vgl_quadric_3d< T > Class Template Reference

#include <vgl_quadric_3d.h>

Public Types
enum	vgl_quadric_type {   no_type =0, real_ellipsoid, imaginary_ellipsoid, hyperboloid_of_one_sheet,   hyperboloid_of_two_sheets, real_elliptic_cone, imaginary_elliptic_cone, elliptic_paraboloid,   hyperbolic_paraboloid, real_elliptic_cylinder, imaginary_elliptic_cylinder, hyperbolic_cylinder,   real_intersecting_planes, imaginary_intersecting_planes, parabolic_cylinder, real_parallel_planes,   imaginary_parallel_planes, coincident_planes, num_quadric_types }

Public Member Functions
vgl_quadric_type	type () const
T	a () const
	Returns the coefficient of . More...
T	b () const
	Returns the coefficient of . More...
T	c () const
	Returns the coefficient of . More...
T	d () const
	Returns the coefficient of . More...
T	e () const
	Returns the coefficient of . More...
T	f () const
	Returns the coefficient of . More...
T	g () const
	Returns the coefficient of . More...
T	h () const
	Returns the coefficient of . More...
T	i () const
	Returns the coefficient of . More...
T	j () const
	Returns the coefficient of . More...
	vgl_quadric_3d ()
	vgl_quadric_3d (T a, T b, T c, T d, T e, T f, T g, T h, T i, T j)
	constructor using polynomial coefficients. More...
	vgl_quadric_3d (T const coeff[])
	constructor from a linear array of polynomial coefficients, given as a C array. More...
	vgl_quadric_3d (std::vector< std::vector< T > > const &Q)
	constructor from a matix of polynomial coefficients (see below). More...
std::vector< std::vector< T > >	coef_matrix () const
	return a matrix of quadric coefficients of the form:. More...
	vgl_quadric_3d (std::vector< std::vector< T > > const &canonical_quadric, std::vector< std::vector< T > > const &H)
	constructor from a canonical 4x4 quadric coefficient matrix and a 4x4 homogeneous matrix, H. More...
	vgl_quadric_3d (std::vector< T > const &diag, std::vector< std::vector< T > > const &H)
	constructor for central quadrics e.g. ellipsoid, ax^2+ by^2+ cz*2 + j = 0, where diag = [a,b,c,j]. More...
void	set (T a, T b, T c, T d, T e, T f, T g, T h, T i, T j)
	set or reset the quadric using polynomial coefficients. More...
void	set (std::vector< std::vector< T > > const &Q)
bool	operator== (vgl_quadric_3d< T > const &c) const
	comparison operator. More...
bool	is_degenerate () const
	Returns true if this quadric is degenerate. More...
T	sampson_dist (vgl_homg_point_3d< T > const &pt) const
	Sampson distance d_sampson(p) = ||p^t Q p||^2/||grad(p*t Q p)||^2. More...
bool	on (vgl_homg_point_3d< T > const &pt, T tol=T(0)) const
	Returns true if the point pt belongs to the quadric surface. More...
bool	center (vgl_point_3d< T > &center) const
	if the upper 3x3 submatrix of Q is full rank then the center of the quadric can be defined. More...
void	upper_3x3_eigensystem (std::vector< T > &eigenvalues, std::vector< std::vector< T > > &eigenvectors) const
	: eigenvalues and eigenvectors of the upper 3x3 quadric matrix. More...
bool	canonical_central_quadric (std::vector< T > &diag, std::vector< std::vector< T > > &H) const
	: The quadric in its canonical frame if the center is defined, i.e. the upper 3x3 quadric matrix is full rank. More...
std::vector< std::vector< T > >	canonical_quadric (std::vector< std::vector< T > > &H) const
	The quadric coefficient matrix in the canonical frame, whether or not the quadric is central. More...

Static Public Member Functions
static vgl_quadric_type	type_by_name (std::string const &name)
	Returns the internal enum value corresponding to the string argument. More...
static std::string	type_by_number (vgl_quadric_type const &type)
	Converts the quadric type from enum (internal representation) to string. More...

Private Member Functions
void	compute_type ()
	set quadric type from polynomial coefficients and store in member type_. More...

