Class to generate a Delaunay triangulation of a 2D set of points.

Algorithm based on Triangle, a two-dimensional quality mesh generator and Delaunay triangulator from Jonathan Richard Shewchuk.

See [http://www.cs.cmu.edu/~quake/triangle.html]

After having found the triangles using the above library, barycentric coordinates are used to test whether a point is inside a triangle (inTriangle test) and for interpolation. All this below is implemented in the DoInterpolateNormalized function.

Given triangle ABC and point P, P can be expressed by

P.x = la * A.x + lb * B.x + lc * C.x P.y = la * A.y + lb * B.y + lc * C.y

with lc = 1 - la - lb

P.x = la * A.x + lb * B.x + (1-la-lb) * C.x P.y = la * A.y + lb * B.y + (1-la-lb) * C.y

Rearranging yields

la * (A.x - C.x) + lb * (B.x - C.x) = P.x - C.x la * (A.y - C.y) + lb * (B.y - C.y) = P.y - C.y

Thus

la = ( (B.y - C.y)*(P.x - C.x) + (C.x - B.x)*(P.y - C.y) ) / ( (B.y - C.y)*(A.x - C.x) + (C.x - B.x)*(A.y - C.y) ) lb = ( (C.y - A.y)*(P.x - C.x) + (A.x - C.x)*(P.y - C.y) ) / ( (B.y - C.y)*(A.x - C.x) + (C.x - B.x)*(A.y - C.y) ) lc = 1 - la - lb

We save the inverse denominator to speedup computation

invDenom = 1 / ( (B.y - C.y)*(A.x - C.x) + (C.x - B.x)*(A.y - C.y) )

P is in triangle (including edges if

0 <= [la, lb, lc] <= 1

The interpolation of P.z is

P.z = la * A.z + lb * B.z + lc * C.z

To speed up localisation of points (to see to which triangle belong) a grid is laid over the internal coordinate space. A reference to triangle ABC is added to all grid cells that include ABC's bounding box. The size of the grid is defined to be 25x25

Optionally (if the compiler macro HAS_GCAL is defined ) the triangle findings and interpolation can be computed using the GCAL library. This is however not supported when using the class within ROOT

Definition at line 115 of file Delaunay2D.h.

Classes
struct	Triangle

Public Types
typedef std::vector< Triangle >	Triangles

Public Member Functions
	Delaunay2D (int n, const double x, const double y, const double *z, double xmin=0, double xmax=0, double ymin=0, double ymax=0)
	class constructor from array of data points

Triangles::const_iterator	begin () const

Triangles::const_iterator	end () const

void	FindAllTriangles ()
	Find all triangles.

double	Interpolate (double x, double y)
	Return the Interpolated z value corresponding to the given (x,y) point.

int	NumberOfTriangles () const
	return the number of triangles

void	SetInputPoints (int n, const double x, const double y, const double *z, double xmin=0, double xmax=0, double ymin=0, double ymax=0)
	set the input points for building the graph

void	SetZOuterValue (double z=0.)
	set z value to be returned for points outside the region

double	XMax () const

double	XMin () const

double	YMax () const

double	YMin () const

double	ZOuterValue () const
	return the user defined Z-outer value

Protected Member Functions
unsigned int	Cell (unsigned int x, unsigned int y) const

int	CellX (double x) const

int	CellY (double y) const

Protected Attributes
std::set< unsigned int >	fCells [(fNCells+1) *(fNCells+1)]
	! grid cells with containing triangles

bool	fInit
	! True if FindAllTriangles() has been performed

int	fNdt
	! Number of Delaunay triangles found

int	fNpoints
	! Number of data points

double	fOffsetX
	! Normalization offset X

double	fOffsetY
	! Normalization offset Y

double	fScaleFactorX
	! Normalization factor X

double	fScaleFactorY
	! Normalization factor Y

Triangles	fTriangles
	! Triangles of Triangulation

const double *	fX
	! Pointer to X array (managed externally)

