DIRECTORY
Matrix3d, RealFns, SV3d, SVLines3d, SVVector3d;

SVLines3dImpl: CEDAR PROGRAM
IMPORTS Matrix3d, RealFns, SVVector3d
EXPORTS SVLines3d
 =
BEGIN

Edge3d: TYPE = SV3d.Edge3d;
Edge3dObj: TYPE = SV3d.Edge3dObj;
Line3d: TYPE = SV3d.Line3d;
Line3dObj: TYPE = SV3d.Line3dObj;
Matrix4by4: TYPE = SV3d.Matrix4by4;
Point3d: TYPE = SV3d.Point3d;
Vector3d: TYPE = SV3d.Vector3d;
CreateEmptyLine: PUBLIC PROC [] RETURNS [line: Line3d] = {
line _ NEW[Line3dObj];
};
CopyLine: PUBLIC PROC [from: Line3d, to: Line3d] = {
to^ _ from^;
};
EqualLine: PUBLIC PROC [a: Line3d, b: Line3d] RETURNS [BOOL] = {
RETURN[a.base = b.base AND a.direction = b.direction];
};
AlmostEqualLine: PUBLIC PROC [a: Line3d, b: Line3d, errorDegrees: REAL, errorDistance: REAL] RETURNS [BOOL] = {
angle: REAL _ SVVector3d.AngleCCWBetweenVectors[a.direction, b.direction];
angleOK: BOOL _ ABS[angle] < errorDegrees OR ABS[angle-180.0] < errorDegrees;
IF NOT angleOK THEN RETURN[FALSE];
RETURN[ABS[
DistancePointToLine[[0.0, 0.0, 0.0], a] -
DistancePointToLine[[0.0, 0.0, 0.0], b]] < errorDistance
];
};
LineFromPoints: PUBLIC PROC [v1, v2: Point3d] RETURNS [line: Line3d] = {
line _ CreateEmptyLine[];
FillLineFromPoints[v1, v2, line];
};
LineFromPointAndVector: PUBLIC PROC  [pt: Point3d, vec: Vector3d] RETURNS [line: Line3d] = {
line _ CreateEmptyLine[];
FillLineFromPointAndVector[pt, vec, line];
};
LineNormalToLineThruPoint: PUBLIC PROC [line: Line3d, pt: Point3d] RETURNS [normalLine: Line3d] = {

};
FillLineFromPoints: PUBLIC PROC [v1, v2: Point3d, line: Line3d] = {
OPEN SVVector3d;
line.direction _ Normalize[Sub[v2, v1]];
line.base _ Add[v1, Scale[line.direction, DotProduct[Sub[[0,0,0], v1], line.direction]]];
};

FillLineFromPointAndVector: PUBLIC PROC [pt: Point3d, vec: Vector3d, line: Line3d] = {
p2: Point3d _ SVVector3d.Add[pt, vec];
FillLineFromPoints[pt, p2, line];
};
FillLineNormalToLineThruPoint: PUBLIC PROC [line: Line3d, pt: Point3d, normalLine: Line3d] = {

};
CreateEmptyEdge: PUBLIC PROC RETURNS [edge: Edge3d] = {
edge _ NEW[Edge3dObj];
edge.line _ CreateEmptyLine[];
};
CopyEdge: PUBLIC PROC [from: Edge3d, to: Edge3d] = {
CopyLine[from.line, to.line];
to.start _ from.start;
to.end _ from.end;
}; -- end of CopyEdge
FillEdge: PUBLIC PROC [v1, v2: Point3d, edge: Edge3d] = {
FillLineFromPoints[v1, v2, edge.line];
edge.start _ v1;
edge.end _ v2;
}; -- end of FillEdge
FillEdgeTransform: PUBLIC PROC [fixed: Edge3d, transform: Matrix4by4, edge: Edge3d] = {
start, end: Point3d;
start _ Matrix3d.Update[fixed.start, transform];
end _ Matrix3d.Update[fixed.end, transform];
FillEdge[start, end, edge];
};
CreateEdge: PUBLIC PROC [v1, v2: Point3d] RETURNS [edge: Edge3d] = {
edge _ CreateEmptyEdge[];
FillEdge[v1, v2, edge];
}; -- end of CreateEdge
DirectionOfLine: PUBLIC PROC [line: Line3d] RETURNS [direction: Vector3d] = {
direction _ line.direction;
};

