}_n\). Bezier curve of order 1 is a linear interpolation curve between two points, defined as

\[ \mathbf{B}_{\mathbf{p}_0\mathbf{p}_1}(t) = (1-t)\mathbf{p}_0 + t\mathbf{p}_1 \]

If we now substitute points \(\mathbf{p_0}\) and \(\mathbf{p_1}\) in this definition by linear interpolations, we get the definition of a Bezier curve of order 2, also called a quadratic Bezier curve.

\begin{align*} \mathbf{B}_{\mathbf{p}_0\mathbf{p}_1\mathbf{p}_2}(t) &= (1-t) \mathbf{B}_{\mathbf{p}_0\mathbf{p}_1}(t) + t \mathbf{B}_{\mathbf{p}_1\mathbf{p}_2}(t) \\ \mathbf{B}_{\mathbf{p}_0\mathbf{p}_1\mathbf{p}_2}(t) &= (1-t)^2\mathbf{p}_0 + 2(1-t)t\mathbf{p}_1 + t^2\mathbf{p}_2 \end{align*}

By substituting points for quadratic Bezier curves in the original definition, we get a Bezier curve of order 3, called a cubic Bezier curve.

\begin{align*} \mathbf{B}_{\mathbf{p}_0\mathbf{p}_1\mathbf{p}_2\mathbf{p}_3}(t) &= (1-t) \mathbf{B}_{\mathbf{p}_0\mathbf{p}_1\mathbf{p}_2}(t) + t \mathbf{B}_{\mathbf{p}_1\mathbf{p}_2\mathbf{p}_3}(t) \\ \mathbf{B}_{\mathbf{p}_0\mathbf{p}_1\mathbf{p}_2\mathbf{p}_3}(t) &= (1-t)^3\mathbf{p}_0+3(1-t)^2t\mathbf{p}_1+3(1-t)t^2\mathbf{p}_2+t^3\mathbf{p}_3 \end{align*}

In general, a Bezier curve or order \(n\) can be recursively defined as

\[ \mathbf{B}_{\mathbf{p}_0\mathbf{p}_1\ldots\mathbf{p}_n}(t) = (1-t) \mathbf{B}_{\mathbf{p}_0\mathbf{p}_1\ldots\mathbf{p}_{n-1}}(t) + t \mathbf{B}_{\mathbf{p}_1\mathbf{p}_2\ldots\mathbf{p}_n}(t) \]

This substitution can be repeated an arbitrary number of times. To picture this, imagine the evaluation of a point on the curve as follows: first, all control points are joined with straight lines, and a point corresponding to the selected time value is marked on them. Then, the marked points are joined with straight lines and the point corresponding to the time value is marked. This is repeated until only one marked point remains, which is the point at the selected time value.

Evaluation of the Bezier curve

An important property of the Bezier curves is that their parameters (control points) have an intuitive geometric interpretation. Because of this, they are frequently used in vector graphics editors.

Every Bezier curve is contained in its control polygon (the convex polygon composed of its control points). This fact is useful for sweepline algorithms and intersection.

Implementation notes: The order of a Bezier curve is immuable once it has been created. Normally, you should know the order at compile time and use the BezierCurveN template. If you need to determine the order at runtime, use the BezierCurve::create() function. It will create a BezierCurveN for orders 1, 2 and 3 (up to cubic Beziers), so you can later dynamic_cast to those types, and for higher orders it will create an instance of BezierCurve.

Definition at line 47 of file bezier-curve.h.

Constructor & Destructor Documentation

◆ BezierCurve() [1/4]

Geom::BezierCurve::BezierCurve ( )

inlineprotected

Definition at line 50 of file bezier-curve.h.

Referenced by create(), derivative(), duplicate(), portion(), and reverse().

◆ BezierCurve() [2/4]

Geom::BezierCurve::BezierCurve	(	Bezier const &	x,
		Bezier const &	y
	)

inlineprotected

Definition at line 51 of file bezier-curve.h.

◆ BezierCurve() [3/4]

Geom::BezierCurve::BezierCurve ( std::vector< Point > const & pts )

protected

Definition at line 111 of file bezier-curve.cpp.

◆ BezierCurve() [4/4]

Geom::BezierCurve::BezierCurve ( D2< Bezier > const & b )

inlineexplicit

Definition at line 55 of file bezier-curve.h.