double	fXCellStep
	! inverse denominator to calculate X cell = fNCells / (fXNmax - fXNmin)

std::vector< double >	fXN
	! normalized X

double	fXNmax
	! Maximum value of fXN

double	fXNmin
	! Minimum value of fXN

const double *	fY
	! Pointer to Y array

double	fYCellStep
	! inverse denominator to calculate X cell = fNCells / (fYNmax - fYNmin)

std::vector< double >	fYN
	! normalized Y

double	fYNmax
	! Maximum value of fYN

double	fYNmin
	! Minimum value of fYN

const double *	fZ
	! Pointer to Z array

double	fZout
	! Height for points lying outside the convex hull

Static Protected Attributes
static const int	fNCells = 25
	! number of cells to divide the normalized space

Private Member Functions
	Delaunay2D (const Delaunay2D &)

void	DoFindTriangles ()
	internal function to find the triangle use Triangle or CGAL if flag is set

double	DoInterpolateNormalized (double x, double y)
	internal method to compute the interpolation

void	DoNormalizePoints ()
	internal function to normalize the points.

double	Linear_transform (double x, double offset, double factor)

Delaunay2D &	operator= (const Delaunay2D &)

#include <Math/Delaunay2D.h>

Member Typedef Documentation

◆ Triangles

typedef std::vector<Triangle> ROOT::Math::Delaunay2D::Triangles

Definition at line 129 of file Delaunay2D.h.

Constructor & Destructor Documentation

◆ Delaunay2D() [1/2]

ROOT::Math::Delaunay2D::Delaunay2D	(	int	n,
		const double *	x,
		const double *	y,
		const double *	z,
		double	xmin = `0`,
		double	xmax = `0`,
		double	ymin = `0`,
		double	ymax = `0`
	)

class constructor from array of data points

Definition at line 36 of file Delaunay2D.cxx.

◆ Delaunay2D() [2/2]

ROOT::Math::Delaunay2D::Delaunay2D ( const Delaunay2D & )

private

Member Function Documentation

◆ begin()

Triangles::const_iterator ROOT::Math::Delaunay2D::begin ( ) const

inline

Definition at line 163 of file Delaunay2D.h.

◆ Cell()

unsigned int ROOT::Math::Delaunay2D::Cell	(	unsigned int	x,
		unsigned int	y
	)		const

inlineprotected

Definition at line 233 of file Delaunay2D.h.

◆ CellX()

int ROOT::Math::Delaunay2D::CellX ( double x ) const

inlineprotected

Definition at line 237 of file Delaunay2D.h.

◆ CellY()

int ROOT::Math::Delaunay2D::CellY ( double y ) const

inlineprotected

Definition at line 241 of file Delaunay2D.h.

◆ DoFindTriangles()

void ROOT::Math::Delaunay2D::DoFindTriangles ( )

private

internal function to find the triangle use Triangle or CGAL if flag is set

Triangle implementation for finding all the triangles.

Definition at line 177 of file Delaunay2D.cxx.

◆ DoInterpolateNormalized()

double ROOT::Math::Delaunay2D::DoInterpolateNormalized	(	double	xx,
		double	yy
	)

private

internal method to compute the interpolation

Triangle implementation for interpolation Finds the Delaunay triangle that the point (xi,yi) sits in (if any) and calculate a z-value for it by linearly interpolating the z-values that make up that triangle.

Relay that all the triangles have been found before see comment in class description (in Delaunay2D.h) for implementation details: finding barycentric coordinates and computing the interpolation

Definition at line 303 of file Delaunay2D.cxx.

◆ DoNormalizePoints()

void ROOT::Math::Delaunay2D::DoNormalizePoints ( )

private

internal function to normalize the points.

Triangle implementation for points normalization.

See the class documentation for the details on how it is computed.

Definition at line 165 of file Delaunay2D.cxx.

◆ end()

Triangles::const_iterator ROOT::Math::Delaunay2D::end ( ) const

inline

Definition at line 164 of file Delaunay2D.h.