Private Attributes
bool	det_zero_
vgl_quadric_type	type_
T	a_
	coefficient of x^2 More...
T	b_
	coefficient of y^2 More...
T	c_
	coefficient of z^2 More...
T	d_
	coefficient of xy More...
T	e_
	coefficient of xz More...
T	f_
	coefficient of yz More...
T	g_
	coefficient of xw More...
T	h_
	coefficient of yw More...
T	i_
	coefficient of zw More...
T	j_
	coefficient of w^2 More...

Related Functions
(Note that these are not member functions.)
template<class T >
std::istream &	operator>> (std::istream &s, vgl_quadric_3d< T > &c)
	Read a b c d e f g h i j from stream. More...

Detailed Description

template<class T>
class vgl_quadric_3d< T >

Definition at line 32 of file vgl_quadric_3d.h.

Member Enumeration Documentation

vgl_quadric_type

template<class T>

enum vgl_quadric_3d::vgl_quadric_type

Enumerator
no_type
real_ellipsoid
imaginary_ellipsoid
hyperboloid_of_one_sheet
hyperboloid_of_two_sheets
real_elliptic_cone
imaginary_elliptic_cone
elliptic_paraboloid
hyperbolic_paraboloid
real_elliptic_cylinder
imaginary_elliptic_cylinder
hyperbolic_cylinder
real_intersecting_planes
imaginary_intersecting_planes
parabolic_cylinder
real_parallel_planes
imaginary_parallel_planes
coincident_planes
num_quadric_types

Definition at line 35 of file vgl_quadric_3d.h.

Constructor & Destructor Documentation

vgl_quadric_3d() [1/6]

template<class T>

vgl_quadric_3d< T >::vgl_quadric_3d ( )

inline

Definition at line 114 of file vgl_quadric_3d.h.

vgl_quadric_3d() [2/6]

template<class T >

vgl_quadric_3d< T >::vgl_quadric_3d	(	T	a,
		T	b,
		T	c,
		T	d,
		T	e,
		T	f,
		T	g,
		T	h,
		T	i,
		T	j
	)

constructor using polynomial coefficients.

Definition at line 45 of file vgl_quadric_3d.hxx.

vgl_quadric_3d() [3/6]

template<class T >

vgl_quadric_3d< T >::vgl_quadric_3d

(

T const

coeff[]

)

constructor from a linear array of polynomial coefficients, given as a C array.

Definition at line 49 of file vgl_quadric_3d.hxx.

vgl_quadric_3d() [4/6]

template<class T >

vgl_quadric_3d< T >::vgl_quadric_3d

(

std::vector< std::vector< T > > const &

Q

)

constructor from a matix of polynomial coefficients (see below).

Definition at line 60 of file vgl_quadric_3d.hxx.

vgl_quadric_3d() [5/6]

template<class T >

vgl_quadric_3d< T >::vgl_quadric_3d	(	std::vector< std::vector< T > > const &	canonical_quadric,
		std::vector< std::vector< T > > const &	H
	)

constructor from a canonical 4x4 quadric coefficient matrix and a 4x4 homogeneous matrix, H.

representing the Euclidean transformation from the canonical frame to the global frame _ _ |R t| where t is a 3x1 translation vector and R is a rotation matrix H = | | 0^t is a 1x3 zero vector. |0^t 1|

• -

Definition at line 65 of file vgl_quadric_3d.hxx.

vgl_quadric_3d() [6/6]

template<class T >

vgl_quadric_3d< T >::vgl_quadric_3d	(	std::vector< T > const &	diag,
		std::vector< std::vector< T > > const &	H
	)

constructor for central quadrics e.g. ellipsoid, ax^2+ by^2+ cz*2 + j = 0, where diag = [a,b,c,j].

are the diagonal elements of the 4x4 quadric coefficient matrix and a 4x4 homogeneous matrix, and H represents the Euclidean transformation from the canonical frame to the global frame (see above)

Definition at line 120 of file vgl_quadric_3d.hxx.

Member Function Documentation

a()

template<class T>

T vgl_quadric_3d< T >::a ( ) const

inline

Returns the coefficient of .

Definition at line 83 of file vgl_quadric_3d.h.

b()

template<class T>

T vgl_quadric_3d< T >::b ( ) const

inline

Returns the coefficient of .

Definition at line 86 of file vgl_quadric_3d.h.

c()

template<class T>

T vgl_quadric_3d< T >::c ( ) const

inline

Returns the coefficient of .

