Inkscape: Geom::EllipticalArc Class Reference

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Elliptical arc curve. More...

#include <elliptical-arc.h>

Inheritance diagram for Geom::EllipticalArc:

Public Member Functions
	EllipticalArc ()
	Creates an arc with all variables set to zero.

	EllipticalArc (Point const &ip, Point const &r, Coord rot_angle, bool large_arc, bool sweep, Point const &fp)
	Create a new elliptical arc.

	EllipticalArc (Point const &ip, Coord rx, Coord ry, Coord rot_angle, bool large_arc, bool sweep, Point const &fp)
	Create a new elliptical arc, giving the ellipse's rays as separate coordinates.

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

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

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

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

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

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

bool	isLineSegment () const override
	Check whether the curve is a line segment.

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

Rect	boundsExact () const override
	Compute bounds of an elliptical arc.

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 int deg) const override

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

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

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

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

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

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

void	operator*= (Translate const &tr) override

void	operator*= (Scale const &s) override

void	operator*= (Rotate const &r) override

void	operator*= (Zoom const &z) override

void	operator*= (Affine const &m) override

std::vector< Point >	pointAndDerivatives (Coord t, unsigned int n) const override

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

Curve *	portion (double f, double 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.

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

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

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

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

Retrieve basic information
Coord	center (Dim2 d) const
	Get a coordinate of the elliptical arc's center.

Point	center () const
	Get the arc's center.

Coord	ray (Dim2 d) const
	Get one of the ellipse's rays.

Point	rays () const
	Get both rays as a point.

Angle	rotationAngle () const
	Get the defining ellipse's rotation.

bool	largeArc () const
	Whether the arc is larger than half an ellipse.

bool	sweep () const
	Whether the arc turns clockwise.

Angle	initialAngle () const

Angle	finalAngle () const

Modify parameters
void	set (Point const &ip, double rx, double ry, double rot_angle, bool large_arc, bool sweep, Point const &fp)
	Change all of the arc's parameters.

void	set (Point const &ip, Point const &r, Angle rot_angle, bool large_arc, bool sweep, Point const &fp)
	Change all of the arc's parameters.

void	setEndpoints (Point const &ip, Point const &fp)
	Change the initial and final point in one operation.

Evaluate the arc as a function
bool	containsAngle (Angle angle) const
	Check whether the arc contains the given angle.

Point	pointAtAngle (Coord t) const
	Evaluate the arc at the specified angular coordinate.

Coord	valueAtAngle (Coord t, Dim2 d) const
	Evaluate one of the arc's coordinates at the specified angle.

Coord	timeAtAngle (Angle a) const
	Compute the curve time value corresponding to the given angular value.

Angle	angleAt (Coord t) const
	Compute the angular domain value corresponding to the given time value.

Coord	sweepAngle () const
	Compute the amount by which the angle parameter changes going from start to end.

Coord	angularExtent () const
	Get the elliptical angle spanned by the arc.

AngleInterval	angularInterval () const
	Get the angular interval of the arc.

Point	pointAt (Coord t) const override
	Evaluate the arc in the curve domain, i.e. \([0, 1]\).

Coord	valueAt (Coord t, Dim2 d) const override
	Evaluate a single coordinate on the arc in the curve domain.

Affine	unitCircleTransform () const
	Compute a transform that maps the unit circle to the arc's ellipse.

Affine	inverseUnitCircleTransform () const
	Compute a transform that maps the arc's ellipse to the unit circle.

Deal with degenerate ellipses.
bool	isChord () const
	Check whether both rays are nonzero.

LineSegment	chord () const
	Get the line segment connecting the arc's endpoints.

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.

virtual std::vector< Point >	pointAndDerivatives (Coord t, unsigned n) const =0
	Evaluate the curve and its derivatives.

virtual OptRect	boundsLocal (OptInterval const &i, unsigned deg) const =0

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.

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

virtual Coord	length (Coord tolerance=0.01) const
	Compute the arc length 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.

Protected Member Functions
bool	_equalTo (Curve const &c) const override

Private Member Functions
void	_updateCenterAndAngles ()

std::vector< ShapeIntersection >	_filterIntersections (std::vector< ShapeIntersection > &&xs, bool is_first) const
	Convert the passed intersections to curve time parametrization and filter out any invalid intersections.

bool	_validateIntersection (ShapeIntersection &xing, bool is_first) const
	Convert the passed intersection to curve time and check whether the intersection is numerically sane.

std::vector< ShapeIntersection >	_intersectSameEllipse (EllipticalArc const *other) const
	Check if two arcs on the same ellipse intersect/overlap.

Private Attributes
Point	_initial_point

Point	_final_point

Ellipse	_ellipse

AngleInterval	_angles

bool	_large_arc

Related Symbols
(Note that these are not member symbols.)
bool	arc_from_sbasis (EllipticalArc &ea, D2< SBasis > const &in, double tolerance=EPSILON, unsigned num_samples=20)
	Fit an elliptical arc to an SBasis fragment.

std::ostream &	operator<< (std::ostream &out, EllipticalArc const &ea)
	Debug output for elliptical arcs.