◆ FindAllTriangles()

void ROOT::Math::Delaunay2D::FindAllTriangles ( )

Find all triangles.

Definition at line 136 of file Delaunay2D.cxx.

◆ Interpolate()

double ROOT::Math::Delaunay2D::Interpolate	(	double	x,
		double	y
	)

Return the Interpolated z value corresponding to the given (x,y) point.

Note that in case no Delaunay triangles are found, for example when the points are aligned, then a default value of zero is always return. See the class documentation for how the interpolation is computed.

Definition at line 106 of file Delaunay2D.cxx.

◆ Linear_transform()

double ROOT::Math::Delaunay2D::Linear_transform	(	double	x,
		double	offset,
		double	factor
	)

inlineprivate

Definition at line 172 of file Delaunay2D.h.

◆ NumberOfTriangles()

int ROOT::Math::Delaunay2D::NumberOfTriangles ( ) const

inline

return the number of triangles

Definition at line 149 of file Delaunay2D.h.

◆ operator=()

Delaunay2D & ROOT::Math::Delaunay2D::operator= ( const Delaunay2D & )

private

◆ SetInputPoints()

void ROOT::Math::Delaunay2D::SetInputPoints	(	int	n,
		const double *	x,
		const double *	y,
		const double *	z,
		double	xmin = `0`,
		double	xmax = `0`,
		double	ymin = `0`,
		double	ymax = `0`
	)

set the input points for building the graph

set the input points

Definition at line 61 of file Delaunay2D.cxx.

◆ SetZOuterValue()

void ROOT::Math::Delaunay2D::SetZOuterValue ( double z = 0. )

inline

set z value to be returned for points outside the region

Definition at line 157 of file Delaunay2D.h.

◆ XMax()

double ROOT::Math::Delaunay2D::XMax ( ) const

inline

Definition at line 152 of file Delaunay2D.h.

◆ XMin()

double ROOT::Math::Delaunay2D::XMin ( ) const

inline

Definition at line 151 of file Delaunay2D.h.

◆ YMax()

double ROOT::Math::Delaunay2D::YMax ( ) const

inline

Definition at line 154 of file Delaunay2D.h.

◆ YMin()

double ROOT::Math::Delaunay2D::YMin ( ) const

inline

Definition at line 153 of file Delaunay2D.h.

◆ ZOuterValue()

double ROOT::Math::Delaunay2D::ZOuterValue ( ) const

inline

return the user defined Z-outer value

Definition at line 160 of file Delaunay2D.h.

Member Data Documentation

◆ fCells

std::set<unsigned int> ROOT::Math::Delaunay2D::fCells[(fNCells+1) *(fNCells+1)]

protected

! grid cells with containing triangles

Definition at line 231 of file Delaunay2D.h.

◆ fInit

bool ROOT::Math::Delaunay2D::fInit

protected

! True if FindAllTriangles() has been performed

Definition at line 216 of file Delaunay2D.h.

◆ fNCells

const int ROOT::Math::Delaunay2D::fNCells = 25

staticprotected

! number of cells to divide the normalized space

Definition at line 228 of file Delaunay2D.h.

◆ fNdt

int ROOT::Math::Delaunay2D::fNdt

protected

! Number of Delaunay triangles found

Definition at line 196 of file Delaunay2D.h.

◆ fNpoints

int ROOT::Math::Delaunay2D::fNpoints

protected

! Number of data points

Definition at line 197 of file Delaunay2D.h.

◆ fOffsetX

double ROOT::Math::Delaunay2D::fOffsetX

protected

! Normalization offset X

Definition at line 208 of file Delaunay2D.h.

◆ fOffsetY

double ROOT::Math::Delaunay2D::fOffsetY

protected

! Normalization offset Y

Definition at line 209 of file Delaunay2D.h.

◆ fScaleFactorX

double ROOT::Math::Delaunay2D::fScaleFactorX

protected

! Normalization factor X

Definition at line 211 of file Delaunay2D.h.

