DIRECTORY G2dBasic, G3dBasic, G3dMatrix, G3dVector, Real, RealFns;

G3dVectorImpl: CEDAR PROGRAM
IMPORTS G2dBasic, G3dMatrix, RealFns 
EXPORTS G3dVector

~ BEGIN
Box:					TYPE ~ G3dBasic.Box;
Pair:					TYPE ~ G3dBasic.Pair;
Triple:				TYPE ~ G3dBasic.Triple;
Quad:				TYPE ~ G3dBasic.Quad;
Ray:					TYPE ~ G3dBasic.Ray;
TripleSequence:	TYPE ~ G3dBasic.TripleSequence;
Matrix:				TYPE ~ G3dMatrix.Matrix;
NearSegment:		TYPE ~ G3dVector.NearSegment;
	
origin:				Triple ~ G3dBasic.origin;
Null: PUBLIC PROC [v: Triple] RETURNS [BOOL] ~ {
RETURN[v.x = 0.0 AND v.y = 0.0 AND v.z = 0.0];
};

Negate: PUBLIC PROC [v: Triple] RETURNS [Triple] ~ {
RETURN[[-v.x, -v.y, -v.z]];
};

Unit: PUBLIC PROC [v: Triple] RETURNS [ret: Triple] ~ {
m: REAL ¬ v.x*v.x+v.y*v.y+v.z*v.z;
SELECT m FROM
0.0, 1.0 => RETURN[v];
ENDCASE => {
m ¬ RealFns.SqRt[m];
RETURN[[v.x/m, v.y/m, v.z/m]];
};
};

Mul: PUBLIC PROC [v: Triple, s: REAL] RETURNS [Triple] ~ {
RETURN[[v.x*s, v.y*s, v.z*s]];
};

Div: PUBLIC PROC [v: Triple, s: REAL] RETURNS [Triple] ~ {
RETURN[IF s # 0.0 THEN [v.x/s, v.y/s, v.z/s] ELSE [v.x, v.y, v.z]];
};

Abs: PUBLIC PROC [v: Triple] RETURNS [Triple] ~ {
RETURN[[ABS[v.x], ABS[v.y], ABS[v.z]]];
};
Add: PUBLIC PROC [v1, v2: Triple] RETURNS [Triple] ~ {
RETURN[[v2.x+v1.x, v2.y+v1.y, v2.z+v1.z]];
};

Sub: PUBLIC PROC [v1, v2: Triple] RETURNS [Triple] ~ {
RETURN[[v1.x-v2.x, v1.y-v2.y, v1.z-v2.z]];
};

Equal: PUBLIC PROC [v1, v2: Triple, epsilon: REAL ¬ 0.001] RETURNS [BOOL] ~ {
RETURN[ABS[v1.x-v2.x]<epsilon AND ABS[v1.y-v2.y]<epsilon AND ABS[v1.z-v2.z] < epsilon];
};

Dot: PUBLIC PROC [v1, v2: Triple] RETURNS [REAL] ~ {
RETURN[v1.x*v2.x + v1.y*v2.y + v1.z*v2.z];
};

Cross: PUBLIC PROC [v1, v2: Triple] RETURNS [Triple] ~ {
RETURN[[v1.y*v2.z-v1.z*v2.y, v1.z*v2.x-v1.x*v2.z, v1.x*v2.y-v1.y*v2.x]];
};

Midpoint: PUBLIC PROC [v1, v2: Triple] RETURNS [Triple] ~ {
RETURN[[0.5*(v1.x+v2.x), 0.5*(v1.y+v2.y), 0.5*(v1.z+v2.z)]];
};
Interp: PUBLIC PROC [t: REAL, v1, v2: Triple] RETURNS [Triple] ~ {
RETURN [[v1.x+t*(v2.x-v1.x), v1.y+t*(v2.y-v1.y), v1.z+t*(v2.z-v1.z)]];
};

InterpQuad: PUBLIC PROC [t: REAL, v1, v2: Quad] RETURNS [Quad] ~ {
RETURN[[v1.x+t*(v2.x-v1.x), v1.y+t*(v2.y-v1.y), v1.z+t*(v2.z-v1.z), v1.w+t*(v2.w-v1.w)]];
};

Combine: PUBLIC PROC [v1: Triple, s1: REAL, v2: Triple, s2: REAL] RETURNS [Triple] ~ {
RETURN[[s1*v1.x+s2*v2.x, s1*v1.y+s2*v2.y, s1*v1.z+s2*v2.z]];
};