Detailed Description

Elliptical arc curve.

Elliptical arc is a curve taking the shape of a section of an ellipse.

The arc function has two forms: the regular one, mapping the unit interval to points on 2D plane (the linear domain), and a second form that maps some interval \(A \subseteq [0,2\pi)\) to the same points (the angular domain). The interval \(A\) determines which part of the ellipse forms the arc. The arc is said to contain an angle if its angular domain includes that angle (and therefore it is defined for that angle).

The angular domain considers each ellipse to be a rotated, scaled and translated unit circle: 0 corresponds to \((1,0)\) on the unit circle, \(\pi/2\) corresponds to \((0,1)\), \(\pi\) to \((-1,0)\) and \(3\pi/2\) to \((0,-1)\). After the angle is mapped to a point from a unit circle, the point is transformed using a matrix of this form

\[ M = \left[ \begin{array}{ccc} r_X \cos(\theta) & -r_Y \sin(\theta) & 0 \\ r_X \sin(\theta) & r_Y \cos(\theta) & 0 \\ c_X & c_Y & 1 \end{array} \right] \]

where \(r_X, r_Y\) are the X and Y rays of the ellipse, \(\theta\) is its angle of rotation, and \(c_X, c_Y\) the coordinates of the ellipse's center - thus mapping the angle to some point on the ellipse. Note that for example the point at angluar coordinate 0, the center and the point at angular coordinate \(\pi/4\) do not necessarily create an angle of \(\pi/4\) radians; it is only the case if both axes of the ellipse are of the same length (i.e. it is a circle).

An illustration of the angular domain

Each arc is defined by five variables: The initial and final point, the ellipse's rays, and the ellipse's rotation. Each set of those parameters corresponds to four different arcs, with two of them larger than half an ellipse and two of them turning clockwise while traveling from initial to final point. The two flags disambiguate between them: "large arc flag" selects the bigger arc, while the "sweep flag" selects the arc going in the direction of positive angles. Angles always increase when going from the +X axis in the direction of the +Y axis, so if Y grows downwards, this means clockwise.

Meaning of arc flags (Y grows downwards)

Definition at line 52 of file elliptical-arc.h.

Constructor & Destructor Documentation

◆ EllipticalArc() [1/3]

Geom::EllipticalArc::EllipticalArc ( )

inline

Creates an arc with all variables set to zero.

Definition at line 56 of file elliptical-arc.h.

Referenced by duplicate(), and portion().

◆ EllipticalArc() [2/3]

Geom::EllipticalArc::EllipticalArc	(	Point const &	ip,
		Point const &	r,
		Coord	rot_angle,
		bool	large_arc,
		bool	sweep,
		Point const &	fp
	)

inline

Create a new elliptical arc.

Parameters

ip	Initial point of the arc
r	Rays of the ellipse as a point
rot	Angle of rotation of the X axis of the ellipse in radians
large	If true, the large arc is chosen (always >= 180 degrees), otherwise the smaller arc is chosen
sweep	If true, the clockwise arc is chosen, otherwise the counter-clockwise arc is chosen
fp	Final point of the arc

Definition at line 70 of file elliptical-arc.h.

References _updateCenterAndAngles().

◆ EllipticalArc() [3/3]

Geom::EllipticalArc::EllipticalArc	(	Point const &	ip,
		Coord	rx,
		Coord	ry,
		Coord	rot_angle,
		bool	large_arc,
		bool	sweep,
		Point const &	fp
	)

inline

Create a new elliptical arc, giving the ellipse's rays as separate coordinates.

Definition at line 84 of file elliptical-arc.h.

References _updateCenterAndAngles().

Member Function Documentation

◆ _equalTo()

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

overrideprotectedvirtual

Implements Geom::Curve.

Definition at line 885 of file elliptical-arc.cpp.

References _final_point, _initial_point, _large_arc, c, rays(), rotationAngle(), and sweep().

◆ _filterIntersections()

std::vector< ShapeIntersection > Geom::EllipticalArc::_filterIntersections	(	std::vector< ShapeIntersection > &&	xs,
		bool	is_first
	)		const

private

Convert the passed intersections to curve time parametrization and filter out any invalid intersections.

Definition at line 604 of file elliptical-arc.cpp.

References _validateIntersection(), and result.

Referenced by intersect().

◆ _intersectSameEllipse()

std::vector< ShapeIntersection > Geom::EllipticalArc::_intersectSameEllipse ( EllipticalArc const * other ) const

private

Check if two arcs on the same ellipse intersect/overlap.

Parameters

other Another elliptical arc on the same ellipse as this one.

Returns: If the arcs overlap, the returned vector contains synthesized intersections at the start and end of the overlap. If the arcs do not overlap, an empty vector is returned.