Member Function Documentation

◆ _equalTo()

bool Geom::BezierCurve::_equalTo ( Curve const & c ) const

overrideprotectedvirtual

Implements Geom::Curve.

Definition at line 261 of file bezier-curve.cpp.

References c, controlPoint(), and size().

◆ boundsExact()

Rect Geom::BezierCurve::boundsExact ( ) const

inlineoverridevirtual

Compute the curve's exact bounding box.

This method can be dramatically slower than boundsFast() depending on the curve type.

Returns: The smallest possible rectangle containing all of the curve's points.

Implements Geom::Curve.

Definition at line 112 of file bezier-curve.h.

References Geom::bounds_exact(), and inner.

◆ boundsFast()

Rect Geom::BezierCurve::boundsFast ( ) const

inlineoverridevirtual

Quickly compute the curve's approximate bounding box.

The resulting rectangle is guaranteed to contain all points belonging to the curve, but it might not be the smallest such rectangle. This method is usually fast.

Returns: A rectangle that contains all points belonging to the curve.

Implements Geom::Curve.

Definition at line 111 of file bezier-curve.h.

References Geom::bounds_fast(), and inner.

Referenced by boundsLocal(), and Geom::Ellipse::intersect().

◆ boundsLocal()

OptRect Geom::BezierCurve::boundsLocal	(	OptInterval const &	i,
		unsigned	deg
	)		const

inlineoverridevirtual

Implements Geom::Curve.

Definition at line 114 of file bezier-curve.h.

References Geom::bounds_local(), boundsFast(), Geom::derivative(), inner, order(), Geom::X, and Geom::Y.

◆ controlPoint()

Point Geom::BezierCurve::controlPoint ( unsigned ix ) const

inline

Access control points of the curve.

Parameters

ix	The (zero-based) index of the control point. Note that the caller is responsible for checking that this value is <= order().

Returns: The control point. No-reference return, use setPoint() to modify control points.

Definition at line 67 of file bezier-curve.h.

References inner, Geom::X, and Geom::Y.

Referenced by _equalTo(), feed(), feed_curve_to_cairo(), isLineSegment(), and operator*=().

◆ controlPoints()

std::vector< Point > Geom::BezierCurve::controlPoints ( ) const

inline

Get the control points.

Returns: Vector with order() + 1 control points.

Definition at line 71 of file bezier-curve.h.

References Geom::bezier_points(), and inner.

Referenced by length(), and wrap_rq_to_cubic_l().

◆ create()

BezierCurve * Geom::BezierCurve::create ( std::vector< Point > const & pts )

static

Construct a curve from a vector of control points.

This will construct the appropriate specialization of BezierCurve (i.e. LineSegment, QuadraticBezier or Cubic Bezier) if the number of control points in the passed vector does not exceed 4.

Definition at line 310 of file bezier-curve.cpp.

References BezierCurve().

◆ degreesOfFreedom()

int Geom::BezierCurve::degreesOfFreedom ( ) const

inlineoverridevirtual

Return the number of independent parameters required to represent all variations of this curve.

For example, for Bezier curves it returns the curve's order multiplied by 2.

Reimplemented from Geom::Curve.

Definition at line 155 of file bezier-curve.h.

References order().

◆ derivative()

Curve * Geom::BezierCurve::derivative ( ) const

inlineoverridevirtual

Create a derivative of this curve.

It's best to think of the derivative in physical terms: if the curve describes the position of some object on the plane from time \(t=0\) to \(t=1\) as said in the introduction, then the curve's derivative describes that object's speed at the same times. The second derivative refers to its acceleration, the third to jerk, etc.

Returns: New curve \(\mathbf{D} = \mathbf{C}'\).

Implements Geom::Curve.

Reimplemented in Geom::BezierCurveN< degree >, Geom::BezierCurveN< 1 >, Geom::BezierCurveN< degree >, Geom::BezierCurveN< 1 >, Geom::BezierCurveN< degree >, and Geom::BezierCurveN< 1 >.

Definition at line 152 of file bezier-curve.h.

References BezierCurve(), Geom::derivative(), inner, Geom::X, and Geom::Y.

◆ duplicate()

Curve * Geom::BezierCurve::duplicate ( ) const

inlineoverridevirtual

Create an exact copy of this curve.

Returns: Pointer to a newly allocated curve, identical to the original

Implements Geom::Curve.

Reimplemented in Geom::BezierCurveN< degree >, Geom::BezierCurveN< 1 >, Geom::Path::ClosingSegment, and Geom::Path::StitchSegment.