ScaleRay: PUBLIC PROC [ray: Ray, d: REAL] RETURNS [Triple] ~ {
RETURN[[ray.base.x+d*ray.axis.x, ray.base.y+d*ray.axis.y, ray.base.z+d*ray.axis.z]];
};

MulVectors: PUBLIC PROC [v1, v2: Triple] RETURNS [Triple] ~ {
RETURN[[v1.x*v2.x, v1.y*v2.y, v1.z*v2.z]];
};

DivVectors: PUBLIC PROC [v1, v2: Triple] RETURNS [Triple] ~ {
ret: Triple ¬ v1;
IF v2.x # 0.0 THEN ret.x ¬ ret.x/v2.x;
IF v2.y # 0.0 THEN ret.y ¬ ret.y/v2.y;
IF v2.z # 0.0 THEN ret.z ¬ ret.z/v2.z;
RETURN[ret];
};
UnitizeSequence: PUBLIC PROC [triples: TripleSequence] ~ {
IF triples # NIL THEN FOR n: NAT IN [0..triples.length) DO
triples[n] ¬ Unit[triples[n]];
ENDLOOP;
};

NegateSequence: PUBLIC PROC [triples: TripleSequence] ~ {
IF triples # NIL THEN FOR n: NAT IN [0..triples.length) DO
triples[n] ¬ Negate[triples[n]];
ENDLOOP;
};

MinMaxSequence: PUBLIC PROC [triples: TripleSequence] RETURNS [b: Box] ~ {
huge: REAL ~ Real.LargestNumber;
b.min ¬ [huge, huge, huge];
b.max ¬ [-huge, -huge, -huge];
IF triples = NIL THEN RETURN[[origin, origin]];
FOR n: NAT IN [0..triples.length) DO
t: Triple ~ triples[n];
IF t.x < b.min.x THEN b.min.x ¬ t.x;
IF t.x > b.max.x THEN b.max.x ¬ t.x;
IF t.y < b.min.y THEN b.min.y ¬ t.y;
IF t.y > b.max.y THEN b.max.y ¬ t.y;
IF t.z < b.min.z THEN b.min.z ¬ t.z;
IF t.z > b.max.z THEN b.max.z ¬ t.z;
ENDLOOP;
};
Length: PUBLIC PROC [v: Triple] RETURNS [REAL] ~ {
RETURN[RealFns.SqRt[v.x*v.x+v.y*v.y+v.z*v.z]];
};

Square: PUBLIC PROC [v: Triple] RETURNS [REAL] ~ {
RETURN[v.x*v.x+v.y*v.y+v.z*v.z];
};

Distance: PUBLIC PROC [p1, p2: Triple] RETURNS [REAL] ~ {
a: REAL ¬ p2.x-p1.x;
b: REAL ¬ p2.y-p1.y;
c: REAL ¬ p2.z-p1.z;
RETURN[RealFns.SqRt[a*a+b*b+c*c]];
};

SquareDistance: PUBLIC PROC [p1, p2: Triple] RETURNS [REAL] ~ {
RETURN[Square[Sub[p1, p2]]];
};

SameLength: PUBLIC PROC [v1, v2: Triple] RETURNS [Triple] ~ {
lengthV2: REAL ¬ Length[v2];
RETURN[IF lengthV2 # 0.0 THEN Mul[v2, Length[v1]/lengthV2] ELSE v2];
};

SetVectorLength: PUBLIC PROC [v: Triple, length: REAL] RETURNS [Triple] ~ {
sqLength: REAL ¬ v.x*v.x+v.y*v.y+v.z*v.z;
IF sqLength = 0.0 THEN RETURN[v];
IF sqLength # 1.0 THEN length ¬ length/RealFns.SqRt[sqLength];
RETURN[[v.x*length, v.y*length, v.z*length]];
};
Collinear: PUBLIC PROC [p1, p2, p3: Triple, epsilon: REAL ¬ 0.01] RETURNS [BOOL] ~ {
RETURN[Parallel[Sub[p1, p2], Sub[p2, p3], epsilon]];
};

VecsCoplanar:	PUBLIC PROC [v1, v2, v3: Triple, epsilon: REAL ¬ 0.01] RETURNS [BOOL] ~ {
e1: REAL ¬ v1.x*(v2.y*v3.z-v3.y*v2.z);
e2: REAL ¬ v1.y*(v2.x*v3.z-v3.x*v2.z);
e3: REAL ¬ v1.z*(v2.x*v3.y-v3.x*v2.y);
RETURN[ABS[e2-e1-e3] < epsilon];
};

