jv.vecmath
Class PuProj

java.lang.Object
  jv.vecmath.PuProj

public class PuProj

extends java.lang.Object

Projection between different models of Euclidean, spherical and hyperbolic space. All vertex projection methods implicitly use the dimension of the argument point which is being projected. The dimension of the resulting projected point depends on the dimension of the target model space.

Coordinate system in Lorentz Rn+1 has signature (1,..,1,-1).

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 in Rn into Lorentz model in Rn+1 of n-dim hyperbolic space.
static void	klein2Poincare(PdVector p, PdVector pk) Projection from Klein ball to Poincare ball of n-dim hyperbolic space.
static void	klein2Uhm(PdVector pu, PdVector p) Projection of H3 from Klein ball to UHM of n-dim hyperbolic space.
static void	lorentz2Klein(PdVector p, PdVector pl) Projection from Lorentz model in Rn+1 into Klein ball in Rn of n-dim hyperbolic space.
static void	lorentz2Poincare(PdVector p, PdVector pl) Projection from Lorentz model in Rn+1 into Poincare ball in Rn of n-dim hyperbolic space.
static void	lorentz2Uhm(PdVector p, PdVector pl) Projection from Lorentz model in Rn+1 into UHM in Rn of n-dim hyperbolic space.
static void	poincare2Klein(PdVector pk, PdVector p) Projection from Poincare ball to Klein ball of n-dim hyperbolic space.
static void	poincare2Lorentz(PdVector pl, PdVector p) Projection from Poincare ball in Rn into Lorentz model in Rn+1 of n-dim hyperbolic space.
static void	poincare2Uhm(PdVector pu, PdVector p) Projection of H3 from Poincare ball to UHM of n-dim hyperbolic space.
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 Rn p=(a,b,c) to Sn 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 of n-dim hyperbolic space.
static void	uhm2Lorentz(PdVector pl, PdVector p) Projection from UHM in Rn into Lorentz model in Rn+1 of n-dim hyperbolic space.
static void	uhm2Poincare(PdVector p, PdVector pu) Projection from UHM to Poincare ball of n-dim hyperbolic space.

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 from Lorentz model in Rn+1 into Klein ball in Rn of n-dim hyperbolic space. For Lorentz R4, a point (x,y,z,t) is mapped to (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 in Klein ball

pl - argument vector in Lorentz model

klein2Lorentz

public static void klein2Lorentz(PdVector pl,
                                 PdVector p)

Projection from Klein ball in Rn into Lorentz model in Rn+1 of n-dim hyperbolic space. For 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 in Klein ball

pl - resulting vector 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:

diffMat - 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 from Lorentz model in Rn+1 into Poincare ball in Rn of n-dim hyperbolic space. For H3, (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 in Poincare ball

pl - argument vector in Lorentz model

poincare2Lorentz

public static void poincare2Lorentz(PdVector pl,
                                    PdVector p)

Projection from Poincare ball in Rn into Lorentz model in Rn+1 of n-dim hyperbolic space. For 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 in Lorentz model

p - argument vector 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:

diffMat - 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 from Lorentz model in Rn+1 into UHM in Rn of n-dim hyperbolic space. For H3, (a, b, c) = (x, y, 1)/(t - z) Coordinate system in Lorentz Rn+1 has signature (1,..,1,-1). The parameters may be identical vectors.

Parameters:

p - resulting vector in Upper Halfspace model

pl - argument vector in Lorentz model

uhm2Lorentz

public static void uhm2Lorentz(PdVector pl,
                               PdVector p)

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

Parameters:

pl - resulting vector in Lorentz model

p - argument vector in Upper Halfspace model

poincare2Klein

public static void poincare2Klein(PdVector pk,
                                  PdVector p)

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

Parameters:

pk - resulting vector in Klein ball

p - argument vector in Poincare ball

klein2Poincare

public static void klein2Poincare(PdVector p,
                                  PdVector pk)

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

Parameters:

p - resulting vector in Poincare ball

pk - argument vector in Klein ball

poincare2Uhm

public static void poincare2Uhm(PdVector pu,
                                PdVector p)

Projection of H3 from Poincare ball to UHM of n-dim hyperbolic space. For H3, (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 in Upper Halfspace model

p - argument vector in Poincare ball

uhm2Poincare

public static void uhm2Poincare(PdVector p,
                                PdVector pu)

Projection from UHM to Poincare ball of n-dim hyperbolic space. For H3, (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 of n-dim hyperbolic space. For H3, (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 in Upper Halfspace model

p - argument vector in Klein ball

uhm2Klein

public static void uhm2Klein(PdVector p,
                             PdVector pu)

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

Parameters:

p - resulting vector in Klein ball

pu - argument vector 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:

diffMat - 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 Rn p=(a,b,c) to Sn 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 in Sn

p - argument vector in Euclidean space Rn

diffStereoInvR3_to_S3

public static void diffStereoInvR3_to_S3(PdMatrix diffMat,
                                         PdVector p)

Compute differential of inverse stereographic projection R3 to S3.

Parameters:

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

p - argument vector of size 3 in euclidean R3