Definition at line 123 of file bezier-curve.h.

References BezierCurve().

Referenced by portion().

◆ expandToTransformed()

void Geom::BezierCurve::expandToTransformed	(	Rect &	bbox,
		Affine const &	transform
	)		const

overridevirtual

Expand the given rectangle to include the transformed curve, assuming it already contains its initial point.

Parameters

bbox[in,out]	bbox The rectangle to expand; it is assumed to already contain (initialPoint() * transform).
transform	The transform to apply to the curve before taking its bounding box.

Implements Geom::Curve.

Reimplemented in Geom::BezierCurveN< degree >, Geom::BezierCurveN< 1 >, Geom::BezierCurveN< degree >, Geom::BezierCurveN< 1 >, Geom::BezierCurveN< degree >, Geom::BezierCurveN< 1 >, Geom::BezierCurveN< degree >, Geom::BezierCurveN< 1 >, Geom::BezierCurveN< degree >, Geom::BezierCurveN< 1 >, Geom::BezierCurveN< degree >, Geom::BezierCurveN< 1 >, Geom::BezierCurveN< degree >, and Geom::BezierCurveN< 1 >.

Definition at line 148 of file bezier-curve.cpp.

References Geom::bounds_exact(), inner, and Geom::Curve::transform().

Referenced by Geom::BezierCurveN< degree >::expandToTransformed().

◆ feed()

void Geom::BezierCurve::feed	(	PathSink &	sink,
		bool	moveto_initial
	)		const

overridevirtual

Feed the curve to a PathSink.

Reimplemented from Geom::Curve.

Definition at line 281 of file bezier-curve.cpp.

References controlPoint(), Geom::PathSink::curveTo(), Geom::Curve::feed(), Geom::PathSink::lineTo(), Geom::PathSink::moveTo(), Geom::PathSink::quadTo(), and size().

Referenced by Geom::BezierCurveN< degree >::feed().

◆ finalPoint()

Point Geom::BezierCurve::finalPoint ( ) const

inlineoverridevirtual

Retrieve the end of the curve.

Returns: The point corresponding to \(\mathbf{C}(1)\).

Implements Geom::Curve.

Definition at line 106 of file bezier-curve.h.

References inner.

Referenced by Inkscape::UI::Tools::EraserTool::_accumulate(), Geom::Path::do_append(), intersect(), Geom::intersection(), Geom::detail::intersection_impl(), Geom::detail::intersection_impl(), length(), Geom::length(), Geom::middle_point(), Geom::projection(), Geom::BezierCurveN< degree >::reverse(), Geom::Path::snapEnds(), and TEST().

◆ fragment()

D2< Bezier > const & Geom::BezierCurve::fragment ( ) const

inline

Definition at line 72 of file bezier-curve.h.

References inner.

◆ initialPoint()

Point Geom::BezierCurve::initialPoint ( ) const

inlineoverridevirtual

Retrieve the start of the curve.

Returns: The point corresponding to \(\mathbf{C}(0)\).

Implements Geom::Curve.

Definition at line 105 of file bezier-curve.h.

References inner.

Referenced by Inkscape::UI::Tools::EraserTool::_accumulate(), Geom::Path::do_append(), Geom::Path::do_update(), intersect(), Geom::intersection(), Geom::detail::intersection_impl(), Geom::detail::intersection_impl(), length(), Geom::length(), Geom::middle_point(), Geom::Path::nearestTime(), Geom::projection(), Geom::BezierCurveN< degree >::reverse(), Geom::Path::stitchTo(), and TEST().

◆ intersect()

std::vector< CurveIntersection > Geom::BezierCurve::intersect	(	Curve const &	other,
		Coord	eps = `EPSILON`
	)		const

overridevirtual

Compute intersections with another curve.

Reimplemented from Geom::Curve.

Definition at line 177 of file bezier-curve.cpp.

References finalPoint(), Geom::find_intersections(), initialPoint(), inner, Geom::Curve::intersect(), Geom::BezierCurveN< degree >::intersect(), isLineSegment(), result, and Geom::transpose_in_place().

Referenced by Geom::BezierCurveN< degree >::intersect(), and Geom::BezierCurveN< degree >::intersect().

◆ isDegenerate()

bool Geom::BezierCurve::isDegenerate ( ) const

overridevirtual

Check whether the curve has exactly zero length.