PointsCoplanar: PUBLIC PROC [p1, p2, p3, p4: Triple, epsilon: REAL ¬ 0.01]
RETURNS [BOOL]
~ {
RETURN[VecsCoplanar[Sub[p4, p1], Sub[p3, p1], Sub[p2, p1], epsilon]];
};

Parallel: PUBLIC PROC [v1, v2: Triple, epsilon: REAL ¬ 0.005] RETURNS [BOOL] ~ {
RETURN[ABS[CosineBetween[v1, v2, TRUE]] > 1.-epsilon];
};

Perpendicular: PUBLIC PROC [v1, v2: Triple, epsilon: REAL ¬ 0.005] RETURNS [BOOL] ~ {
RETURN[ABS[CosineBetween[v1, v2]] < epsilon];
};

FrontFacing: PUBLIC PROC [vector, base: Triple, view: Matrix] RETURNS [BOOL] ~ {
IF G3dMatrix.HasPerspective[view]
THEN RETURN[FrontFacingWithPerspective[vector, base, G3dMatrix.Invert[view]]]
ELSE RETURN[FrontFacingNoPerspective[vector, view]];
};

FrontFacingNoPerspective: PUBLIC PROC [vector: Triple, view: Matrix] RETURNS [BOOL] ~ {
RETURN[vector.x*view[0][2]+vector.y*view[1][2]+vector.z*view[2][2] < 0.0];
};

FrontFacingWithPerspective: PUBLIC PROC [vector, base: Triple, invView: Matrix]
RETURNS [BOOL] -- New, improved!  Faster!!  This should always work.
~ {
q: Quad ¬ [invView[3][0], invView[3][1], invView[3][2], invView[3][3]];
camera: Triple ¬ IF q.w = 1.0 THEN [q.x, q.y, q.z] ELSE [q.x/q.w, q.y/q.w, q.z/q.w];
RETURN[Dot[Sub[camera, base], vector] > 0.0];
};
NearnessAccelerator: PUBLIC PROC [p0, p1: Triple] RETURNS [n: Quad] ~ {
n.x ¬ p1.x-p0.x;
n.y ¬ p1.y-p0.y;
n.z ¬ p1.z-p0.z;
n.w ¬ n.x*n.x+n.y*n.y+n.z*n.z;
IF n.w = 0.0 THEN RETURN;
n.x ¬ n.x/n.w;
n.y ¬ n.y/n.w;
n.z ¬ n.z/n.w;
};

NearestToSegment: PUBLIC PROC [p0, p1, q: Triple, acc: Quad ¬ [0.0, 0.0, 0.0, 0.0]]
RETURNS [nearest: NearSegment]
~ {
IF acc # [0.0, 0.0, 0.0, 0.0]
THEN {
alpha: REAL ¬ (q.x-p0.x)*acc.x+(q.y-p0.y)*acc.y+(q.z-p0.z)*acc.z;
SELECT alpha FROM
<= 0.0 => {
nearest.inside ¬ FALSE;
nearest.point ¬ p0;
nearest.w0 ¬ 1.0;
nearest.w1 ¬ 0.0;
};
>= 1.0 => {
nearest.inside ¬ FALSE;
nearest.point ¬ p1;
nearest.w0 ¬ 0.0;
nearest.w1 ¬ 1.0;
};
ENDCASE => {
nearest.inside ¬ TRUE;
nearest.w1 ¬ alpha;
nearest.w0 ¬ 1.0-nearest.w1;
alpha ¬ alpha*acc.w;
nearest.point ¬ [p0.x+alpha*acc.x, p0.y+alpha*acc.y, p0.z+alpha*acc.z];
};
}
ELSE {
delta: Triple ¬ [p1.x-p0.x, p1.y-p0.y, p1.z-p0.z];
nearest.point ¬ p0;
IF delta = origin
THEN RETURN
ELSE {
ua: Triple ~ [q.x-p0.x, q.y-p0.y, q.z-p0.z];
deltaSq: REAL ~ delta.x*delta.x+delta.y*delta.y+delta.z*delta.z;
alpha: REAL ¬ (ua.x*delta.x+ua.y*delta.y+ua.z*delta.z)/deltaSq;
SELECT alpha FROM
< 0.0 => {nearest.inside ¬ FALSE; alpha ¬ 0.0};
> 1.0 => {nearest.inside ¬ FALSE; alpha ¬ 1.0};
ENDCASE => nearest.inside ¬ TRUE;
nearest.point ¬ [p0.x+alpha*delta.x, p0.y+alpha*delta.y, p0.z+alpha*delta.z];
nearest.w1 ¬ alpha;
nearest.w0 ¬ 1.0-alpha;
};
};
};

