jv.vecmath
Class PuProj

java.lang.Object
  |
  +--jv.vecmath.PuProj

public class PuProj

extends java.lang.Object

Projections between different models of three dimensional space forms. Coordinate system in Lorentz R4 has signature (1,1,1,-1).

Version:

27.12.99, 1.10 revised (kp) Converted to Java.
00.00.92, 1.00 created (kp, bo) Original C code, S3 methods from B. Oberknapp.

Author:

Konrad Polthier

Constructor Summary

PuProj()

Method Summary

static void	diffKlein2Lorentz(PdMatrix diffMat, PdVector p) Compute differential of projection map of H3 from Klein ball into Lorentz model at given point in Klein ball.
static void	diffPoincare2Lorentz(PdMatrix diffMat, PdVector p) Compute differential of projection map of H3 from Poincare ball into Lorentz model at given point in Poincare ball.
static void	diffStereoInvR3_to_S3(PdMatrix diffMat, PdVector p) Compute differential of inverse stereographic projection R3 to S3.
static void	diffStereoS3_to_R3(PdMatrix diffMat, PdVector p) Compute differential of stereographic projection S3 to R3 with projection center (1,0,0,0).
static void	klein2Lorentz(PdVector pl, PdVector p) Projection from Klein ball into Lorentz model of H3.
static void	klein2Poincare(PdVector p, PdVector pk) Projection from Klein ball to Poincare ball.
static void	klein2Uhm(PdVector pu, PdVector p) Projection of H3 from Klein ball to UHM.
static void	lorentz2Klein(PdVector p, PdVector pl) Projection of H3 from Lorentz model into Klein ball.
static void	lorentz2Poincare(PdVector p, PdVector pl) Projection of H3 from Lorentz model into Poincare ball.
static void	lorentz2Uhm(PdVector p, PdVector pl) Projection of H3 from Lorentz model into UHM.
static void	poincare2Klein(PdVector pk, PdVector p) Projection of H3 from Poincare ball to Klein ball.
static void	poincare2Lorentz(PdVector pl, PdVector p) Projection from Poincare ball into Lorentz model of H3.
static void	poincare2Uhm(PdVector pu, PdVector p) Projection of H3 from Poincare ball to UHM.
static void	stereographic(PdVector p, PdVector pl) Stereographic projection of a sphere Sn into Rn with projection center (0,..,0,1).
static void	stereoInvR3_to_S3(PdVector pl, PdVector p) Projection from R3 p=(a,b,c) to S4 pl=(x,y,z,t).
static void	stereoS3_to_R3(PdVector p, PdVector pl) Deprecated. use stereographic(PdVector, PdVector)
static void	uhm2Klein(PdVector p, PdVector pu) Projection from UHM to Klein ball.
static void	uhm2Lorentz(PdVector pl, PdVector p) Projection from UHM into Lorentz model of H3.
static void	uhm2Poincare(PdVector p, PdVector pu) Projection from UHM to Poincare ball.

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

PuProj

public PuProj()

Method Detail

lorentz2Klein

public static void lorentz2Klein(PdVector p,
                                 PdVector pl)

Projection of H3 from Lorentz model into Klein ball. (a, b, c) = (x, y, z)/t Coordinate system in Lorentz R4 has signature (1,1,1,-1). The parameters may be identical vectors.

Parameters:

p - resulting vector of size 3 in Klein ball

pl - argument vector of size 4 in Lorentz model

klein2Lorentz

public static void klein2Lorentz(PdVector pl,
                                 PdVector p)

Projection from Klein ball into Lorentz model of H3. (t, x, y, z) = (t, t * a, t * b, t * c) with t = 1/sqrt(1 - |p|^2) Coordinate system in Lorentz R4 has signature (1,1,1,-1). The parameters may be identical vectors.

Parameters:

p - argument vector of size 3 in Klein ball

pl - resulting vector of size 4 in Lorentz model

diffKlein2Lorentz

public static void diffKlein2Lorentz(PdMatrix diffMat,
                                     PdVector p)

Compute differential of projection map of H3 from Klein ball into Lorentz model at given point in Klein ball. Coordinate system in Lorentz R4 has signature (1,1,1,-1).

Parameters:

dp - resulting differential matrix of size 4*3

p - argument vector of size 3 in Klein ball

lorentz2Poincare

public static void lorentz2Poincare(PdVector p,
                                    PdVector pl)

Projection of H3 from Lorentz model into Poincare ball. (a, b, c) = (x, y, z)/(1 + t) Coordinate system in Lorentz R4 has signature (1,1,1,-1). The parameters may be identical vectors.

Parameters:

p - resulting vector of size 3 in Poincare ball

pl - argument vector of size 4 in Lorentz model

poincare2Lorentz

public static void poincare2Lorentz(PdVector pl,
                                    PdVector p)

Projection from Poincare ball into Lorentz model of H3. (t, x, y, z) = (t, (1 + t) * a, (1 + t) * b, (1 + t) * c) with t = (1 + |p|^2)/(1 - |p|^2), 1 + t = 2/(1 - |p|^2). Coordinate system in Lorentz R4 has signature (1,1,1,-1). The parameters may be identical vectors.

Parameters:

pl - resulting vector of size 4 in Lorentz model

p - argument vector of size 3 in Poincare ball

diffPoincare2Lorentz

public static void diffPoincare2Lorentz(PdMatrix diffMat,
                                        PdVector p)

Compute differential of projection map of H3 from Poincare ball into Lorentz model at given point in Poincare ball. Coordinate system in Lorentz R4 has signature (1,1,1,-1).