A closure to create an "intersection" at the prescribed angle.

Definition at line 686 of file elliptical-arc.cpp.

References _angles, _ellipse, angularInterval(), Geom::AngleInterval::contains(), Geom::AngleInterval::finalAngle(), Geom::AngleInterval::initialAngle(), Geom::Ellipse::pointAt(), result, and timeAtAngle().

Referenced by intersect().

◆ _updateCenterAndAngles()

void Geom::EllipticalArc::_updateCenterAndAngles ( )

private

Definition at line 717 of file elliptical-arc.cpp.

References _angles, _ellipse, _large_arc, Geom::angle_between(), Geom::atan2(), c, finalPoint(), initialPoint(), Geom::Rotate::inverse(), isChord(), Geom::L2(), Geom::middle_point(), ray(), rays(), rotationAngle(), Geom::AngleInterval::setAngles(), Geom::Ellipse::setCenter(), Geom::AngleInterval::setFinal(), Geom::AngleInterval::setInitial(), Geom::Ellipse::setRays(), Geom::Ellipse::setRotationAngle(), Geom::AngleInterval::setSweep(), sweep(), Geom::X, and Geom::Y.

Referenced by EllipticalArc(), EllipticalArc(), set(), set(), setEndpoints(), setFinal(), and setInitial().

◆ _validateIntersection()

bool Geom::EllipticalArc::_validateIntersection	(	ShapeIntersection &	xing,
		bool	is_first
	)		const

private

Convert the passed intersection to curve time and check whether the intersection is numerically sane.

Parameters

xing	The intersection to convert to curve time and to validate.
is_first	If true, this arc is the first of the intersected curves; if false, it's second.

Returns: Whether the intersection is valid.

Note that the intersection is guaranteed to be converted only if the return value is true.

Definition at line 626 of file elliptical-arc.cpp.

References _ellipse, Geom::are_near_rel(), Geom::GenericInterval< C >::contains(), Geom::Intersection< TimeA, TimeB >::first, Geom::Intersection< TimeA, TimeB >::point(), Geom::Ellipse::pointAt(), pointAt(), Geom::Intersection< TimeA, TimeB >::second, and timeAtAngle().

Referenced by _filterIntersections().

◆ allNearestTimes()

std::vector< double > Geom::EllipticalArc::allNearestTimes	(	Point const &	p,
		double	from = `0`,
		double	to = `1`
	)		const

overridevirtual

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

Parameters

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

Returns: Vector of points closest and equally far away from the query point

Reimplemented from Geom::Curve.

Definition at line 373 of file elliptical-arc.cpp.

References Geom::are_near(), center(), Geom::distanceSq(), Geom::BezierCurveN< degree >::nearestTime(), Geom::BezierCurve::pointAt(), pointAt(), pointAtAngle(), ray(), result, roots(), rotationAngle(), Geom::sincos(), Geom::solve_reals(), timeAtAngle(), Geom::X, and Geom::Y.

Referenced by nearestTime().

◆ angleAt()

Angle Geom::EllipticalArc::angleAt ( Coord t ) const

inline

Compute the angular domain value corresponding to the given time value.

Definition at line 204 of file elliptical-arc.h.

References _angles, and Geom::AngleInterval::angleAt().

Referenced by pointAndDerivatives(), pointAt(), portion(), and valueAt().

◆ angularExtent()

Coord Geom::EllipticalArc::angularExtent ( ) const

inline

Get the elliptical angle spanned by the arc.

This is basically the absolute value of sweepAngle().

Definition at line 213 of file elliptical-arc.h.

References _angles, and Geom::AngleInterval::extent().

Referenced by portion().

◆ angularInterval()

AngleInterval Geom::EllipticalArc::angularInterval ( ) const

inline

Get the angular interval of the arc.

Definition at line 216 of file elliptical-arc.h.

References _angles.

Referenced by _intersectSameEllipse().

◆ boundsExact()

Rect Geom::EllipticalArc::boundsExact ( ) const

overridevirtual

Compute bounds of an elliptical arc.

The bounds computation works as follows. The extreme X and Y points are either the endpoints or local minima / maxima of the ellipse. We already have endpoints, and we compute the local extremes. The local extremes correspond to two angles separated by \(\pi\). Once we compute these angles, we check whether they belong to the arc, and if they do, we evaluate the ellipse at these angles. The bounding box of the arc is equal to the bounding box of the endpoints and the local extrema that belong to the arc.

Implements Geom::Curve.

Definition at line 106 of file elliptical-arc.cpp.

References _angles, _ellipse, _final_point, _initial_point, Geom::Ellipse::boundsExact(), Geom::AngleInterval::contains(), isChord(), Geom::AngleInterval::isFull(), result, unitCircleTransform(), Geom::X, and Geom::Y.

Referenced by boundsFast().