Returns: True if the curve's initial point is exactly the same as its final point, and it contains no other points (its value set contains only one element).

Implements Geom::Curve.

Reimplemented in Geom::BezierCurveN< degree >, Geom::BezierCurveN< 1 >, Geom::BezierCurveN< degree >, and Geom::BezierCurveN< 1 >.

Definition at line 119 of file bezier-curve.cpp.

References inner, and size().

Referenced by Geom::BezierCurveN< degree >::isDegenerate().

◆ isLineSegment()

bool Geom::BezierCurve::isLineSegment ( ) const

overridevirtual

Return false if there are at least 3 distinct control points, true otherwise.

Reimplemented from Geom::Curve.

Reimplemented in Geom::BezierCurveN< degree >, Geom::BezierCurveN< 1 >, Geom::BezierCurveN< degree >, and Geom::BezierCurveN< 1 >.

Definition at line 131 of file bezier-curve.cpp.

References controlPoint(), end, size(), and start.

Referenced by intersect(), and Geom::BezierCurveN< degree >::isLineSegment().

◆ isNear()

bool Geom::BezierCurve::isNear	(	Curve const &	c,
		Coord	precision
	)		const

overridevirtual

Test whether two curves are approximately the same.

Implements Geom::Curve.

Definition at line 207 of file bezier-curve.cpp.

References Geom::are_near(), c, inner, isNear(), order(), and size().

Referenced by isNear().

◆ length()

Coord Geom::BezierCurve::length ( Coord tolerance ) const

overridevirtual

Compute the arc length of this curve.

For a curve \(\mathbf{C}(t) = (C_x(t), C_y(t))\), arc length is defined for 2D curves as

\[ \ell = \int_{0}^{1} \sqrt { [C_x'(t)]^2 + [C_y'(t)]^2 }\, \text{d}t \]

In other words, we divide the curve into infinitely small linear segments and add together their lengths. Of course we can't subdivide the curve into infinitely many segments on a computer, so this method returns an approximation. Not that there is usually no closed form solution to such integrals, so this method might be slow.

Parameters

tolerance Maximum allowed error

Returns: Total distance the curve's value travels on the plane when going from 0 to 1

Reimplemented from Geom::Curve.

Definition at line 153 of file bezier-curve.cpp.

References bezier_length(), controlPoints(), Geom::distance(), finalPoint(), initialPoint(), and order().

Referenced by Geom::Ellipse::intersect(), and Geom::Path::snapEnds().

◆ nearestTime()

Coord Geom::BezierCurve::nearestTime	(	Point const &	p,
		Coord	a = `0`,
		Coord	b = `1`
	)		const

overridevirtual

Compute a time value at which the curve comes closest to a specified point.

The first value with the smallest distance is returned if there are multiple such points.

Parameters

p	Query point
a	Minimum time value to consider
b	Maximum time value to consider; \(a < b\)

Returns: \(q \in [a, b]: ||\mathbf{C}(q) - \mathbf{p}|| = \inf(\{r \in \mathbb{R} : ||\mathbf{C}(r) - \mathbf{p}||\})\)

Reimplemented from Geom::Curve.

Reimplemented in Geom::BezierCurveN< degree >, Geom::BezierCurveN< 1 >, Geom::BezierCurveN< degree >, Geom::BezierCurveN< 1 >, Geom::BezierCurveN< degree >, and Geom::BezierCurveN< 1 >.

Definition at line 276 of file bezier-curve.cpp.

References inner, and Geom::nearest_time().

Referenced by Geom::BezierCurveN< degree >::nearestTime().

◆ operator*=() [1/10]

void Geom::BezierCurve::operator*= ( Affine const & m )

inlineoverridevirtual

Implements Geom::Curve.

Definition at line 146 of file bezier-curve.h.

References controlPoint(), setPoint(), and size().

◆ operator*=() [2/10]

virtual void Geom::Curve::operator*= ( Affine const & m )

virtual

Implements Geom::Curve.

◆ operator*=() [3/10]

virtual void Geom::Curve::operator*= ( HShear const & hs )

inlinevirtual

Reimplemented from Geom::Curve.

Definition at line 200 of file curve.h.

◆ operator*=() [4/10]

virtual void Geom::Curve::operator*= ( Rotate const & r )

inlinevirtual

Reimplemented from Geom::Curve.

Definition at line 199 of file curve.h.

