Quadric curves in the plane. More...

#include <curve.h>

Public Member Functions
	QuadricCurve2 ()
	Empty.

	QuadricCurve2 (const Quadric &, const Quadric &)
	Creates a quadric curve. More...

	QuadricCurve2 (const QuadricCurve &)
	Creates a quadric curve in the plane. More...

	~QuadricCurve2 ()
	Empty.

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

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

QuadricCurve2	operator- () const
	Negates the coefficients of a quadric curve.

Vector2	operator() (const double &) const
	Computes the point on the curve. More...

Vector2	Eval (const double &) const
	Computes the point on the curve. More...

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

Vector2	Tangent (const double &) const
	Computes the tangent to the curve. More...

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

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

void	Translate (const Vector2 &)
	Translate the curve. More...

double	R (const Vector2 &, double &) const
	Compute the squared distance between a point and the quadric 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...

double	LengthIntegral (const double &=0.0, const double &=1.0) const
	Computes the length of the quadric curve over interval [a,b] using a closed form integral. More...

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

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

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

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

int	Intersect (const Ray2 &, double[2]) const
	Compute the intersection between the curve and a ray. More...

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

Static Public Member Functions
static QuadricCurve2	Bezier (const Vector2 &, const Vector2 &, const Vector2 &)
	Creates a quadratic Bezier curve on interval [0,1]. More...

static QVector< QuadricCurve2 >	Bezier (const QVector< Vector2 > &)
	Create a set of quadratic Bezier curves. More...

Protected Attributes
Quadric	y
	Quadric functions for every coordinate.

Friends
std::ostream &	operator<< (std::ostream &, const QuadricCurve2 &)
	Overloaded. More...

Detailed Description

Quadric curves in the plane.

Quadric curves are implemented with two quadric polynomials defining the coordinates of the points on the curve. It is a minimum storage class in the sense that it does not keep extra constant data which may be useful for computing the bounding box of the quadric curve parameterized over the unit interval [0,1] or the distance between a point an the curve.

Constructor & Destructor Documentation

◆ QuadricCurve2() [1/2]

QuadricCurve2::QuadricCurve2	(	const Quadric &	x,
		const Quadric &	y
	)

explicit

Creates a quadric curve.

Parameters

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

◆ QuadricCurve2() [2/2]

QuadricCurve2::QuadricCurve2 ( const QuadricCurve & c )

explicit

Creates a quadric curve in the plane.

Parameters

c	Quadric curve.

Member Function Documentation

◆ Bezier() [1/2]

QVector< QuadricCurve2 > QuadricCurve2::Bezier ( const QVector< Vector2 > & c )

static

Create a set of quadratic Bezier curves.

Parameters

c	Control points of the Bezier curve.

◆ Bezier() [2/2]

QuadricCurve2 QuadricCurve2::Bezier	(	const Vector2 &	a,
		const Vector2 &	b,
		const Vector2 &	c
	)

static

Creates a quadratic Bezier curve on interval [0,1].

Parameters

a,b,c The control points of the Bezier curve.

◆ Curvature()

double QuadricCurve2::Curvature ( const double & t ) const

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

Parameters

t	Parameter.

◆ Draw()

void QuadricCurve2::Draw	(	QGraphicsScene &	scene,
		const QPen &	pen = `QPen()`,
		const QBrush &	brush = `QBrush()`
	)		const

Draw a quadric curve.

Parameters

scene	Graphics scene.
pen	The pen.
brush	The brush, should the triangle be filled.

◆ Eval()

Vector2 QuadricCurve2::Eval ( const double & t ) const

Computes the point on the curve.

Parameters

t	Parameter.

◆ Intersect()

int QuadricCurve2::Intersect	(	const Ray2 &	ray,
		double	t[2]
	)		const

Compute the intersection between the curve and a ray.

Parameters

ray	The ray.
t	Intersection depths.

◆ Length()

double QuadricCurve2::Length	(	const double &	a,
		const double &	b,
		int	n = `256`
	)		const

Computes the length of the quadric 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].

s = Integral Sqrt((dx/dt)² + (dy/dt)²) dt,

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.

◆ LengthIntegral()

double QuadricCurve2::LengthIntegral	(	const double &	a = `0.0`,
		const double &	b = `1.0`
	)		const

Computes the length of the quadric curve over interval [a,b] using a closed form integral.

Parameters

a,b Interval.

◆ Normal() [1/2]

Vector2 QuadricCurve2::Normal ( const double & t ) const

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

The normal is the obtained by evaluating the second derivative.

Parameters

t	Position (not used).

◆ Normal() [2/2]

Vector2 QuadricCurve2::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 QuadricCurve2::operator() ( const double & t ) const

Computes the point on the curve.

Parameters

t	Parameter.

◆ R()

double QuadricCurve2::R	(	const Vector2 &	p,
		double &	u
	)		const

Compute the squared distance between a point and the quadric curve.

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

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

Warning This function does not work if the quadric degenerates to a linear curve.

Parameters

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

◆ S()

double QuadricCurve2::S	(	const double &	t,
		int	n = `256`
	)		const

Compute the curvilign absisca of a point on the curve.

Parameters

t	Parameter.
n	Discretization of the integration interval.

◆ Tangent()

Vector2 QuadricCurve2::Tangent ( const double & t ) const

Computes the tangent to the curve.

Parameters

t	Parameter.

◆ Translate()

void QuadricCurve2::Translate ( const Vector2 & t )

Translate the curve.

Parameters

t	Translation vector.

◆ U()

double QuadricCurve2::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()

double QuadricCurve2::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.

Friends And Related Function Documentation

◆ operator<<

std::ostream & operator<<	(	std::ostream &	s,
		const QuadricCurve2 &	curve
	)

friend

Overloaded.

Parameters

s	Stream.
curve	The quadric curve.

Public Member Functions

Static Public Member Functions

Protected Attributes

Friends

Detailed Description

Constructor & Destructor Documentation

◆ QuadricCurve2() [1/2]

◆ QuadricCurve2() [2/2]

Member Function Documentation

◆ Bezier() [1/2]

◆ Bezier() [2/2]

◆ Curvature()

◆ Draw()

◆ Eval()

◆ Intersect()

◆ Length()

◆ LengthIntegral()

◆ Normal() [1/2]

◆ Normal() [2/2]

◆ operator()()

◆ R()

◆ S()

◆ Tangent()

◆ Translate()

◆ U()

◆ UV()

Friends And Related Function Documentation

◆ operator<<