◆ boundsFast()

Rect Geom::EllipticalArc::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 269 of file elliptical-arc.h.

References boundsExact().

◆ boundsLocal()

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

inlineoverride

Definition at line 275 of file elliptical-arc.h.

References Geom::SBasisCurve::boundsLocal(), and toSBasis().

◆ center() [1/2]

Point Geom::EllipticalArc::center ( ) const

inline

Get the arc's center.

Returns: The arc's center, situated on the intersection of the ellipse's rays

Definition at line 107 of file elliptical-arc.h.

References _ellipse, and Geom::Ellipse::center().

Referenced by allNearestTimes(), nearestTime(), roots(), and toSBasis().

◆ center() [2/2]

Coord Geom::EllipticalArc::center ( Dim2 d ) const

inline

Get a coordinate of the elliptical arc's center.

Parameters

d	The dimension to retrieve

Returns: The selected coordinate of the center

Definition at line 103 of file elliptical-arc.h.

References _ellipse, and Geom::Ellipse::center().

◆ chord()

LineSegment Geom::EllipticalArc::chord ( ) const

inline

Get the line segment connecting the arc's endpoints.

Returns: A linear segment with initial and final point corresponding to those of the arc.

Definition at line 250 of file elliptical-arc.h.

References _final_point, and _initial_point.

Referenced by derivative(), isNear(), pointAndDerivatives(), pointAt(), roots(), toSBasis(), and valueAt().

◆ containsAngle()

bool Geom::EllipticalArc::containsAngle ( Angle angle ) const

inline

Check whether the arc contains the given angle.

Parameters

t	The angle to check

Returns: True if the arc contains the angle, false otherwise

Definition at line 187 of file elliptical-arc.h.

References _angles, and Geom::AngleInterval::contains().

◆ degreesOfFreedom()

int Geom::EllipticalArc::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 289 of file elliptical-arc.h.

◆ derivative()

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

overridevirtual

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.

Definition at line 261 of file elliptical-arc.cpp.

References _ellipse, chord(), Geom::BezierCurveN< degree >::derivative(), duplicate(), isChord(), result, and Geom::Ellipse::setCenter().

◆ duplicate()

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

inlineoverridevirtual

Create an exact copy of this curve.

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

Implements Geom::Curve.

Definition at line 256 of file elliptical-arc.h.

References EllipticalArc().

Referenced by derivative(), pointAndDerivatives(), portion(), reverse(), and TEST().

◆ expandToTransformed()

void Geom::EllipticalArc::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.

Definition at line 142 of file elliptical-arc.cpp.

References _angles, _final_point, Geom::AngleInterval::contains(), Geom::contains(), Geom::GenericRect< C >::expandTo(), isChord(), Geom::AngleInterval::isFull(), Geom::Curve::transform(), Geom::GenericRect< C >::unionWith(), unitCircleTransform(), Geom::X, and Geom::Y.

◆ feed()

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

overridevirtual

Feed the curve to a PathSink.

Reimplemented from Geom::Curve.

Definition at line 920 of file elliptical-arc.cpp.

References _final_point, _initial_point, _large_arc, Geom::PathSink::arcTo(), Geom::PathSink::moveTo(), ray(), rotationAngle(), sweep(), Geom::X, and Geom::Y.

◆ finalAngle()

Angle Geom::EllipticalArc::finalAngle ( ) const

inline

Definition at line 134 of file elliptical-arc.h.

References _angles, and Geom::AngleInterval::finalAngle().

Referenced by Geom::CairoPathSink::arcTo(), and winding().

◆ finalPoint()

Point Geom::EllipticalArc::finalPoint ( ) const

inlineoverridevirtual

Retrieve the end of the curve.

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

Implements Geom::Curve.

Definition at line 255 of file elliptical-arc.h.

References _final_point.

Referenced by _updateCenterAndAngles(), operator<<(), pointAt(), TEST(), and toSBasis().

◆ initialAngle()

Angle Geom::EllipticalArc::initialAngle ( ) const

inline

Definition at line 133 of file elliptical-arc.h.

References _angles, and Geom::AngleInterval::initialAngle().

Referenced by Geom::CairoPathSink::arcTo(), toSBasis(), and winding().

◆ initialPoint()

Point Geom::EllipticalArc::initialPoint ( ) const

inlineoverridevirtual

Retrieve the start of the curve.

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

Implements Geom::Curve.

Definition at line 254 of file elliptical-arc.h.

References _initial_point.

Referenced by _updateCenterAndAngles(), operator<<(), pointAt(), TEST(), and toSBasis().

◆ intersect()

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

overridevirtual

Compute intersections with another curve.

Reimplemented from Geom::Curve.

Definition at line 646 of file elliptical-arc.cpp.

References _ellipse, _filterIntersections(), _final_point, _initial_point, _intersectSameEllipse(), Geom::crossings(), Geom::Curve::finalPoint(), Geom::Curve::initialPoint(), Geom::Curve::intersect(), Geom::BezierCurveN< degree >::intersect(), Geom::Ellipse::intersect(), Geom::Curve::isLineSegment(), isLineSegment(), result, and Geom::transpose_in_place().