DistancePointToLine: PUBLIC PROC  [pt: Point3d, line: Line3d] RETURNS [d: REAL] = {
U: Point3d;
U _ DropPerpendicular[pt, line];
d _ SVVector3d.Distance[pt, U];
};

Distance2PointToLine: PUBLIC PROC  [pt: Point3d, line: Line3d] RETURNS [d2: REAL] = {
U: Point3d;
U _ DropPerpendicular[pt, line];
d2 _ SVVector3d.DistanceSquared[pt, U];
};

DropPerpendicular: PUBLIC PROC  [pt: Point3d, line: Line3d] RETURNS [projectedPt: Point3d] = {
OPEN SVVector3d;
projectedPt _ Add[line.base, Scale[line.direction, DotProduct[Sub[pt,line.base], line.direction]]];
};

NearLinePoints: PUBLIC PROC [l1, l2: Line3d] RETURNS [p1, p2: Point3d, parallel: BOOL _ FALSE] = {
OPEN SVVector3d;
A, B: Point3d;
N, M, P: Vector3d;
k: REAL;
A _ l1.base;  N _ l1.direction;  B _ l2.base;  M _ l2.direction;
k _ DotProduct[M,N];
IF ABS[1.0-ABS[k]] < 3.808737e-5 -- within 0.5 degrees of parallel.  Numerically stable this ain't.
THEN {parallel _ TRUE; RETURN};
P _ Sub[N, Scale[M, k]];
p1 _ Add[A, Scale[N, DotProduct[Sub[B,A], P]/(1.0-k*k)]];
p2 _ Add[B, Scale[M, DotProduct[Sub[p1,B], M]]];
};

DistanceLineToLine: PUBLIC PROC [l1, l2: Line3d] RETURNS [d: REAL] = {
OPEN SVVector3d;
U, V: Point3d;
parallel: BOOL;
[U, V, parallel] _ NearLinePoints[l1, l2];
IF parallel THEN RETURN[DistancePointToLine[l1.base, l2]];
d _ Distance[U,V];
};
NearEndpointToPoint: PUBLIC PROC  [pt: Point3d, edge: Edge3d] RETURNS [endpoint: Point3d] = {
IF ABS[pt[1]-edge.start[1]] <= ABS[pt[1]-edge.end[1]] THEN {
IF ABS[pt[2]-edge.start[2]] <= ABS[pt[2]-edge.end[2]] AND
    ABS[pt[3]-edge.start[3]] <= ABS[pt[3]-edge.end[3]] THEN RETURN[edge.start]
ELSE GOTO DoMath
}
ELSE {
IF ABS[pt[2]-edge.start[2]] > ABS[pt[2]-edge.end[2]] AND
    ABS[pt[3]-edge.start[3]] > ABS[pt[3]-edge.end[3]] THEN RETURN[edge.end]
ELSE GOTO DoMath;
};
EXITS
DoMath =>
IF SVVector3d.DistanceSquared[pt, edge.start] <= SVVector3d.DistanceSquared[pt, edge.end]
THEN endpoint _ edge.start
ELSE endpoint _ edge.end;
};

Distance2PointToEndpoint: PUBLIC PROC [pt: Point3d, edge: Edge3d] RETURNS [distanceSquared: REAL] = {
distance2ToPLo, distance2ToPHi: REAL;
distance2ToPLo _ SVVector3d.DistanceSquared[pt, edge.start];
distance2ToPHi _ SVVector3d.DistanceSquared[pt, edge.end];
RETURN[MIN[distance2ToPLo, distance2ToPHi]];
};

NearEdgePointToPoint: PUBLIC PROC [pt: Point3d, edge: Edge3d] RETURNS [onEdge: Point3d] = {
projectedPt: Point3d _ DropPerpendicular[pt, edge.line];
IF LinePointOnEdge[projectedPt, edge] THEN onEdge _ projectedPt
ELSE onEdge _ NearEndpointToPoint[pt, edge];
};

