Cubic curves in the plane. More...

#include <curve.h>

Public Member Functions
	CubicCurve2 ()
	Empty.

	CubicCurve2 (const Cubic &, const Cubic &)
	Creates a cubic curve. More...

	CubicCurve2 (const CubicCurve &)
	Creates a cubic curve. More...

	~CubicCurve2 ()
	Empty.

Cubic &	operator[] (int)
	Access curve components.

Cubic	operator[] (int) const
	Access curve components.

CubicCurve	ToCubicCurve (const Cubic &=Cubic(0.0, 0.0, 0.0, 0.0)) const
	Convert a planar cubic curve to a cubic curve. More...

CubicCurve2	operator- () const
	Negative curve unary operator.

QuadricCurve2	Tangent () const
	Computes the quadric curve whose components are the derivatives of the curve. More...

Vector2	Tangent (const double &) const
	Computes the tangent to the cubic curve at a given position on the curve. More...

Vector2	Normal (const double &) const
	Computes the normal vector to the cubic curve at a given position on the curve. More...

double	S (const double &, int=256) const
	Compute the curvilign absisca of a point on the curve. More...

double	U (const double &, int=256) const
	Compute the parameter corresponding to the argument curvilign absisca of a point on the curve. More...

int	Intersect (const Ray2 &, double[3]) const
	Compute the intersection between the curve and a ray. More...

double	Length (int=256) const
	Computes the length of the cubic curve. Integration is performed over interval [0,1]. More...

double	Length (const double &, const double &, int=256) const
	Computes the length of the cubic curve over interval [a,b]. More...

double	InverseArcLength (const double &, int=256) const
	Compute the parameter t corresponding to a given input arc length. More...

Vector2	operator() (const double &) const
	Evaluates curve point at a given location. More...

Vector2	Eval (const double &) const
	Evaluates curve point at a given location. More...

double	Curvature (const double &) const
	Computes the signed curvature of the curve at a given point p(t). More...

double	Curvature (const double &, const double &) const
	Compute the maximum curvature over a given interval. More...

Box2	GetBox () const
	Compute the bounding box of the cubic curve with parameter interval [0,1].

double	R (const Vector2 &, double &) const
	Compute the squared distance between a point in the plane and the cubic curve. More...

Vector2	Normal (const Vector2 &, double &) const
	Compute the projection of a point on the curve. More...

double	UV (const Vector2 &, double &, double &) const
	Compute the inverse mapping for a given input point. More...

Vector2	UV (const Vector2 &) const
	Compute the inverse mapping for a given input point. More...

Matrix2	GetMatrix (const double &) const
	Compute the Frenet vectors (tangent and normal) at a given point on the curve. More...

Frame2	GetFrame (const double &) const
	Compute the frame at a given point on the curve. More...

Vector2	BezierControl (int) const
	Compute the i-th B�zier control point of the curve. More...

void	Draw (QGraphicsScene &, const QPen &=QPen(), const QBrush &=QBrush()) const
	Draw a cubic curve. More...

Static Public Member Functions
static CubicCurve2	Hermite (const Vector2 &, const Vector2 &, const Vector2 &, const Vector2 &)
	Creates an Hermite cubic curve on interval [0,1] given vertex locations and tangent vectors (in that order). More...

static CubicCurve2	Bezier (const Vector2 &, const Vector2 &, const Vector2 &, const Vector2 &)
	Creates a Bezier cubic curve on interval [0,1] given four vertex locations. More...

Protected Attributes
Cubic	y
	Cubic polynomial functions for every coordinate.

Friends
std::ostream &	operator<< (std::ostream &, const CubicCurve2 &)
	Overloaded. More...

Detailed Description

Cubic curves in the plane.

Constructor & Destructor Documentation

◆ CubicCurve2() [1/2]

CubicCurve2::CubicCurve2	(	const Cubic &	x,
		const Cubic &	y
	)

explicit

Creates a cubic curve.

Parameters

x,y	Cubic parametric equations of the corresponding x, y coordinates.