Referenced by TEST(), TEST(), and TEST().

◆ inverseUnitCircleTransform()

Affine Geom::EllipticalArc::inverseUnitCircleTransform ( ) const

inline

Compute a transform that maps the arc's ellipse to the unit circle.

Definition at line 234 of file elliptical-arc.h.

References _ellipse, Geom::Ellipse::inverseUnitCircleTransform(), and result.

◆ isChord()

bool Geom::EllipticalArc::isChord ( ) const

inline

Check whether both rays are nonzero.

If they are not, the arc is represented as a line segment instead.

Definition at line 244 of file elliptical-arc.h.

References ray(), Geom::X, and Geom::Y.

Referenced by _updateCenterAndAngles(), boundsExact(), derivative(), expandToTransformed(), isLineSegment(), isNear(), operator*=(), pointAndDerivatives(), pointAt(), roots(), toSBasis(), and valueAt().

◆ isDegenerate()

bool Geom::EllipticalArc::isDegenerate ( ) const

inlineoverridevirtual

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.

Definition at line 265 of file elliptical-arc.h.

References _final_point, and _initial_point.

◆ isLineSegment()

bool Geom::EllipticalArc::isLineSegment ( ) const

inlineoverridevirtual

Check whether the curve is a line segment.

Reimplemented from Geom::Curve.

Definition at line 268 of file elliptical-arc.h.

References isChord().

Referenced by intersect().

◆ isNear()

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

overridevirtual

Test whether two curves are approximately the same.

Implements Geom::Curve.

Definition at line 901 of file elliptical-arc.cpp.

References _ellipse, _final_point, _initial_point, Geom::are_near(), c, chord(), isChord(), isNear(), and sweep().

Referenced by isNear().

◆ largeArc()

bool Geom::EllipticalArc::largeArc ( ) const

inline

Whether the arc is larger than half an ellipse.

Returns: True if the arc is larger than \(\pi\), false otherwise

Definition at line 126 of file elliptical-arc.h.

References _large_arc.

Referenced by operator<<(), and winding().

◆ nearestTime()

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

inlineoverridevirtual

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.

Definition at line 281 of file elliptical-arc.h.

References allNearestTimes(), Geom::are_near(), center(), ray(), Geom::X, and Geom::Y.

◆ operator*=() [1/12]

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

overridevirtual

Implements Geom::Curve.

Definition at line 860 of file elliptical-arc.cpp.

References _angles, _ellipse, _final_point, _initial_point, Geom::Affine::det(), isChord(), Geom::middle_point(), Geom::Ellipse::setCenter(), Geom::AngleInterval::setFinal(), Geom::AngleInterval::setInitial(), Geom::Ellipse::setRays(), Geom::Ellipse::setRotationAngle(), Geom::AngleInterval::setSweep(), sweep(), and Geom::Ellipse::timeAt().

◆ operator*=() [2/12]

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

virtual

Implements Geom::Curve.

◆ operator*=() [3/12]

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

inlinevirtual

Reimplemented from Geom::Curve.

Definition at line 200 of file curve.h.

◆ operator*=() [4/12]

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

inlinevirtual

Reimplemented from Geom::Curve.

Definition at line 199 of file curve.h.

◆ operator*=() [5/12]

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

overridevirtual

Reimplemented from Geom::Curve.

Definition at line 846 of file elliptical-arc.cpp.

References _ellipse, _final_point, and _initial_point.

◆ operator*=() [6/12]

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

inlinevirtual

Reimplemented from Geom::Curve.

Definition at line 198 of file curve.h.

◆ operator*=() [7/12]

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

overridevirtual

Reimplemented from Geom::Curve.

Definition at line 839 of file elliptical-arc.cpp.

References _ellipse, _final_point, and _initial_point.

◆ operator*=() [8/12]

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

inlinevirtual

Reimplemented from Geom::Curve.

Definition at line 197 of file curve.h.

◆ operator*=() [9/12]

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

overridevirtual

Reimplemented from Geom::Curve.

Definition at line 832 of file elliptical-arc.cpp.

References _ellipse, _final_point, and _initial_point.

◆ operator*=() [10/12]

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

inlinevirtual

Reimplemented from Geom::Curve.

Definition at line 201 of file curve.h.

◆ operator*=() [11/12]

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

inlinevirtual

Reimplemented from Geom::Curve.

Definition at line 202 of file curve.h.

◆ operator*=() [12/12]

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

overridevirtual

Reimplemented from Geom::Curve.

Definition at line 853 of file elliptical-arc.cpp.

References _ellipse, _final_point, and _initial_point.

◆ pointAndDerivatives()

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

override

Definition at line 278 of file elliptical-arc.cpp.

