jv.vecmath

Class PuVectorGeom

java.lang.Object

jv.vecmath.PuVectorGeom

public class PuVectorGeom
extends java.lang.Object

Static methods for vector geometry. Most methods work in Euclidean vector spaces of arbitrary dimension.

Author:

Konrad Polthier

Version:

17.10.01, 1.30 revised (ep) Method added that calculates a sequence of rotations around y-,x-,z-axis to transform a given frame to the standard Euclidean frame.
03.10.99, 1.10 revised (kp) Renamed from PdVectorGeom to PuVectorGeom.
28.08.99, 1.00 revised (kp) Several beautifications.
28.08.99, 1.00 created (kp) Extracted from PdVector, methods made static.

Constructor Summary

Constructors

Constructor and Description
PuVectorGeom()

Method Summary

Modifier and Type	Method and Description
static boolean	circleThruPoints(PdVector center, double radius, PdVector p, PdVector q, PdVector r) Compute circle through three given points.
static void	ctg(double[] ctg, PdVector p, PdVector q, PdVector r) Compute cotangent of the vertex angles at all vertices of the triangle (p, q, r).
static double	ctg(PdVector p, PdVector q1, PdVector q2) Compute cotangent of the vertex angle at vertex p in the triangle (p, q1, q2).
static double	distOfLineToLine(PdVector base1, PdVector dir1, PdVector base2, PdVector dir2) Compute distance of line to line.
static double	distOfPointToLine(PdVector p, PdVector base, PdVector dir) Compute distance of point to line.
static double	distOfPointToPlane(PdVector p, PdVector base, PdVector normal) Compute distance of point to plane.
static double	distVectorOfLineToLine(PdVector lot, PdVector base1, PdVector dir1, PdVector base2, PdVector dir2) Compute shortest distance vector of line to line.
static void	distVectorOfPointToLine(PdVector lot, PdVector p, PdVector base, PdVector dir) Compute shortest distance vector of point to line.
static void	distVectorOfPointToPlane(PdVector lot, PdVector p, PdVector base, PdVector normal) Compute shortest distance vector of point to plane.
static boolean	evalCircle(PdVector p, PdVector mid, PdVector orient, PdVector start, PdVector end, double t) Compute a point between the two end points of a circle segment.
static boolean	evalHelix(PdVector p, PdVector axisBot, PdVector axisDir, PdVector start, PdVector end, double t) Compute a point between the two end points of a segment of a helix.
static double[]	frameToStandardFrame(PdVector first, PdVector second, PdVector third) Computes the necessary rotations around the y, x and z axis, to transform a given orthonormal frame into the standard basis of R^3.
static double	intersectionOfLineAndLine(PdVector p, PdVector base1, PdVector dir1, PdVector base2, PdVector dir2) Check for intersection of line and line by solving equation b1+sd1 = b2+td2.
static double	intersectionOfLineAndPlane(PdVector p, PdVector base1, PdVector dir, PdVector base2, PdVector normal) Compute intersection line and plane.
static boolean	intersectionOfPlaneAndPlane(PdVector lineBase, PdVector lineDir, PdVector base1, PdVector normal1, PdVector base2, PdVector normal2) Compute intersection of plane and plane, and return the intersection line.
static void	projectOntoLine(PdVector v, PdVector dir) Project a given vector to a line through the origin.
static void	projectOntoLine(PdVector vProj, PdVector v, PdVector dir) Project a given vector to a line through the origin.
static void	projectOntoPlane(PdVector v, PdVector normal) Project a given vector to a plane through the origin.
static void	projectOntoPlane(PdVector vProj, PdVector v, PdVector normal) Project a given vector to a plane through the origin.
static void	projectPointToCircle(PdVector proj, PdVector p, PdVector mid, PdVector normal, double radius) Project 3d-point onto circle in 3d by project point onto plane of circle adjust distance to mid point.
static void	projectPointToLine(PdVector proj, PdVector p, PdVector base, PdVector dir) Project a given point to line.
static void	projectPointToPlane(PdVector proj, PdVector p, PdVector base, PdVector normal) Project a given point to plane.
static boolean	rotatePointAroundLine(PdVector pRot, PdVector p, PdVector axisBase, PdVector axisDir, double alpha) Rotate a point around an arbitrary axis by a given angle.
static boolean	rotatePointAroundVector(PdVector pRot, PdVector p, PdVector axisDir, double alpha) Rotate a point around an axis through the origin by a given angle.
static double	sphericalAngle(PdVector p, PdVector q1, PdVector q2) Compute spherical angle at vertex p or a triangle on the unit sphere in S^n.
static double	sphericalArea(PdVector p, PdVector q, PdVector r) Compute area of a spherical triangle on the unit sphere in S^n.