◆ CubicCurve2() [2/2]

CubicCurve2::CubicCurve2 ( const CubicCurve & c )

explicit

Creates a cubic curve.

Parameters

c	Cubic curve.

Member Function Documentation

◆ Bezier()

CubicCurve2 CubicCurve2::Bezier	(	const Vector2 &	a,
		const Vector2 &	b,
		const Vector2 &	c,
		const Vector2 &	d
	)

static

Creates a Bezier cubic curve on interval [0,1] given four vertex locations.

Parameters

a,b,c,d Vertices of the Bezier curve.

◆ BezierControl()

Vector2 CubicCurve2::BezierControl ( int i ) const

inline

Compute the i-th B�zier control point of the curve.

Computations have been optimized.

Parameters

i Index.

◆ Curvature() [1/2]

double CubicCurve2::Curvature ( const double & t ) const

Computes the signed curvature of the curve at a given point p(t).

Parameters

t	Parameter.

◆ Curvature() [2/2]

double CubicCurve2::Curvature	(	const double &	a,
		const double &	b
	)		const

Compute the maximum curvature over a given interval.

Compute the analytic equation of the numerator of the curvature expression, search the roots, find extrema and returns the maximum.

Parameters

a,b Interval

◆ Draw()

void CubicCurve2::Draw	(	QGraphicsScene &	scene,
		const QPen &	pen = `QPen()`,
		const QBrush &	= `QBrush()`
	)		const

Draw a cubic curve.

The brush is not used.

Parameters

scene	Graphics scene.
pen	The pen.

◆ Eval()

Vector2 CubicCurve2::Eval ( const double & t ) const

Evaluates curve point at a given location.

Parameters

t	Parameter.

◆ GetFrame()

Frame2 CubicCurve2::GetFrame ( const double & u ) const

Compute the frame at a given point on the curve.

Parameters

u	Parameter.

◆ GetMatrix()

Matrix2 CubicCurve2::GetMatrix ( const double & u ) const

Compute the Frenet vectors (tangent and normal) at a given point on the curve.

Parameters

u	Parameter.

◆ Hermite()

CubicCurve2 CubicCurve2::Hermite	(	const Vector2 &	a,
		const Vector2 &	b,
		const Vector2 &	ta,
		const Vector2 &	tb
	)

static

Creates an Hermite cubic curve on interval [0,1] given vertex locations and tangent vectors (in that order).

Parameters

a,b	End vertices of the Hermite cubic curve.
ta,tb	Tangent vectors at the end vertices.

◆ Intersect()

int CubicCurve2::Intersect	(	const Ray2 &	ray,
		double	t[3]
	)		const

Compute the intersection between the curve and a ray.

Parameters

ray	The ray.
t	Intersection depths.

◆ InverseArcLength()

double CubicCurve2::InverseArcLength	(	const double &	s,
		int	n = `256`
	)		const

Compute the parameter t corresponding to a given input arc length.

Parameters

s	Arc length, which should be positive.
n	Discretization of the curve.

◆ Length() [1/2]

double CubicCurve2::Length	(	const double &	a,
		const double &	b,
		int	n = `256`
	)		const

Computes the length of the cubic curve over interval [a,b].

Since we have the parametric equations of the coordinates of the points on the curve, then the length is the integral of the square root of the sum of the squared derivatives where the limits of integration are [a,b].

In the general case, the function we end up with is not integrable in closed form, leading to an elliptic integral or some such. Thus we evaluate the integral numerically.

Parameters

a,b	Interval.
n	Discretization of the integration interval.

◆ Length() [2/2]

double CubicCurve2::Length ( int n = 256 ) const

Computes the length of the cubic curve. Integration is performed over interval [0,1].

Simply call CubicCurve::Length(0.0,1.0,n) with n as the passed argument parameter.

Parameters

n	Discretization of the curve.

◆ Normal() [1/2]

Vector2 CubicCurve2::Normal ( const double & t ) const

Computes the normal vector to the cubic curve at a given position on the curve.

The normal is the obtained by evaluating the second derivative.

Parameters

t Position.