References angleAt(), chord(), duplicate(), isChord(), Geom::BezierCurve::pointAndDerivatives(), pointAtAngle(), result, and sweep().

◆ pointAt()

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

overridevirtual

Evaluate the arc in the curve domain, i.e. \([0, 1]\).

Reimplemented from Geom::Curve.

Definition at line 313 of file elliptical-arc.cpp.

References _ellipse, angleAt(), chord(), finalPoint(), initialPoint(), isChord(), Geom::Ellipse::pointAt(), and Geom::BezierCurve::pointAt().

Referenced by _validateIntersection(), allNearestTimes(), portion(), TEST(), and TEST().

◆ pointAtAngle()

Point Geom::EllipticalArc::pointAtAngle ( Coord t ) const

Evaluate the arc at the specified angular coordinate.

Parameters

t Angle

Returns: Point corresponding to the given angle

Definition at line 173 of file elliptical-arc.cpp.

References _ellipse, and Geom::Ellipse::pointAt().

Referenced by allNearestTimes(), pointAndDerivatives(), TEST(), and winding().

◆ portion()

Curve * Geom::EllipticalArc::portion	(	double	a,
		double	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.

Definition at line 333 of file elliptical-arc.cpp.

References _angles, _final_point, _initial_point, _large_arc, angleAt(), angularExtent(), duplicate(), EllipticalArc(), pointAt(), reverse(), Geom::AngleInterval::setAngles(), Geom::AngleInterval::setSweep(), and sweep().

◆ ray()

Coord Geom::EllipticalArc::ray ( Dim2 d ) const

inline

Get one of the ellipse's rays.

Parameters

d	Dimension to retrieve

Returns: The selected ray of the ellipse

Definition at line 112 of file elliptical-arc.h.

References _ellipse, and Geom::Ellipse::ray().

Referenced by _updateCenterAndAngles(), allNearestTimes(), feed(), isChord(), nearestTime(), operator<<(), roots(), toSBasis(), and winding().

◆ rays()

Point Geom::EllipticalArc::rays ( ) const

inline

Get both rays as a point.

Returns: Point with X equal to the X ray and Y to Y ray

Definition at line 116 of file elliptical-arc.h.

References _ellipse, and Geom::Ellipse::rays().

Referenced by _equalTo(), and _updateCenterAndAngles().

◆ reverse()

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

overridevirtual

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.

Definition at line 363 of file elliptical-arc.cpp.

References _angles, _final_point, _initial_point, duplicate(), and Geom::AngleInterval::reverse().

Referenced by portion().

◆ roots()

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

overridevirtual

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 184 of file elliptical-arc.cpp.

References c, center(), chord(), Geom::GenericInterval< C >::contains(), delta, isChord(), ray(), Geom::BezierCurve::roots(), rotationAngle(), Geom::sincos(), timeAtAngle(), Geom::X, and Geom::Y.

Referenced by allNearestTimes().

◆ rotationAngle()

Angle Geom::EllipticalArc::rotationAngle ( ) const

inline

Get the defining ellipse's rotation.

Returns: Angle between the +X ray of the ellipse and the +X axis

Definition at line 120 of file elliptical-arc.h.

References _ellipse, and Geom::Ellipse::rotationAngle().

Referenced by _equalTo(), _updateCenterAndAngles(), allNearestTimes(), feed(), operator<<(), roots(), toSBasis(), and winding().

◆ set() [1/2]

void Geom::EllipticalArc::set	(	Point const &	ip,
		double	rx,
		double	ry,
		double	rot_angle,
		bool	large_arc,
		bool	sweep,
		Point const &	fp
	)

inline

Change all of the arc's parameters.

Definition at line 141 of file elliptical-arc.h.

References _angles, _ellipse, _final_point, _initial_point, _large_arc, _updateCenterAndAngles(), Geom::Ellipse::setRays(), Geom::Ellipse::setRotationAngle(), Geom::AngleInterval::setSweep(), and sweep().

Referenced by Geom::make_elliptical_arc().

◆ set() [2/2]

void Geom::EllipticalArc::set	(	Point const &	ip,
		Point const &	r,
		Angle	rot_angle,
		bool	large_arc,
		bool	sweep,
		Point const &	fp
	)

inline

Change all of the arc's parameters.

Definition at line 156 of file elliptical-arc.h.

◆ setEndpoints()

void Geom::EllipticalArc::setEndpoints	(	Point const &	ip,
		Point const &	fp
	)

inline

Change the initial and final point in one operation.

This method exists because modifying any of the endpoints causes rather costly recalculations of the center and extreme angles.

Parameters

ip	New initial point
fp	New final point

Definition at line 175 of file elliptical-arc.h.

References _final_point, _initial_point, and _updateCenterAndAngles().

◆ setFinal()

void Geom::EllipticalArc::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 261 of file elliptical-arc.h.

References _final_point, and _updateCenterAndAngles().