Methods inherited from class java.lang.Object

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

Constructor Detail

PuVectorGeom

public PuVectorGeom()

Method Detail

intersectionOfLineAndLine

public static double intersectionOfLineAndLine(PdVector p,
                                               PdVector base1,
                                               PdVector dir1,
                                               PdVector base2,
                                               PdVector dir2)

Check for intersection of line and line by solving equation b1+sd1 = b2+td2. Multiply both sides with d1 and with d2 to obtain two scalar equations

 
   <b1,d1> + s = <b2,d1> + t<d2,d1>
   <b1,d2> + s<d1,d2> = <b2,d2> + t
   => s = <b2-b1, d1-<d1,d2>d2>/(1.-<d1,d2>**2)
 

Method works in Euclidean vector spaces of arbitrary dimension.

Parameters:

p - computed intersection point

base1 - any point on the first line

dir1 - unit direction of first line

base2 - any point on the second line

dir2 - unit direction of second line

Returns:

double distance s from base1 to intersection point, or Double.MAX_VALUE if line and plane are parallel.

Author:

Konrad Polthier

Version:

04.03.01, 1.15 revised (kp) Return value changed to double from boolean.
28.08.99, 1.10 revised (kp) Several beautifications.

intersectionOfLineAndPlane

public static double intersectionOfLineAndPlane(PdVector p,
                                                PdVector base1,
                                                PdVector dir,
                                                PdVector base2,
                                                PdVector normal)

Compute intersection line and plane. Line is given by base point and direction, plane is given by base point and plane normal. The intersection point x=ray+s*dir on the line must fulfill plane equation <n, x> = <n, b2>. Use this equation to derive a value for s by solving <n, b1+sd> = <n,b2> leading to s = <n, b2-b1>/<n, d>. Direction of ray must be normalized, normal of plane need not be normalized.

Method works in Euclidean vector spaces of arbitrary dimension.

Parameters:

p - computed intersection point

base1 - any point on the line

dir - unit direction of line

base2 - any point on the plane

normal - normal of plane, need not be normalized

Returns:

double distance s from base1 to intersection point, or Double.MAX_VALUE if line and plane are parallel.

Author:

Konrad Polthier

Version:

28.08.99, 1.10 revised (kp) Several beautifications.

intersectionOfPlaneAndPlane

public static boolean intersectionOfPlaneAndPlane(PdVector lineBase,
                                                  PdVector lineDir,
                                                  PdVector base1,
                                                  PdVector normal1,
                                                  PdVector base2,
                                                  PdVector normal2)

Compute intersection of plane and plane, and return the intersection line.

   lineDir parallel to n1xn2
   lineBase = s*n1 + t*n2.
 

Insert in both plane equations and solve for s and t:

 
   <n1, sn1 + tn2> = <n1, b1>
   <n2, sn1 + tn2> = <n2, b2>
   s + t<n1,n2> = <n1,b1>
   s<n1,n2> + t = <n2,b2>
   => s = (<n1,b1>-<n2,b2><n1,n2>)/(1.-<n1,n2>**2)
   => t = (<n2,b2>-<n1,b1><n1,n2>)/(1.-<n1,n2>**2)
 

Method only works in Euclidean 3-space.