◆ operator*=() [5/10]

virtual void Geom::Curve::operator*= ( Scale const & s )

inlinevirtual

Reimplemented from Geom::Curve.

Definition at line 198 of file curve.h.

◆ operator*=() [6/10]

void Geom::BezierCurve::operator*= ( Scale const & s )

inlineoverridevirtual

Reimplemented from Geom::Curve.

Definition at line 140 of file bezier-curve.h.

References inner, size(), Geom::X, and Geom::Y.

◆ operator*=() [7/10]

virtual void Geom::Curve::operator*= ( Translate const & tr )

inlinevirtual

Reimplemented from Geom::Curve.

Definition at line 197 of file curve.h.

◆ operator*=() [8/10]

void Geom::BezierCurve::operator*= ( Translate const & tr )

inlineoverridevirtual

Reimplemented from Geom::Curve.

Definition at line 134 of file bezier-curve.h.

References inner, size(), Geom::X, and Geom::Y.

◆ operator*=() [9/10]

virtual void Geom::Curve::operator*= ( VShear const & vs )

inlinevirtual

Reimplemented from Geom::Curve.

Definition at line 201 of file curve.h.

◆ operator*=() [10/10]

virtual void Geom::Curve::operator*= ( Zoom const & z )

inlinevirtual

Reimplemented from Geom::Curve.

Definition at line 202 of file curve.h.

◆ operator[]()

Point Geom::BezierCurve::operator[] ( unsigned ix ) const

inline

Definition at line 68 of file bezier-curve.h.

References inner, Geom::X, and Geom::Y.

◆ order()

unsigned Geom::BezierCurve::order ( ) const

inline

Get the order of the Bezier curve.

A Bezier curve has order() + 1 control points.

Definition at line 61 of file bezier-curve.h.

References inner, and Geom::X.

Referenced by Inkscape::UI::PathManipulator::_createControlPointsFromGeometry(), boundsLocal(), degreesOfFreedom(), isNear(), length(), setFinal(), and setPoints().

◆ pointAndDerivatives()

std::vector< Point > Geom::BezierCurve::pointAndDerivatives	(	Coord	t,
		unsigned	n
	)		const

inlineoverridevirtual

Evaluate the curve and its derivatives.

This will return a vector that contains the value of the curve and the specified number of derivatives. However, the returned vector might contain less elements than specified if some derivatives do not exist.

Parameters

t	Time value
n	The number of derivatives to compute

Returns: Vector of at most \(n+1\) elements of the form \([\mathbf{C}(t), \mathbf{C}'(t), \mathbf{C}''(t), \ldots]\)

Implements Geom::Curve.

Definition at line 165 of file bezier-curve.h.

References inner.

Referenced by Geom::BezierFitter::NewtonRaphsonRootFind(), and Geom::EllipticalArc::pointAndDerivatives().

◆ pointAt()

Point Geom::BezierCurve::pointAt ( Coord t ) const

inlineoverridevirtual

Evaluate the curve at a specified time value.

Parameters

t	Time value

Returns: \(\mathbf{C}(t)\)

Reimplemented from Geom::Curve.

Definition at line 164 of file bezier-curve.h.

References inner.

Referenced by GrDrag::addStopNearPoint(), Geom::EllipticalArc::allNearestTimes(), Geom::BezierFitter::compute_hook(), Geom::distance(), Inkscape::distance_to_segment(), draw_ratquad(), gr_knot_moved_midpoint_handler(), RectHandle::hit(), Inkscape::intersect_line_rectangle(), Geom::BezierFitter::NewtonRaphsonRootFind(), Inkscape::LivePathEffect::path_from_piecewise_fix_cusps(), Geom::EllipticalArc::pointAt(), Geom::BezierCurveN< degree >::portion(), GrDrag::selected_move(), sp_gradient_reset_to_userspace(), and Geom::xAx::toCurve().

◆ portion()

Curve * Geom::BezierCurve::portion	(	Coord	a,
		Coord	b
	)		const

overridevirtual

Create a curve that corresponds to a part of this curve.

For \(a > b\), the returned portion will be reversed with respect to the original. The returned curve will always be of the same type.

Parameters

a	Beginning of the interval specifying the portion of the curve
b	End of the interval

Returns

New curve \(\mathbf{D}\) such that:

\(\mathbf{D}(0) = \mathbf{C}(a)\)
\(\mathbf{D}(1) = \mathbf{C}(b)\)
\(\mathbf{D}[ [0, 1] ] = \mathbf{C}[ [a?b] ]\), where \([a?b] = [\min(a, b), \max(a, b)]\)

Implements Geom::Curve.

