QuadricCurve Class Reference

Quadric curves. More...

#include <curve.h>

Public Member Functions
	QuadricCurve ()
	Empty.
	QuadricCurve (const Quadric &, const Quadric &, const Quadric &)
	Creates a quadric curve.
	~QuadricCurve ()
	Empty.
Quadric &	operator[] (int)
	Access curve components.
Quadric	operator[] (int) const
	Access curve components.
QuadricCurve	operator- () const
	Negates the coefficients of a quadric curve.
Vector	operator() (const double &) const
	Computes the point on the curve.
Vector	Eval (const double &) const
	Computes the point on the curve.
double	Curvature (const double &) const
	Computes the curvature of the curve at a given positon.
Vector	Tangent (const double &) const
	Computes the tangent to the curve.
Vector	Normal (const double &) const
	Computes the secondary derivative to the curve.
Box	GetBox () const
	Compute the bounding box of the quadric curve with parameter interval [0,1].
double	R (const Vector &, double &) const
	Compute the squared distance between a point and the quadric curve.
double	Signed (const Vector &) const
	Compute signed squared distance between a point and the quadric curve.
double	S (const double &, int=256) const
	Compute the curvilign absisca of a point on the curve.
double	U (const double &, int=256) const
	Compute the parameter corresponding to the argument curvilign absisca of a point on the curve.
double	LengthIntegral (const double &, const double &) const
	Computes the length of the quadric curve over interval [a,b] using a closed form integral.
double	Length (int=256) const
	Computes the length of the cubic curve.
double	Length (const double &, const double &, int=256) const
	Computes the length of the quadric curve over interval [a,b].
Matrix	GetMatrix (const double &, bool=false) const
	Compute the Frenet vectors at a given point on the curve.
Frame	GetFrame (const double &, bool=false) const
	Compute the Frenet frame at a given point on the curve.
QVector< Frame >	GetFrames (const double &, bool=false) const
	Compute the set of frames along the curve.

Static Public Member Functions
static QuadricCurve	Bezier (const Vector &, const Vector &, const Vector &)
	Create a quadratic Bezier curve on interval [0,1].
static QVector< QuadricCurve >	Bezier (const QVector< Vector > &)
	Create a set of quadratic Bezier curves.

Protected Attributes
Quadric	z
	Quadric functions for every coordinate.

Friends
std::ostream &	operator<< (std::ostream &s, const QuadricCurve &curve)
	Overloaded.

Detailed Description

Quadric curves.

Quadric curves are implemented with three 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 for accelerating the computation of 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

QuadricCurve()

QuadricCurve::QuadricCurve ( const Quadric & x, const Quadric & y, const Quadric & z )

explicit

Creates a quadric curve.

Parameters

x,y,z

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

Member Function Documentation

Bezier() [1/2]

QVector< QuadricCurve > QuadricCurve::Bezier ( const QVector< Vector > & c )

static

Create a set of quadratic Bezier curves.

Parameters

c	Control points of the Bezier curve.

Bezier() [2/2]

QuadricCurve QuadricCurve::Bezier ( const Vector & a, const Vector & b, const Vector & c )

static

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

Parameters

a,b,c

The control points of the Bezier curve.

Curvature()

double QuadricCurve::Curvature

(

const double &

t

)

const

Computes the curvature of the curve at a given positon.

Parameters

t	Parameter.

Eval()

Vector QuadricCurve::Eval

(

const double &

t

)

const

Computes the point on the curve.

This function is redundant with the operator().

Parameters

t	Parameter.

GetFrame()

Frame QuadricCurve::GetFrame	(	const double &	u,
		bool	vertical = false ) const

Compute the Frenet frame at a given point on the curve.

Parameters

u	Parameter.
vertical	Boolean, set to true to compute frame along the horizontal curve first, false (default) otherwise.

GetFrames()

QVector< Frame > QuadricCurve::GetFrames	(	const double &	step,
		bool	vertical = false ) const

Compute the set of frames along the curve.

Parameters

step	Distance beween samples.
vertical	Boolean, set to true to compute frame along the horizontal curve first, false (default) otherwise.

GetMatrix()

Matrix QuadricCurve::GetMatrix	(	const double &	u,
		bool	vertical = false ) const

Compute the Frenet vectors at a given point on the curve.

The columns of the returned matrix contain the tangent, normal and binormal vectors.

Parameters

u	Parameter.
vertical	Boolean, set to true to compute frame along the horizontal curve first, false (default) otherwise.

Length() [1/2]

double QuadricCurve::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.

Length() [2/2]

double QuadricCurve::Length

(

int

n = 256

)

const

Computes the length of the cubic curve.

Integration is performed over interval [0,1] using a uniform discretization.

Parameters

n	Discretization.

LengthIntegral()

double QuadricCurve::LengthIntegral	(	const double &	a,
		const double &	b ) const

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

Parameters

a,b

Interval.

Normal()

Vector QuadricCurve::Normal

(

const double &

)

const

Computes the secondary derivative to the curve.

Note that although the parameter has no influence over the secondary derivative for quadric curves, it is still there out of closed form consistency.

operator()()

Vector QuadricCurve::operator()

(

const double &

t

)

const

Computes the point on the curve.

Parameters

t	Parameter.

R()

double QuadricCurve::R	(	const Vector &	p,
		double &	u ) const

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

This function computes the projection of a 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	Point.
u	Parameter defining the coordinate of the projection of the argument vertex onto the curve.

S()

double QuadricCurve::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.

Signed()

double QuadricCurve::Signed

(

const Vector &

p

)

const

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

Simply compute the square root of the squared distance.

See also

double QuadricCurve::R(const Vector& , double& ) const

Parameters

p

Point.

Tangent()

Vector QuadricCurve::Tangent

(

const double &

t

)

const

Computes the tangent to the curve.

Parameters

t	Parameter.

U()

double QuadricCurve::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.

Friends And Related Symbol Documentation

operator<<

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

friend

Overloaded.

Parameters

s	Stream.
curve	The quadric curve.