Parameters:

lineBase - computed point on the intersection line

lineDir - computed direction of the intersection line

base1 - base point of plane1

base2 - base point of plane2

normal1 - unit normal of plane1

normal2 - unit normal of plane2

Returns:

true if both planes intersect

Author:

Konrad Polthier

Version:

28.08.99, 1.10 revised (kp) Several beautifications.

distVectorOfPointToLine

public static void distVectorOfPointToLine(PdVector lot,
                                           PdVector p,
                                           PdVector base,
                                           PdVector dir)

Compute shortest distance vector of point to line.

Method works in Euclidean vector spaces of arbitrary dimension.

Parameters:

lot - resulting distance vector

p - given point

base - some point of the line

dir - unit direction of the line

See Also:

projectPointToLine(jv.vecmath.PdVector, jv.vecmath.PdVector, jv.vecmath.PdVector, jv.vecmath.PdVector)

Author:

Konrad Polthier

Version:

28.08.99, 1.10 revised (kp) Several beautifications.

distVectorOfPointToPlane

public static void distVectorOfPointToPlane(PdVector lot,
                                            PdVector p,
                                            PdVector base,
                                            PdVector normal)

Compute shortest distance vector of point to plane.

Method works in Euclidean vector spaces of arbitrary dimension.

Parameters:

lot - resulting distance vector

p - given point

base - some point on the plane

normal - unit normal of plane

See Also:

projectPointToPlane(jv.vecmath.PdVector, jv.vecmath.PdVector, jv.vecmath.PdVector, jv.vecmath.PdVector)

Author:

Konrad Polthier

Version:

28.08.99, 1.10 revised (kp) Several beautifications.

distVectorOfLineToLine

public static double distVectorOfLineToLine(PdVector lot,
                                            PdVector base1,
                                            PdVector dir1,
                                            PdVector base2,
                                            PdVector dir2)

Compute shortest distance vector of line to line. Method only works in Euclidean 3-spaces.

Parameters:

lot - resulting distance vector

base1 - base point of line1

base2 - base point of line2

dir1 - direction of line1

dir2 - direction of line2

Returns:

distance of both lines, or Double.MAX_VALUE if parallel.

See Also:

distOfLineToLine(jv.vecmath.PdVector, jv.vecmath.PdVector, jv.vecmath.PdVector, jv.vecmath.PdVector)

Author:

Konrad Polthier

Version:

28.08.99, 1.10 revised (kp) Several beautifications.

distOfPointToLine

public static double distOfPointToLine(PdVector p,
                                       PdVector base,
                                       PdVector dir)

Compute distance of point to line. Use formula dist = |(p-base) x dir|.

Method works in Euclidean vector spaces of arbitrary dimension.

Parameters:

p - given point

base - some point of the line

dir - direction of the line with unit length 1.

Returns:

distance of point to line, or Double.MAX_VALUE if line direction not normalized.

Author:

Konrad Polthier

Version:

15.09.10, 1.20 revised (fk) Avoid PdVector.subNew.
18.07.00, 1.10 revised (kp) Return Double.MAX_VALUE if line direction not normalized.
28.08.99, 1.00 revised (kp) Several beautifications.

distOfPointToPlane

public static double distOfPointToPlane(PdVector p,
                                        PdVector base,
                                        PdVector normal)

Compute distance of point to plane. Use formula dist = <p-base, normal>, that means, we use the signed distance which is positive if p lies in direction of the normal.

Method works in Euclidean vector spaces of arbitrary dimension.

Parameters:

p - given point

base - some point of the line

normal - unit normal of plane with length 1.