◆ fScaleFactorY

double ROOT::Math::Delaunay2D::fScaleFactorY

protected

! Normalization factor Y

Definition at line 212 of file Delaunay2D.h.

◆ fTriangles

Triangles ROOT::Math::Delaunay2D::fTriangles

protected

! Triangles of Triangulation

Definition at line 219 of file Delaunay2D.h.

◆ fX

const double* ROOT::Math::Delaunay2D::fX

protected

! Pointer to X array (managed externally)

Definition at line 199 of file Delaunay2D.h.

◆ fXCellStep

double ROOT::Math::Delaunay2D::fXCellStep

protected

! inverse denominator to calculate X cell = fNCells / (fXNmax - fXNmin)

Definition at line 229 of file Delaunay2D.h.

◆ fXN

std::vector<double> ROOT::Math::Delaunay2D::fXN

protected

! normalized X

Definition at line 225 of file Delaunay2D.h.

◆ fXNmax

double ROOT::Math::Delaunay2D::fXNmax

protected

! Maximum value of fXN

Definition at line 204 of file Delaunay2D.h.

◆ fXNmin

double ROOT::Math::Delaunay2D::fXNmin

protected

! Minimum value of fXN

Definition at line 203 of file Delaunay2D.h.

◆ fY

const double* ROOT::Math::Delaunay2D::fY

protected

! Pointer to Y array

Definition at line 200 of file Delaunay2D.h.

◆ fYCellStep

double ROOT::Math::Delaunay2D::fYCellStep

protected

! inverse denominator to calculate X cell = fNCells / (fYNmax - fYNmin)

Definition at line 230 of file Delaunay2D.h.

◆ fYN

std::vector<double> ROOT::Math::Delaunay2D::fYN

protected

! normalized Y

Definition at line 226 of file Delaunay2D.h.

◆ fYNmax

double ROOT::Math::Delaunay2D::fYNmax

protected

! Maximum value of fYN

Definition at line 206 of file Delaunay2D.h.

◆ fYNmin

double ROOT::Math::Delaunay2D::fYNmin

protected

! Minimum value of fYN

Definition at line 205 of file Delaunay2D.h.

◆ fZ

const double* ROOT::Math::Delaunay2D::fZ

protected

! Pointer to Z array

Definition at line 201 of file Delaunay2D.h.

◆ fZout

double ROOT::Math::Delaunay2D::fZout

protected

! Height for points lying outside the convex hull

Definition at line 214 of file Delaunay2D.h.

Libraries for ROOT::Math::Delaunay2D:

[legend]

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

math/mathcore/inc/Math/Delaunay2D.h
math/mathcore/src/Delaunay2D.cxx

Classes

Public Types

Public Member Functions

Protected Member Functions

Protected Attributes

Static Protected Attributes

Private Member Functions

Member Typedef Documentation

◆ Triangles

Constructor & Destructor Documentation

◆ Delaunay2D() [1/2]

◆ Delaunay2D() [2/2]

Member Function Documentation

◆ begin()

◆ Cell()

◆ CellX()

◆ CellY()

◆ DoFindTriangles()

◆ DoInterpolateNormalized()

◆ DoNormalizePoints()

◆ end()

◆ FindAllTriangles()

◆ Interpolate()

◆ Linear_transform()

◆ NumberOfTriangles()

◆ operator=()

◆ SetInputPoints()

◆ SetZOuterValue()

◆ XMax()

◆ XMin()

◆ YMax()

◆ YMin()

◆ ZOuterValue()

Member Data Documentation

◆ fCells

◆ fInit

◆ fNCells

◆ fNdt

◆ fNpoints

◆ fOffsetX

◆ fOffsetY

◆ fScaleFactorX

◆ fScaleFactorY

◆ fTriangles

◆ fX

◆ fXCellStep

◆ fXN

◆ fXNmax

◆ fXNmin

◆ fY

◆ fYCellStep

◆ fYN

◆ fYNmax

◆ fYNmin

◆ fZ

◆ fZout