Definition at line 89 of file vgl_quadric_3d.h.

canonical_central_quadric()

template<class T >

bool vgl_quadric_3d< T >::canonical_central_quadric	(	std::vector< T > &	diag,
		std::vector< std::vector< T > > &	H
	)		const

: The quadric in its canonical frame if the center is defined, i.e. the upper 3x3 quadric matrix is full rank.

In this case the quadric coefficient matrix in the canonical frame is a 4x4 diagonal matrix, e.g. ax^2 + by^2 + cz^2 + j = 0. Note that the canonical frame is not unique as alignment of quadric axes with the orthogonal frame has numerous possible arrangements. H is a homogenous(4x4)transformation from canonical coordinate space back to the original space.

Definition at line 398 of file vgl_quadric_3d.hxx.

canonical_quadric()

template<class T >

std::vector< std::vector< T > > vgl_quadric_3d< T >::canonical_quadric

(

std::vector< std::vector< T > > &

H

)

const

The quadric coefficient matrix in the canonical frame, whether or not the quadric is central.

H is a homogenous(4x4)transformation from canonical coordinate space back to the original space.

Definition at line 424 of file vgl_quadric_3d.hxx.

center()

template<class T >

bool vgl_quadric_3d< T >::center

(

vgl_point_3d< T > &

center

)

const

if the upper 3x3 submatrix of Q is full rank then the center of the quadric can be defined.

otherwise the center is not defined for degenerate quadrics

Definition at line 166 of file vgl_quadric_3d.hxx.

coef_matrix()

template<class T >

std::vector< std::vector< T > > vgl_quadric_3d< T >::coef_matrix

(

)

const

return a matrix of quadric coefficients of the form:.

_ _ | a d/2 e/2 g/2 | | d/2 b f/2 h/2 | Q = | e/2 f/2 c i/2 | | g/2 h/2 i/2 j |

• - Note that X^t Q X = 0 , where X^t =[x y z w] is the same as implicit equation 1) above.

Definition at line 139 of file vgl_quadric_3d.hxx.

compute_type()

template<class T >

void vgl_quadric_3d< T >::compute_type ( )

private

set quadric type from polynomial coefficients and store in member type_.

Definition at line 205 of file vgl_quadric_3d.hxx.

d()

template<class T>

T vgl_quadric_3d< T >::d ( ) const

inline

Returns the coefficient of .

Definition at line 92 of file vgl_quadric_3d.h.

e()

template<class T>

T vgl_quadric_3d< T >::e ( ) const

inline

Returns the coefficient of .

Definition at line 95 of file vgl_quadric_3d.h.

f()

template<class T>

T vgl_quadric_3d< T >::f ( ) const

inline

Returns the coefficient of .

Definition at line 98 of file vgl_quadric_3d.h.

g()

template<class T>

T vgl_quadric_3d< T >::g ( ) const

inline

Returns the coefficient of .

Definition at line 101 of file vgl_quadric_3d.h.

h()

template<class T>

T vgl_quadric_3d< T >::h ( ) const

inline

Returns the coefficient of .

Definition at line 104 of file vgl_quadric_3d.h.

i()

template<class T>

T vgl_quadric_3d< T >::i ( ) const

inline

Returns the coefficient of .

Definition at line 107 of file vgl_quadric_3d.h.

is_degenerate()

template<class T>

bool vgl_quadric_3d< T >::is_degenerate ( ) const

inline

Returns true if this quadric is degenerate.

Definition at line 164 of file vgl_quadric_3d.h.

j()

template<class T>

T vgl_quadric_3d< T >::j ( ) const

inline

Returns the coefficient of .

Definition at line 110 of file vgl_quadric_3d.h.

on()

template<class T >

bool vgl_quadric_3d< T >::on	(	vgl_homg_point_3d< T > const &	pt,
		T	tol = T(0)
	)		const

Returns true if the point pt belongs to the quadric surface.

I.e., if it satisfies the quadric equation within algebraic distance, i.e. pt^t Q pt < tol;

Definition at line 159 of file vgl_quadric_3d.hxx.

operator==()

template<class T >

bool vgl_quadric_3d< T >::operator==

(

vgl_quadric_3d< T > const &

c

)

const

comparison operator.

Comparison is on the quadric, not the equation coefficients. Hence two quadrics are identical if their coefficient vectors are multiples of each other.