Returns:

distance signed distance is positive if p is in direction of normal

Author:

Konrad Polthier

Version:

15.09.10, 1.30 revised (fk) Avoid PdVector.subNew.
05.09.02, 1.20 revised (kp) Return signed distance as written in documentation.
18.07.00, 1.10 revised (kp) Return Double.MAX_VALUE if line direction not normalized.
28.08.99, 1.00 revised (kp) Several beautifications.

distOfLineToLine

public static double distOfLineToLine(PdVector base1,
                                      PdVector dir1,
                                      PdVector base2,
                                      PdVector dir2)

Compute distance of line to line. Use formula dist = |<base1-base2, dir1xdir2>|/|dir1xdir2|. Special care is taken if directions are degenerate.

Currently, method works in Euclidean 3-space only.

Parameters:

base1 - base point of line1

base2 - base point of line2

dir1 - direction of line1

dir2 - direction of line2

Returns:

distance

Author:

Konrad Polthier

Version:

02.12.00, 1.10 revised (kp) Catch degenerate directions.
28.08.99, 1.00 revised (kp) Several beautifications.

projectPointToLine

public static void projectPointToLine(PdVector proj,
                                      PdVector p,
                                      PdVector base,
                                      PdVector dir)

Project a given point to line. Use formula proj = base + <p-base, dir>dir. Point and projected point may be the same vector instances.

Method works in Euclidean vector spaces of arbitrary dimension.

Parameters:

proj - projection of p to the line

p - given point

base - some point of the line

dir - unit direction of the line

Author:

Konrad Polthier

Version:

28.08.99, 1.00 revised (kp) Several beautifications.

projectPointToPlane

public static void projectPointToPlane(PdVector proj,
                                       PdVector p,
                                       PdVector base,
                                       PdVector normal)

Project a given point to plane. Use formula p - <p-base, normal>normal. Point and projected point may be the same vector instances.

Method works in Euclidean vector spaces of arbitrary dimension.

Parameters:

proj - projection of p to the plane

p - given point

base - some point of the plane

normal - unit normal of plane

Author:

Konrad Polthier

Version:

28.08.99, 1.00 revised (kp) Several beautifications.

projectPointToCircle

public static void projectPointToCircle(PdVector proj,
                                        PdVector p,
                                        PdVector mid,
                                        PdVector normal,
                                        double radius)

Project 3d-point onto circle in 3d by

1. project point onto plane of circle
2. adjust distance to mid point.

Point and projected point may be the same vector instances.

Method works in Euclidean vector spaces of arbitrary dimension.

Parameters:

proj - projection of p to the line

p - given point

mid - some point of the plane

normal - unit normal of plane

Author:

Konrad Polthier

Version:

15.09.10, 1.10 revised (fk) Avoid PdVector.subNew.
28.08.99, 1.00 revised (kp) Several beautifications.

projectOntoLine

public static void projectOntoLine(PdVector v,
                                   PdVector dir)

Project a given vector to a line through the origin. Use formula v = <v,dir>dir/|dir|^2.

Method works in Euclidean vector spaces of arbitrary dimension.

Parameters:

v - given vector, also contains the result

dir - unit direction of the line

Author:

Konrad Polthier

Version:

28.08.99, 1.00 revised (kp) Several beautifications.

projectOntoLine

public static void projectOntoLine(PdVector vProj,
                                   PdVector v,
                                   PdVector dir)

Project a given vector to a line through the origin. Use formula vProj = <v,dir>dir/|dir|^2. Vector and projected vector may be the same vector instances.

Method works in Euclidean vector spaces of arbitrary dimension.

Parameters:

vProj - projection of v to the line through the origin

v - given vector

dir - unit direction of the line

Author:

Konrad Polthier

Version:

28.08.99, 1.00 revised (kp) Several beautifications.

projectOntoPlane

public static void projectOntoPlane(PdVector v,
                                    PdVector normal)

Project a given vector to a plane through the origin. Use formula v = v-<v,normal>normal/|normal|^2.

Method works in Euclidean vector spaces of arbitrary dimension.

Parameters:

v - given vector, also contains the result

normal - unit normal of the plane

Author:

Konrad Polthier

Version:

28.08.99, 1.00 revised (kp) Several beautifications.

projectOntoPlane

public static void projectOntoPlane(PdVector vProj,
                                    PdVector v,
                                    PdVector normal)

Project a given vector to a plane through the origin. Use formula vProj = v-<v,normal>normal/|normal|^2. Vector and projected vector may be the same vector instances.