NearestToLine: PUBLIC PROC [line: Ray, q: Triple] RETURNS [Triple] ~ {
u: Triple ¬ Unit[line.axis];
RETURN[Add[line.base, Mul[u, Dot[Sub[q, line.base], u]]]];
};

PointOnLine: PUBLIC PROC [p: Triple, line: Ray, epsilon: REAL ¬ 0.005] RETURNS [BOOL] ~ {
RETURN[Distance[p, NearestToLine[line, p]] < epsilon];
};

NearestToSequence: PUBLIC PROC [p: Triple, points: TripleSequence] RETURNS [index: NAT ¬ 0]
~ {
minSqDist: REAL ¬ Real.LargestNumber;
IF points # NIL THEN FOR n: NAT IN [0..points.length) DO
sqDist: REAL ¬ SquareDistance[p, points[n]];
IF sqDist < minSqDist THEN {index ¬ n; minSqDist ¬ sqDist};
ENDLOOP;
};

ClosestApproach2Lines: PUBLIC PROC [line1, line2: Ray] RETURNS [p1, p2: Triple] ~ {
axis1: Triple ¬ Unit[line1.axis];
axis2: Triple ¬ Unit[line2.axis];
den: REAL ¬ Dot[axis1, axis2];
densq: REAL ¬ den*den;
dif: Triple ¬ Sub[line1.base, line2.base];
dot1: REAL ¬ Dot[axis1, dif];
dot2: REAL ¬ Dot[axis2, dif];
s: REAL ¬ (dot1/densq-dot2/den)/(1.0-1.0/densq);
t: REAL ¬ (s+dot1)/den;
p1 ¬ Add[line1.base, Mul[axis1, s]];
p2 ¬ Add[line2.base, Mul[axis2, t]];
};

DistanceBetween2Lines: PUBLIC PROC [line1, line2: Ray] RETURNS [REAL] ~ {
x21: REAL ¬ line2.base.x-line1.base.x;
y21: REAL ¬ line2.base.y-line1.base.y;
z21: REAL ¬ line2.base.z-line1.base.z;
ab: REAL ¬ line1.axis.x*line2.axis.y-line2.axis.x*line1.axis.y;
bc: REAL ¬ line1.axis.y*line2.axis.z-line2.axis.y*line1.axis.z;
ca: REAL ¬ line1.axis.z*line2.axis.x-line2.axis.z*line1.axis.x;
den: REAL ¬ ab*ab+bc*bc+ca*ca;
IF den < 0.0001
THEN RETURN[Distance[line1.base, NearestToLine[line2, line1.base]]]
ELSE RETURN[ABS[x21*bc+y21*ca+z21*ab]/RealFns.SqRt[den]];
};

V90: PUBLIC PROC [v0, v1: Triple, unitize: BOOL ¬ TRUE] RETURNS [t: Triple] ~ { 
dot: REAL ¬ Dot[v0, v1];
t ¬ IF ABS[dot] > 0.000001 THEN [v1.x-dot*v0.x, v1.y-dot*v0.y, v1.z-dot*v0.z] ELSE v1;
IF unitize THEN t ¬ Unit[t];
};

Ortho: PUBLIC PROC [v: Triple, crosser: Triple ¬ [0.0, 0.0, 0.0]] RETURNS [Triple] ~ {
length: REAL ¬ Length[v];
IF crosser = [0.0, 0.0, 0.0] THEN crosser ¬ [v.z, -v.x, v.y];
RETURN[IF length = 0.0 THEN [0.0, 0.0, 0.0] ELSE Mul[Unit[Cross[v, crosser]], length]];
};