◆ setInitial()

void Geom::EllipticalArc::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 257 of file elliptical-arc.h.

References _initial_point, and _updateCenterAndAngles().

◆ sweep()

bool Geom::EllipticalArc::sweep ( ) const

inline

Whether the arc turns clockwise.

Returns: True if the arc makes a clockwise turn when going from initial to final point, false otherwise

Definition at line 131 of file elliptical-arc.h.

References _angles, and Geom::AngleInterval::sweep().

Referenced by _equalTo(), _updateCenterAndAngles(), feed(), isNear(), operator*=(), operator<<(), pointAndDerivatives(), portion(), set(), set(), and winding().

◆ sweepAngle()

Coord Geom::EllipticalArc::sweepAngle ( ) const

inline

Compute the amount by which the angle parameter changes going from start to end.

This has range \((-2\pi, 2\pi)\) and thus cannot be represented as instance of the class Angle. Add this to the initial angle to obtain the final angle.

Definition at line 209 of file elliptical-arc.h.

References _angles, and Geom::AngleInterval::sweepAngle().

Referenced by toSBasis().

◆ timeAtAngle()

Coord Geom::EllipticalArc::timeAtAngle ( Angle a ) const

inline

Compute the curve time value corresponding to the given angular value.

Definition at line 201 of file elliptical-arc.h.

References _angles, and Geom::AngleInterval::timeAtAngle().

Referenced by _intersectSameEllipse(), _validateIntersection(), allNearestTimes(), and roots().

◆ toSBasis()

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

overridevirtual

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 800 of file elliptical-arc.cpp.

References center(), chord(), Geom::cos(), finalPoint(), initialAngle(), initialPoint(), isChord(), Geom::Angle::radians(), ray(), rotationAngle(), Geom::sin(), Geom::sincos(), sweepAngle(), Geom::BezierCurve::toSBasis(), Geom::X, and Geom::Y.

Referenced by boundsLocal(), and Inkscape::LivePathEffect::path_from_piecewise_fix_cusps().

◆ unitCircleTransform()

Affine Geom::EllipticalArc::unitCircleTransform ( ) const

inline

Compute a transform that maps the unit circle to the arc's ellipse.

Each ellipse can be interpreted as a translated, scaled and rotate unit circle. This function returns the transform that maps the unit circle to the arc's ellipse.

Returns: Transform from unit circle to the arc's ellipse

Definition at line 228 of file elliptical-arc.h.

References _ellipse, result, and Geom::Ellipse::unitCircleTransform().

Referenced by Geom::CairoPathSink::arcTo(), boundsExact(), and expandToTransformed().

◆ valueAt()

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

overridevirtual

Evaluate a single coordinate on the arc in the curve domain.

Reimplemented from Geom::Curve.

Definition at line 327 of file elliptical-arc.cpp.

References angleAt(), chord(), isChord(), Geom::BezierCurve::valueAt(), and valueAtAngle().

◆ valueAtAngle()

Coord Geom::EllipticalArc::valueAtAngle	(	Coord	t,
		Dim2	d
	)		const

Evaluate one of the arc's coordinates at the specified angle.

Parameters

t	Angle
d	The dimension to retrieve

Returns: Selected coordinate of the arc at the specified angle

Definition at line 179 of file elliptical-arc.cpp.

References _ellipse, and Geom::Ellipse::valueAt().

Referenced by valueAt().

◆ winding()

int Geom::EllipticalArc::winding ( Point const & p ) const

overridevirtual

Compute the partial winding number of this curve.

The partial winding number is equal to the difference between the number of roots at which the curve goes in the +Y direction and the number of roots at which the curve goes in the -Y direction. This method is mainly useful for implementing path winding calculation. It will ignore roots which are local maxima on the Y axis.

Parameters

p	Point where the winding number should be determined

Returns: Winding number contribution at p

Reimplemented from Geom::Curve.

Definition at line 928 of file elliptical-arc.cpp.

References _angles, _ellipse, _final_point, _initial_point, Geom::AngleInterval::contains(), Geom::Ellipse::contains(), Geom::cross(), finalAngle(), initialAngle(), Geom::AngleInterval::isFull(), largeArc(), Geom::Interval::lowerContains(), pointAtAngle(), ray(), rotationAngle(), Geom::sincos(), sweep(), Geom::X, and Geom::Y.

Friends And Related Symbol Documentation

◆ arc_from_sbasis()

bool arc_from_sbasis	(	EllipticalArc &	ea,
		D2< SBasis > const &	in,
		double	tolerance = `EPSILON`,
		unsigned	num_samples = `20`
	)

Fit an elliptical arc to an SBasis fragment.

Definition at line 323 of file elliptical-arc-from-sbasis.cpp.

References Geom::arc_from_sbasis(), and Geom::make_elliptical_arc().

◆ operator<<()

std::ostream & operator<<	(	std::ostream &	out,
		EllipticalArc const &	ea
	)

Debug output for elliptical arcs.