Reimplemented in Geom::BezierCurveN< degree >, and Geom::BezierCurveN< 1 >.

Definition at line 250 of file bezier-curve.cpp.

References BezierCurve(), duplicate(), inner, Geom::portion(), and reverse().

◆ reverse()

Curve * Geom::BezierCurve::reverse ( ) const

inlineoverridevirtual

Create a reversed version of this curve.

The result corresponds to portion(1, 0), but this method might be faster.

Returns: Pointer to a new curve \(\mathbf{D}\) such that \(\forall_{x \in [0, 1]} \mathbf{D}(x) = \mathbf{C}(1-x)\)

Reimplemented from Geom::Curve.

Reimplemented in Geom::BezierCurveN< degree >, Geom::BezierCurveN< 1 >, Geom::Path::ClosingSegment, and Geom::Path::StitchSegment.

Definition at line 129 of file bezier-curve.h.

References BezierCurve(), inner, and Geom::reverse().

Referenced by portion().

◆ roots()

std::vector< Coord > Geom::BezierCurve::roots	(	Coord	v,
		Dim2	d
	)		const

inlineoverridevirtual

Computes time values at which the curve intersects an axis-aligned line.

Parameters

v	The coordinate of the line
d	Which axis the coordinate is on. X means a vertical line, Y a horizontal line.

Implements Geom::Curve.

Definition at line 158 of file bezier-curve.h.

References inner.

Referenced by Geom::EllipticalArc::roots().

◆ setFinal()

void Geom::BezierCurve::setFinal ( Point const & v )

inlineoverridevirtual

Change the ending point of the curve.

After calling this method, it is guaranteed that \(\mathbf{C}(0) = \mathbf{p}\), and the curve is still continuous. The precise new shape of the curve varies with curve type.

Parameters

p	New ending point of the curve

Implements Geom::Curve.

Definition at line 110 of file bezier-curve.h.

References order(), and setPoint().

Referenced by Geom::Path::clear(), Geom::Path::do_append(), Geom::Path::do_update(), draw_ray(), Geom::Path::Path(), Geom::Path::setInitial(), and Geom::Path::start().

◆ setInitial()

void Geom::BezierCurve::setInitial ( Point const & v )

inlineoverridevirtual

Change the starting point of the curve.

After calling this method, it is guaranteed that \(\mathbf{C}(0) = \mathbf{p}\), and the curve is still continuous. The precise new shape of the curve varies with curve type.

Parameters

p	New starting point of the curve

Implements Geom::Curve.

Definition at line 109 of file bezier-curve.h.

References setPoint().

Referenced by Geom::Path::clear(), Geom::Path::close(), Geom::Path::do_append(), Geom::Path::do_update(), Inkscape::LivePathEffect::LPEFillBetweenMany::doEffect(), draw_ray(), Geom::Path::Path(), Geom::Path::setFinal(), Geom::Path::snapEnds(), and Geom::Path::start().

◆ setPoint()

void Geom::BezierCurve::setPoint	(	unsigned	ix,
		Point const &	v
	)

inline

Modify a control point.

Parameters

ix	The zero-based index of the point to modify. Note that the caller is responsible for checking that this value is <= order().
v	The new value of the point

Definition at line 77 of file bezier-curve.h.

References inner, Geom::X, and Geom::Y.

Referenced by SPCurve::last_point_additive_move(), operator*=(), setFinal(), setInitial(), and setPoints().

◆ setPoints()

virtual void Geom::BezierCurve::setPoints ( std::vector< Point > const & ps )

inlinevirtual

Set new control points.

Parameters

ps	Vector which must contain order() + 1 points. Note that the caller is responsible for checking the size of this vector.

Exceptions

LogicalError Thrown when the size of the vector does not match the order.

Definition at line 85 of file bezier-curve.h.

References order(), and setPoint().

◆ size()

unsigned Geom::BezierCurve::size ( ) const

inline

Get the number of control points.

Definition at line 63 of file bezier-curve.h.

References inner, and Geom::X.

Referenced by _equalTo(), feed(), isDegenerate(), isLineSegment(), isNear(), operator*=(), operator*=(), operator*=(), TEST(), TEST_F(), TEST_F(), and timesWithRadiusOfCurvature().