Parameters:

dp - resulting differential matrix of size 4*3

p - argument vector of size 3 in Klein ball

lorentz2Uhm

public static void lorentz2Uhm(PdVector p,
                               PdVector pl)

Projection of H3 from Lorentz model into UHM. (a, b, c) = (x, y, 1)/(t - z) Coordinate system in Lorentz R4 has signature (1,1,1,-1). The parameters may be identical vectors.

Parameters:

p - resulting vector of size 3 in Upper Halfspace model

pl - argument vector of size 4 in Lorentz model

uhm2Lorentz

public static void uhm2Lorentz(PdVector pl,
                               PdVector p)

Projection from UHM into Lorentz model of H3. (x, y, z, t) = (2 x, 2 y, t - 1, 1 + t)/(2 * z) with t = |p|^2. Coordinate system in Lorentz R4 has signature (1,1,1,-1). The parameters may be identical vectors.

Parameters:

pl - resulting vector of size 4 in Lorentz model

p - argument vector of size 3 in Upper Halfspace model

poincare2Klein

public static void poincare2Klein(PdVector pk,
                                  PdVector p)

Projection of H3 from Poincare ball to Klein ball. (a, b, c) = 2/(1-|p|^2) * (x, y, z) The parameters may be identical vectors.

Parameters:

p - resulting vector of size 3 in Klein ball

pl - argument vector of size 3 in Poincare ball

klein2Poincare

public static void klein2Poincare(PdVector p,
                                  PdVector pk)

Projection from Klein ball to Poincare ball. (a, b, c) = (x, y, z) / (1+sqrt(1-|p|^2)) The parameters may be identical vectors.

Parameters:

p - resulting vector of size 3 in Poincare ball

pk - argument vector of size 3 in Klein ball

poincare2Uhm

public static void poincare2Uhm(PdVector pu,
                                PdVector p)

Projection of H3 from Poincare ball to UHM. (a, b, c) = (2 x, 2 y, 1 - |p|^2)/(x^2 + y^2 + (z - 1)^2). The parameters may be identical vectors.

Parameters:

pu - resulting vector of size 3 in Upper Halfspace model

p - argument vector of size 3 in Poincare ball

uhm2Poincare

public static void uhm2Poincare(PdVector p,
                                PdVector pu)

Projection from UHM to Poincare ball. (a, b, c) = (2 x, 2 y, |p|^2 - 1)/(x^2 + y^2 + (z + 1)^2). The parameters may be identical vectors.

Parameters:

p - resulting vector of size 3 in Poincare ball

pu - argument vector of size 3 in Upper Halfspace model

klein2Uhm

public static void klein2Uhm(PdVector pu,
                             PdVector p)

Projection of H3 from Klein ball to UHM. (a, b, c) = (x, y, 1 - |p|^2/(1 + sqrt(1 - |p|^2)))/(1 - z). The parameters may be identical vectors.

Parameters:

pu - resulting vector of size 3 in Upper Halfspace model

p - argument vector of size 3 in Klein ball

uhm2Klein

public static void uhm2Klein(PdVector p,
                             PdVector pu)

Projection from UHM to Klein ball. (a, b, c) = (2 x, 2 y, |p|^2 - 1)/(1 + |p|^2). The parameters may be identical vectors.

Parameters:

p - resulting vector of size 3 in Klein ball

pu - argument vector of size 3 in Upper Halfspace model

stereoS3_to_R3

public static void stereoS3_to_R3(PdVector p,
                                  PdVector pl)

Deprecated. use stereographic(PdVector, PdVector)

Stereographic projection of S3 into R3 with projection center (0,0,0,1).

stereographic

public static void stereographic(PdVector p,
                                 PdVector pl)

Stereographic projection of a sphere Sn into Rn with projection center (0,..,0,1). (a_1, .., a_n) = (x_1, ..., x_n)/(1 - x_n+1). The parameters may be identical vectors of any dimension.

Parameters:

p - resulting vector of size n in euclidean space

pl - argument vector of size n+1 in Sn

diffStereoS3_to_R3

public static void diffStereoS3_to_R3(PdMatrix diffMat,
                                      PdVector p)

Compute differential of stereographic projection S3 to R3 with projection center (1,0,0,0).

Parameters:

dp - resulting 3*4 differential matrix of size 4*3

p - argument vector of size 3 in Klein ball

stereoInvR3_to_S3

public static void stereoInvR3_to_S3(PdVector pl,
                                     PdVector p)

Projection from R3 p=(a,b,c) to S4 pl=(x,y,z,t). (x, y, z, t) = ((1 + t) * a, (1 + t) * b, (1 + t) * c, t) with t = (1 - |p|^2)/(1 + |p|^2), 1 + t = 2/(1 + |p|^2). The parameters may be identical vectors.

Parameters:

pl - resulting vector of size 4 in S3

p - argument vector of size 3 in euclidean space

diffStereoInvR3_to_S3

public static void diffStereoInvR3_to_S3(PdMatrix diffMat,
                                         PdVector p)

Compute differential of inverse stereographic projection R3 to S3.

Parameters:

dp - resulting 4*3 differential matrix of size 4*3

p - argument vector of size 3 in euclidean R3