◆ Normal() [2/2]

Vector2 CubicCurve2::Normal	(	const Vector2 &	p,
		double &	u
	)		const

Compute the projection of a point on the curve.

Parameters

p	Point.
u	Parameter defining the projected point.

◆ operator()()

Vector2 CubicCurve2::operator() ( const double & t ) const

Evaluates curve point at a given location.

Parameters

t	Parameter.

◆ R()

double CubicCurve2::R	(	const Vector2 &	p,
		double &	u
	)		const

Compute the squared distance between a point in the plane and the cubic curve.

This function computes the projection of a point p onto the curve by minimizing the distance function along the curve, expressed as a degree 5 polynomial.

Most of the coefficients of this polynomial are constant, and could be pre-processed in the constructor.

Parameters

p	Point.
u	Parameter defining the coordinate of the projection of the argument vertex onto the curve.

◆ S()

double CubicCurve2::S	(	const double &	t,
		int	n = `256`
	)		const

Compute the curvilign absisca of a point on the curve.

Parameters

t	Parameter that should be within [0,1] interval.
n	Discretization of the integration interval.

◆ Tangent() [1/2]

QuadricCurve2 CubicCurve2::Tangent ( ) const

Computes the quadric curve whose components are the derivatives of the curve.

This function is useful for preprocessing tangent computations.

◆ Tangent() [2/2]

Vector2 CubicCurve2::Tangent ( const double & t ) const

Computes the tangent to the cubic curve at a given position on the curve.

The tangent is the obtained by evaluating the first derivative.

Parameters

t Position.

◆ ToCubicCurve()

CubicCurve CubicCurve2::ToCubicCurve ( const Cubic & z = Cubic(0.0, 0.0, 0.0, 0.0) ) const

inline

Convert a planar cubic curve to a cubic curve.

This is a convenience function, which is basically the same as:

CubicCurve2 c; // Curve in the plane
Cubic z;       // Cubic for the z coordinate
CubicCurve3 c3=CubicCurve(c[0],c[1],z);

Parameters

z	Missing cubic.

◆ U()

double CubicCurve2::U	(	const double &	s,
		int	n = `256`
	)		const

Compute the parameter corresponding to the argument curvilign absisca of a point on the curve.

Parameters

s	Argument curvilign absisca (should be between 0.0 and the length of the curve).
n	Discretization of the integration interval.

◆ UV() [1/2]

Vector2 CubicCurve2::UV ( const Vector2 & p ) const

Compute the inverse mapping for a given input point.

See also: CubicCurve2::UV(const Vector2&, double&, double&)

Parameters

p Point.

Returns: Inverse uv coordinates compacted into a Vector2.

◆ UV() [2/2]

double CubicCurve2::UV	(	const Vector2 &	p,
		double &	u,
		double &	v
	)		const

Compute the inverse mapping for a given input point.

Parameters

p	Point.
u,v	Inverse Coordinates, u will denote the position along the curve and v the signed distance to the curve.

Returns: Squared distance to the curve.

Friends And Related Function Documentation

◆ operator<<

std::ostream & operator<<	(	std::ostream &	s,
		const CubicCurve2 &	curve
	)

friend

Overloaded.

Parameters

s	Stream.
curve	The cubic curve.

Public Member Functions

Static Public Member Functions

Protected Attributes

Friends

Detailed Description

Constructor & Destructor Documentation

◆ CubicCurve2() [1/2]

◆ CubicCurve2() [2/2]

Member Function Documentation

◆ Bezier()

◆ BezierControl()

◆ Curvature() [1/2]

◆ Curvature() [2/2]

◆ Draw()

◆ Eval()

◆ GetFrame()

◆ GetMatrix()

◆ Hermite()

◆ Intersect()

◆ InverseArcLength()

◆ Length() [1/2]

◆ Length() [2/2]

◆ Normal() [1/2]

◆ Normal() [2/2]

◆ operator()()

◆ R()

◆ S()

◆ Tangent() [1/2]

◆ Tangent() [2/2]

◆ ToCubicCurve()

◆ U()

◆ UV() [1/2]

◆ UV() [2/2]

Friends And Related Function Documentation

◆ operator<<