RotateAbout: PUBLIC PROC [v, axis: Triple, a: REAL, degrees: BOOL ¬ TRUE]
RETURNS [t: Triple] ~ {
rotate: Matrix ¬ G3dMatrix.MakeRotate[axis, a, degrees, G3dMatrix.ObtainMatrix[]];
t ¬ G3dMatrix.TransformVec[v, rotate];
G3dMatrix.ReleaseMatrix[rotate];
};
PolarFromCartesian: PUBLIC PROC [cartesian: Triple] RETURNS [Triple] ~ {
xySum: REAL ¬ cartesian.x*cartesian.x+cartesian.y*cartesian.y;
lng: REAL ¬ RealFns.ArcTanDeg[cartesian.y, cartesian.x];
lat: REAL ¬ RealFns.ArcTanDeg[cartesian.z, RealFns.SqRt[xySum]];
mag: REAL ¬ RealFns.SqRt[xySum+cartesian.z*cartesian.z];
RETURN[[lng, lat, mag]];
};

CartesianFromPolar: PUBLIC PROC [polar: Triple] RETURNS [Triple] ~ {
cosmag: REAL ¬ RealFns.CosDeg[polar.y]*polar.z;
RETURN[[
cosmag*RealFns.CosDeg[polar.x],
cosmag*RealFns.SinDeg[polar.x],
polar.z*RealFns.SinDeg[polar.y]]];
};
Clip: PUBLIC PROC [p: Triple, box: Box] RETURNS [clipped: Triple] ~ {
clipped ¬ [
MIN[MAX[box.min.x, p.x], box.max.x],
MIN[MAX[box.min.y, p.y], box.max.y],
MIN[MAX[box.min.z, p.z], box.max.z]];
};

Project: PUBLIC PROC [v1, v2: Triple] RETURNS [Triple] ~ {
RETURN[IF v2 = origin THEN v2 ELSE Mul[v2, Dot[v1, v2]/Square[v2]]];
};

Average: PUBLIC PROC [triples: TripleSequence] RETURNS [average: Triple] ~ {
average ¬ origin;
FOR n: NAT IN [0..triples.length) DO
average ¬ Add[average, triples[n]];
ENDLOOP;
IF triples.length # 0 THEN average ¬ Div[average, triples.length];
};

CosineBetween: PUBLIC PROC [v0, v1: Triple, unitize: BOOL ¬ FALSE]
RETURNS [r: REAL]
~ {
IF v0 = origin OR v1 = origin THEN RETURN[0.0];
r ¬ (v0.x*v1.x+v0.y*v1.y+v0.z*v1.z);
IF unitize THEN r ¬
r/RealFns.SqRt[(v0.x*v0.x+v0.y*v0.y+v0.z*v0.z)*(v1.x*v1.x+v1.y*v1.y+v1.z*v1.z)]
};
AngleBetween: PUBLIC PROC [
v0, v1: Triple,
degrees: BOOL ¬ TRUE,
unitize: BOOL ¬ FALSE]
RETURNS [REAL] ~ {
RETURN[IF degrees
THEN G2dBasic.ArcCosDeg[CosineBetween[v0, v1, unitize]]
ELSE G2dBasic.ArcCos[CosineBetween[v0, v1, unitize]]];
};
END.


0
G3dVectorImpl.mesa
Copyright Ó 1984, 1992 by Xerox Corporation.  All rights reserved.
Bloomenthal, April 9, 1993 4:11 pm PDT
Heckbert, June 23, 1988 1:49:10 am PDT
Glassner, February 14, 1991 9:02 am PST
Types
Basic Operations on a Single Vector
Basic Operations on Two Vectors

Basic Operations on Sequence of Vectors
Length and Distance Operations
Directional Tests
Compute determinant of matrix of vectors:
The following can be completely bogus (in particular, it can ignore epsilon):
ratio: REAL _ Real.LargestNumber;
IF v1 = v2 THEN RETURN[TRUE];
IF (v1.x=0) # (v2.x=0) OR (v1.y=0) # (v2.y=0) OR (v1.z=0) # (v2.z=0)
THEN RETURN[FALSE]; 
IF v1.x # 0.0 THEN ratio _ v2.x/v1.x;
IF v1.y # 0.0 THEN {
IF ratio = Real.LargestNumber
THEN ratio _ v2.y/v1.y
ELSE IF ABS[ratio-(v2.y/v1.y)] > epsilon THEN RETURN[FALSE];
};
IF ratio = Real.LargestNumber OR v1.z = 0.0 THEN RETURN[TRUE];
RETURN[ABS[ratio-(v2.z/v1.z)] <= epsilon];
This presumes invView to be the INVERSE of the actual world-to-view matrix.
same as: camera: Triple ¬ G3dMatrix.Transform[[0, 0, 0], invView]:
the following way is fine, and equivalent to the above, but takes more multiplies:
plane: Plane ¬ G3dPlane.FromPointAndNormal[base, vector];
RETURN[
invView[2][0]*plane.x+
invView[2][1]*plane.y+
invView[2][2]*plane.z+
invView[2][3]*plane.w < 0.0];