◆ timesWithRadiusOfCurvature()

std::vector< Coord > Geom::BezierCurve::timesWithRadiusOfCurvature ( double radius ) const

Computes the times where the radius of curvature of the bezier curve equals the given radius.

The algorithm works as follows: Find the solutions of curvature of curve at t = curvatureValue This is equivalent to (dx*ddy-ddx*dy)/curvatureValue = (dx**2+dy**2)**(3/2) When the left side is positive, taking the square gives ((dx*ddy-ddx*dy)/curvatureValue)**2 - (dx**2+dy**2)**3 = 0 This is a polyomial for BezierCurves and can be solved with root finding algos.

Definition at line 331 of file bezier-curve.cpp.

References Geom::derivative(), inner, size(), Geom::X, and Geom::Y.

◆ toSBasis()

D2< SBasis > Geom::BezierCurve::toSBasis ( ) const

inlineoverridevirtual

Convert the curve to a symmetric power basis polynomial.

Symmetric power basis polynomials (S-basis for short) are numerical representations of curves with excellent numerical properties. Most high level operations provided by 2Geom are implemented in terms of S-basis operations, so every curve has to provide a method to convert it to an S-basis polynomial on two variables. See SBasis class reference for more information.

Implements Geom::Curve.

Definition at line 169 of file bezier-curve.h.

References inner.

Referenced by Geom::xAx::crossings(), Geom::EllipticalArc::toSBasis(), wrap_rq_to_cubic_sb(), and wrap_rq_to_cubic_sb_l().

◆ valueAt()

Coord Geom::BezierCurve::valueAt	(	Coord	t,
		Dim2	d
	)		const

inlineoverridevirtual

Evaluate one of the coordinates at the specified time value.

Parameters

t	Time value
d	The dimension to evaluate

Returns: The specified coordinate of \(\mathbf{C}(t)\)

Reimplemented from Geom::Curve.

Definition at line 168 of file bezier-curve.h.

References inner.

Referenced by Geom::EllipticalArc::valueAt().

Friends And Related Symbol Documentation

◆ bezier_length()

Coord bezier_length	(	std::vector< Point > const &	points,
		Coord	tolerance
	)

Compute the length of a bezier curve given by a vector of its control points.

Definition at line 746 of file bezier-curve.cpp.

References Geom::bezier_length_internal().

Referenced by length().

Member Data Documentation

◆ inner

D2<Bezier> Geom::BezierCurve::inner

protected

Definition at line 49 of file bezier-curve.h.

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

/builds/inkscape/inkscape/src/3rdparty/2geom/include/2geom/bezier-curve.h
/builds/inkscape/inkscape/src/3rdparty/2geom/src/2geom/bezier-curve.cpp

Public Member Functions

Static Public Member Functions

Protected Member Functions

Protected Attributes

Related Symbols

Detailed Description

Constructor & Destructor Documentation

◆ BezierCurve() [1/4]

◆ BezierCurve() [2/4]

◆ BezierCurve() [3/4]

◆ BezierCurve() [4/4]

Member Function Documentation

◆ _equalTo()

◆ boundsExact()

◆ boundsFast()

◆ boundsLocal()

◆ controlPoint()

◆ controlPoints()

◆ create()

◆ degreesOfFreedom()

◆ derivative()

◆ duplicate()

◆ expandToTransformed()

◆ feed()

◆ finalPoint()

◆ fragment()

◆ initialPoint()

◆ intersect()

◆ isDegenerate()

◆ isLineSegment()

◆ isNear()

◆ length()

◆ nearestTime()

◆ operator*=() [1/10]

◆ operator*=() [2/10]

◆ operator*=() [3/10]

◆ operator*=() [4/10]

◆ operator*=() [5/10]

◆ operator*=() [6/10]

◆ operator*=() [7/10]

◆ operator*=() [8/10]

◆ operator*=() [9/10]

◆ operator*=() [10/10]

◆ operator[]()

◆ order()

◆ pointAndDerivatives()

◆ pointAt()

◆ portion()

◆ reverse()

◆ roots()

◆ setFinal()

◆ setInitial()

◆ setPoint()

◆ setPoints()

◆ size()

◆ timesWithRadiusOfCurvature()

◆ toSBasis()

◆ valueAt()

Friends And Related Symbol Documentation

◆ bezier_length()

Member Data Documentation

◆ inner