Method works in Euclidean vector spaces of arbitrary dimension.

Parameters:

vProj - projection of v to the plane through the origin

v - given vector

normal - unit normal of the plane

Author:

Konrad Polthier

Version:

28.08.99, 1.00 revised (kp) Several beautifications.

circleThruPoints

public static boolean circleThruPoints(PdVector center,
                                       double radius,
                                       PdVector p,
                                       PdVector q,
                                       PdVector r)

Compute circle through three given points. Use formula dist = |<base1-base2, dir1xdir2>|/|dir1xdir2|.

Method works in Euclidean vector spaces of arbitrary dimension.

Parameters:

center - resulting center of circle

radius - resulting radius of circle

p - given point

q - given point

r - given point

Returns:

true if circle could be computed.

Author:

Konrad Polthier

Version:

28.08.99, 1.00 revised (kp) Several beautifications.

evalCircle

public static boolean evalCircle(PdVector p,
                                 PdVector mid,
                                 PdVector orient,
                                 PdVector start,
                                 PdVector end,
                                 double t)

Compute a point between the two end points of a circle segment. The three points mid, start and end determine the plane of the circle. The vector orient determines the orientation of the circle segment from start to end, i.e. how to run around the circle. The value t specifies the relative position on the segment, t=0 corresponds to start and t=1 corresponds to end.

Method only works in Euclidean 3-space.

Parameters:

p - calculated point on circle segment

mid - center of circle

orient - normal vector determines which angle to take

start - first point on circle segment

end - second point on circle segment

t - relative position in [0.,1.] between start and end on segment

Returns:

true if circle could be evaluated.

See Also:

PdVector.angleWithOrientation(PdVector, PdVector, PdVector), rotatePointAroundLine(PdVector, PdVector, PdVector, PdVector, double)

Author:

Konrad Polthier

Version:

28.08.99, 1.00 revised (kp) Several beautifications.

evalHelix

public static boolean evalHelix(PdVector p,
                                PdVector axisBot,
                                PdVector axisDir,
                                PdVector start,
                                PdVector end,
                                double t)

Compute a point between the two end points of a segment of a helix. The point axisBot and direction axisDir determine an axis which generates a helix when rotating and translating the point start around the axis to the point end. The value t specifies the relative position on the segment, t=0 corresponds to start and t=1 corresponds to end.

Method only works in Euclidean 3-space.

Parameters:

p - calculated point on helix segment

axisBot - point on axis of helix

axisDir - unit direction of axis of helix

start - first point on helix

end - second point on helix

t - relative position in [0.,1.] between start and end on segment

Returns:

true if helix could be evaluated.

See Also:

PdVector.angleWithOrientation(PdVector, PdVector, PdVector), rotatePointAroundLine(PdVector, PdVector, PdVector, PdVector, double)

Author:

Konrad Polthier

Version:

28.08.99, 1.00 revised (kp) Several beautifications.

rotatePointAroundVector

public static boolean rotatePointAroundVector(PdVector pRot,
                                              PdVector p,
                                              PdVector axisDir,
                                              double alpha)

Rotate a point around an axis through the origin by a given angle.

Method only works in Euclidean 3-space.

Parameters:

pRot - rotated point

p - point to rotate. May be equal to pRot.

axisDir - unit direction of rotation axis through origin

alpha - rotation angle, in radians

Returns:

true if rotation could be performed.

See Also:

rotatePointAroundLine(PdVector,PdVector,PdVector,PdVector,double)

Author:

Konrad Polthier

Version:

28.08.99, 1.00 revised (kp) Several beautifications.

rotatePointAroundLine

public static boolean rotatePointAroundLine(PdVector pRot,
                                            PdVector p,
                                            PdVector axisBase,
                                            PdVector axisDir,
                                            double alpha)

Rotate a point around an arbitrary axis by a given angle. Axis need not go through the origin.