FrontFacingWithPerspective: PUBLIC PROC [vector, base: Triple, view: Matrix]
RETURNS [BOOL] -- Way it was yesterday . . .  (this is faulty)
~ {
end: Triple ~ Add[base, vector];
z0: REAL _ base.x*view[0][2]+base.y*view[1][2]+base.z*view[2][2]+view[3][2];
z1: REAL _ end.x*view[0][2]+end.y*view[1][2]+end.z*view[2][2]+view[3][2];
RETURN[z1 < z0];
};

FrontFacingWithPerspective: PUBLIC PROC [vector, base: Triple, view: Matrix]
RETURNS [BOOL] -- Way it was a few minutes ago . . .
(this probably corrects above as long as vector doesn't go behind the eye!)
~ {
end: Triple ~ Add[base, vector];
z0: REAL _ base.x*view[0][2]+base.y*view[1][2]+base.z*view[2][2]+view[3][2];
w0: REAL _ base.x*view[0][3]+base.y*view[1][3]+base.z*view[2][3]+view[3][3];
z1: REAL _ end.x*view[0][2]+end.y*view[1][2]+end.z*view[2][2]+view[3][2];
w1: REAL _ end.x*view[0][3]+end.y*view[1][3]+end.z*view[2][3]+view[3][3];
RETURN[z1/w1 < z0/w0];
};
Nearness Operations
 [Artwork node; type 'Artwork on' to command tool] 
Presumes p1 and p2 form a line segment that is perpendicular to each of the lines.
p1 and p2 are given as line1.base+s(line1.axis) and line2.base+t(line2.axis);
(p2-p1).line1.axis = 0 and (p2-p1).line2.axis = 0.
From Bowyer and Woodwark (A Programmer's Geometry)
P. Heckbert writes
It looks like the Bowyer and Woodwark formula is equivalent to
abs((P2-P1).D1xD2) / ||D1XD2||
where A.BXC is a triple scalar product, equivalent to the determinant of the 3x3 matrix
with 3-vectors A, B, and C as rows.  Their variables ab, bc, ca are components of vector
D1xD2.  Since ||D1xD2|| = ||D1|| * ||D2|| * |sin(theta)|, and A.BxC is the volume of the
parallelepiped with sides A, B, and C, imagine this parallelepiped with one edge parallel to
and sliding along line 1, one edge sliding along line 2, and the other edge connecting one
point of line 1 (P1) to one point of line 2 (P2).  If you kept the edge on line 1 fixed and
changed P2, the origin point of line 2, stretching the parallelogram like a block of jello,
the volume of the block would stay fixed because its base has constant area of ||D1xD2||
and its altitude is constant (equal to the distance between the lines). So you can pick any
points P1 and P2 that you like and the volume is determined by the sin of the angles
between the lines and the distance between the lines.  If you divide the volume of the
parallelepiped by its base area, you get the height, which is the distance between the
lines.  Cool!
ClosestApproach2Lines: PUBLIC PROC [line1, line2: Ray] RETURNS [p1, p2: Triple] ~ {
This works, but ....
a, aa: REAL _ 1000000.0;
sense: BOOL _ TRUE;
DO
IF (sense _ NOT sense)
THEN p1 _ NearestToLine[line1, p2]
ELSE  p2 _ NearestToLine[line2, p1];
IF (a _ Distance[p1, p2]) = 0.0 OR ABS[aa-a]/a < 0.0000001 THEN EXIT;
aa _ a;
ENDLOOP;
};
Simple Geometric Operations
IF Parallel[crosser, v] THEN crosser _ [crosser.z, crosser.y, crosser.x];
Polar/Cartesian Coordinates
 [Artwork node; type 'Artwork on' to command tool] 
These procedures assume right-handed coordinate system: x to the right, y away, and z up. 
The polar triple consists of longitude, latitude, and magnitude.
Latitude is deviation from the horizon; i.e., latitude = 90 is North Pole, -90 is South Pole.
Longitude is rotation about the vertical; i.e., longitude = 0 is the xAxis, 90 is the yAxis.

Miscellany


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Produced by version 8811.30

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