Definition at line 363 of file vgl_quadric_3d.hxx.

sampson_dist()

template<class T >

T vgl_quadric_3d< T >::sampson_dist

(

vgl_homg_point_3d< T > const &

pt

)

const

Sampson distance d_sampson(p) = ||p^t Q p||^2/||grad(p*t Q p)||^2.

a first order approximation to Euclidean distance

Definition at line 148 of file vgl_quadric_3d.hxx.

set() [1/2]

template<class T >

void vgl_quadric_3d< T >::set	(	T	a,
		T	b,
		T	c,
		T	d,
		T	e,
		T	f,
		T	g,
		T	h,
		T	i,
		T	j
	)

set or reset the quadric using polynomial coefficients.

Definition at line 127 of file vgl_quadric_3d.hxx.

set() [2/2]

template<class T >

void vgl_quadric_3d< T >::set

(

std::vector< std::vector< T > > const &

Q

)

Definition at line 133 of file vgl_quadric_3d.hxx.

type()

template<class T>

vgl_quadric_type vgl_quadric_3d< T >::type ( ) const

inline

Definition at line 73 of file vgl_quadric_3d.h.

type_by_name()

template<class T >

vgl_quadric_3d< T >::vgl_quadric_type vgl_quadric_3d< T >::type_by_name ( std::string const &  name )

static

Returns the internal enum value corresponding to the string argument.

Useful for comparison purposes, or for use in "case" statements.

Definition at line 36 of file vgl_quadric_3d.hxx.

type_by_number()

template<class T >

std::string vgl_quadric_3d< T >::type_by_number ( vgl_quadric_type const &  type )

static

Converts the quadric type from enum (internal representation) to string.

Definition at line 33 of file vgl_quadric_3d.hxx.

upper_3x3_eigensystem()

template<class T >

void vgl_quadric_3d< T >::upper_3x3_eigensystem	(	std::vector< T > &	eigenvalues,
		std::vector< std::vector< T > > &	eigenvectors
	)		const

: eigenvalues and eigenvectors of the upper 3x3 quadric matrix.

Definition at line 381 of file vgl_quadric_3d.hxx.

Friends And Related Function Documentation

operator>>()

template<class T >

std::istream & operator>> ( std::istream &  s, vgl_quadric_3d< T > &  c  )

related

Read a b c d e f g h i j from stream.

Member Data Documentation

a_

template<class T>

T vgl_quadric_3d< T >::a_

private

coefficient of x^2

Definition at line 61 of file vgl_quadric_3d.h.

b_

template<class T>

T vgl_quadric_3d< T >::b_

private

coefficient of y^2

Definition at line 62 of file vgl_quadric_3d.h.

c_

template<class T>

T vgl_quadric_3d< T >::c_

private

coefficient of z^2

Definition at line 63 of file vgl_quadric_3d.h.

d_

template<class T>

T vgl_quadric_3d< T >::d_

private

coefficient of xy

Definition at line 64 of file vgl_quadric_3d.h.

det_zero_

template<class T>

bool vgl_quadric_3d< T >::det_zero_

private

Definition at line 59 of file vgl_quadric_3d.h.

e_

template<class T>

T vgl_quadric_3d< T >::e_

private

coefficient of xz

Definition at line 65 of file vgl_quadric_3d.h.

f_

template<class T>

T vgl_quadric_3d< T >::f_

private

coefficient of yz

Definition at line 66 of file vgl_quadric_3d.h.

g_

template<class T>

T vgl_quadric_3d< T >::g_

private

coefficient of xw

Definition at line 67 of file vgl_quadric_3d.h.

h_

template<class T>

T vgl_quadric_3d< T >::h_

private

coefficient of yw

Definition at line 68 of file vgl_quadric_3d.h.

i_

template<class T>

T vgl_quadric_3d< T >::i_

private

coefficient of zw

Definition at line 69 of file vgl_quadric_3d.h.

j_

template<class T>

T vgl_quadric_3d< T >::j_

private

coefficient of w^2

Definition at line 70 of file vgl_quadric_3d.h.

type_

template<class T>

vgl_quadric_type vgl_quadric_3d< T >::type_

private

Definition at line 60 of file vgl_quadric_3d.h.

---

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

• core/vgl/vgl_quadric_3d.h
• core/vgl/vgl_quadric_3d.hxx