Method only works in Euclidean 3-space.

Parameters:

pRot - rotated point

p - point to rotate

axisBase - some point on the rotation axis

axisDir - unit direction of rotation axis

alpha - rotation angle

Returns:

true if rotation could be performed.

See Also:

rotatePointAroundVector(PdVector,PdVector,PdVector,double)

Author:

Konrad Polthier

Version:

28.08.99, 1.00 revised (kp) Several beautifications.

ctg

public static double ctg(PdVector p,
                         PdVector q1,
                         PdVector q2)

Compute cotangent of the vertex angle at vertex p in the triangle (p, q1, q2).

Method works in Euclidean vector spaces of arbitrary dimension.

Parameters:

p - vertex where to compute the cotangent

q1 - other vertex of the triangle

q2 - other vertex of the triangle

Returns:

cotangent of vertex angle

See Also:

ctg(double[],PdVector,PdVector,PdVector)

Author:

Konrad Polthier

Version:

15.09.10, 1.10 revised (fk) Avoid PdVector.subNew.
28.08.99, 1.00 revised (kp) Several beautifications.

ctg

public static void ctg(double[] ctg,
                       PdVector p,
                       PdVector q,
                       PdVector r)

Compute cotangent of the vertex angles at all vertices of the triangle (p, q, r).

Method works in Euclidean vector spaces of arbitrary dimension.

Parameters:

ctg - cotangent of all vertex angles

p - vertex of the triangle

q - vertex of the triangle

r - vertex of the triangle

See Also:

ctg(PdVector,PdVector,PdVector), PuMath.ctg(double[],double,double,double)

Author:

Konrad Polthier

Version:

28.08.99, 1.00 revised (kp) Several beautifications.

sphericalAngle

public static double sphericalAngle(PdVector p,
                                    PdVector q1,
                                    PdVector q2)

Compute spherical angle at vertex p or a triangle on the unit sphere in S^n. All vertices must lie on a unit sphere, i.e. they must have unit length.

Method works in Euclidean vector spaces of arbitrary dimension.

Parameters:

p - vertex where to compute the angle

q1 - other vertex of the triangle

q2 - other vertex of the triangle

Returns:

spherical vertex angle at p

See Also:

PdVector.angle(PdVector,PdVector,PdVector)

Author:

Konrad Polthier

Version:

28.08.99, 1.00 revised (kp) Several beautifications.

sphericalArea

public static double sphericalArea(PdVector p,
                                   PdVector q,
                                   PdVector r)

Compute area of a spherical triangle on the unit sphere in S^n. All vertices must lie on a unit sphere, i.e. they must have unit length.

Method works in Euclidean vector spaces of arbitrary dimension.

Parameters:

p - vertex of the triangle

q - vertex of the triangle

r - vertex of the triangle

Returns:

area of the spherical triangle

See Also:

sphericalAngle(PdVector,PdVector,PdVector)

Author:

Konrad Polthier

Version:

28.08.99, 1.00 revised (kp) Several beautifications.

frameToStandardFrame

public static double[] frameToStandardFrame(PdVector first,
                                            PdVector second,
                                            PdVector third)

Computes the necessary rotations around the y, x and z axis, to transform a given orthonormal frame into the standard basis of R^3. The first vector of the frame is mapped to the x-axis, the second to the y-axis and so on.

So if you want to get these rotation angles for e.g. the camera position, then you should call this method with normalized first = viewdir x upvector, second = upvector and third = -viewdir.

Parameters:

first - First vector of frame becomes x-axis. FRAME MUST BE ORTHONORMAL! Orthonormality is not checked by this method!

second - Second vector of frame becomes y-axis. See first.

third - Third vector of frame becomes z-axis. See first.

Returns:

Array of angles (RAD): angle to rotate around y-axis, around x-axis and around z-axis in this order.

Author:

Eike Preuß

Version:

17.10.01, 1.00 created (ep)