Definition at line 1023 of file elliptical-arc.cpp.

References finalPoint(), Geom::format_coord_nice(), initialPoint(), largeArc(), Geom::operator<<(), ray(), rotationAngle(), sweep(), Geom::X, and Geom::Y.

Member Data Documentation

◆ _angles

AngleInterval Geom::EllipticalArc::_angles

private

Definition at line 318 of file elliptical-arc.h.

Referenced by _intersectSameEllipse(), _updateCenterAndAngles(), angleAt(), angularExtent(), angularInterval(), boundsExact(), containsAngle(), expandToTransformed(), finalAngle(), initialAngle(), operator*=(), portion(), reverse(), set(), set(), sweep(), sweepAngle(), timeAtAngle(), and winding().

◆ _ellipse

Ellipse Geom::EllipticalArc::_ellipse

private

Definition at line 317 of file elliptical-arc.h.

Referenced by _intersectSameEllipse(), _updateCenterAndAngles(), _validateIntersection(), boundsExact(), center(), center(), derivative(), intersect(), inverseUnitCircleTransform(), isNear(), operator*=(), operator*=(), operator*=(), operator*=(), operator*=(), pointAt(), pointAtAngle(), ray(), rays(), rotationAngle(), set(), set(), unitCircleTransform(), valueAtAngle(), and winding().

◆ _final_point

Point Geom::EllipticalArc::_final_point

private

Definition at line 316 of file elliptical-arc.h.

Referenced by _equalTo(), boundsExact(), chord(), expandToTransformed(), feed(), finalPoint(), intersect(), isDegenerate(), isNear(), operator*=(), operator*=(), operator*=(), operator*=(), operator*=(), portion(), reverse(), set(), set(), setEndpoints(), setFinal(), and winding().

◆ _initial_point

Point Geom::EllipticalArc::_initial_point

private

Definition at line 316 of file elliptical-arc.h.

Referenced by _equalTo(), boundsExact(), chord(), feed(), initialPoint(), intersect(), isDegenerate(), isNear(), operator*=(), operator*=(), operator*=(), operator*=(), operator*=(), portion(), reverse(), set(), set(), setEndpoints(), setInitial(), and winding().

◆ _large_arc

bool Geom::EllipticalArc::_large_arc

private

Definition at line 319 of file elliptical-arc.h.

Referenced by _equalTo(), _updateCenterAndAngles(), feed(), largeArc(), portion(), set(), and set().

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

/builds/inkscape/inkscape/src/3rdparty/2geom/include/2geom/elliptical-arc.h
/builds/inkscape/inkscape/src/3rdparty/2geom/src/2geom/elliptical-arc.cpp

Public Member Functions

Protected Member Functions

Private Member Functions

Private Attributes

Related Symbols

Detailed Description

Constructor & Destructor Documentation

◆ EllipticalArc() [1/3]

◆ EllipticalArc() [2/3]

◆ EllipticalArc() [3/3]

Member Function Documentation

◆ _equalTo()

◆ _filterIntersections()

◆ _intersectSameEllipse()

◆ _updateCenterAndAngles()

◆ _validateIntersection()

◆ allNearestTimes()

◆ angleAt()

◆ angularExtent()

◆ angularInterval()

◆ boundsExact()

◆ boundsFast()

◆ boundsLocal()

◆ center() [1/2]

◆ center() [2/2]

◆ chord()

◆ containsAngle()

◆ degreesOfFreedom()

◆ derivative()

◆ duplicate()

◆ expandToTransformed()

◆ feed()

◆ finalAngle()

◆ finalPoint()

◆ initialAngle()

◆ initialPoint()

◆ intersect()

◆ inverseUnitCircleTransform()

◆ isChord()

◆ isDegenerate()

◆ isLineSegment()

◆ isNear()

◆ largeArc()

◆ nearestTime()

◆ operator*=() [1/12]

◆ operator*=() [2/12]

◆ operator*=() [3/12]

◆ operator*=() [4/12]

◆ operator*=() [5/12]

◆ operator*=() [6/12]

◆ operator*=() [7/12]

◆ operator*=() [8/12]

◆ operator*=() [9/12]

◆ operator*=() [10/12]

◆ operator*=() [11/12]

◆ operator*=() [12/12]

◆ pointAndDerivatives()

◆ pointAt()

◆ pointAtAngle()

◆ portion()

◆ ray()

◆ rays()

◆ reverse()

◆ roots()

◆ rotationAngle()

◆ set() [1/2]

◆ set() [2/2]

◆ setEndpoints()

◆ setFinal()

◆ setInitial()

◆ sweep()

◆ sweepAngle()

◆ timeAtAngle()

◆ toSBasis()

◆ unitCircleTransform()

◆ valueAt()

◆ valueAtAngle()

◆ winding()

Friends And Related Symbol Documentation

◆ arc_from_sbasis()