Distance2PointToEdge: PUBLIC PROC  [pt: Point3d, edge: Edge3d] RETURNS [distanceSquared: REAL] = {
projectedPt: Point3d _ DropPerpendicular[pt, edge.line];
IF LinePointOnEdge[projectedPt, edge]
THEN distanceSquared _ SVVector3d.DistanceSquared[pt, projectedPt]
ELSE distanceSquared _ Distance2PointToEndpoint[pt, edge];
};

DistancePointToEdge: PUBLIC PROC [pt: Point3d, edge: Edge3d] RETURNS [distance: REAL] = {
projectedPt: Point3d _ DropPerpendicular[pt, edge.line];
nearEndpoint: Point3d;
IF LinePointOnEdge[projectedPt, edge]
THEN distance _ SVVector3d.Distance[pt, projectedPt]
ELSE {
nearEndpoint _ NearEndpointToPoint[pt, edge];
distance _ SVVector3d.Distance[pt, nearEndpoint];
};
};

LinePointOnEdge: PUBLIC PROC [pt: Point3d, edge: Edge3d] RETURNS [BOOL] = {
dx, dy, dz, max: REAL;
dx _ ABS[edge.end[1] - edge.start[1]];
dy _ ABS[edge.end[2] - edge.start[2]];
dz _ ABS[edge.end[3] - edge.start[3]];
max _ MAX[dx, dy, dz];
IF dx = max THEN RETURN[Between[pt[1], edge.start[1], edge.end[1]]]
ELSE IF dy = max THEN RETURN[Between[pt[2], edge.start[2], edge.end[2]]]
ELSE RETURN[Between[pt[3], edge.start[3], edge.end[3]]];
}; -- end of LinePointOnEdge
NearEndpointToLine: PUBLIC PROC  [line: Line3d, edge: Edge3d] RETURNS [endpoint: Point3d] = {
distance2ToPLo, distance2ToPHi: REAL;
distance2ToPLo _ Distance2PointToLine[edge.start, line];
distance2ToPHi _ Distance2PointToLine[edge.end, line];
IF distance2ToPLo <= distance2ToPHi THEN endpoint _ edge.start
ELSE endpoint _ edge.end;
};

DistanceLineToEndpoint: PROC [line: Line3d, edge: Edge3d] RETURNS [distance: REAL] = {
distance2ToPLo, distance2ToPHi: REAL;
distance2ToPLo _ Distance2PointToLine[edge.start, line];
distance2ToPHi _ Distance2PointToLine[edge.end, line];
IF distance2ToPLo <= distance2ToPHi THEN distance _ RealFns.SqRt[distance2ToPLo]
ELSE distance _ RealFns.SqRt[distance2ToPHi];
};

DistanceAndEndpointLineToEdge: PROC [line: Line3d, edge: Edge3d] RETURNS [distance: REAL, endpoint: Point3d] = {
distance2ToPLo, distance2ToPHi: REAL;
distance2ToPLo _ Distance2PointToLine[edge.start, line];
distance2ToPHi _ Distance2PointToLine[edge.end, line];
IF distance2ToPLo <= distance2ToPHi THEN {
endpoint _ edge.start;
distance _ RealFns.SqRt[distance2ToPLo];
}
ELSE {
endpoint _ edge.end;
distance _ RealFns.SqRt[distance2ToPHi];
};
};

DistanceLineToEdge: PUBLIC PROC [line: Line3d, edge: Edge3d] RETURNS [distance: REAL] = {
U, V: Point3d;
parallel: BOOL;
[U, V, parallel] _ NearLinePoints[line, edge.line];
IF parallel THEN RETURN[DistancePointToLine[line.base, edge.line]];
IF LinePointOnEdge[V, edge] THEN distance _ SVVector3d.Distance[U,V]
ELSE distance _ DistanceLineToEndpoint[line, edge];
};

NearEdgePointToLine: PUBLIC PROC [line: Line3d, edge: Edge3d] RETURNS [edgePoint: Point3d] = {
U, V: Point3d;
parallel: BOOL;
[U, V, parallel] _ NearLinePoints[line, edge.line];
IF parallel THEN RETURN[edge.start]; -- any point on edge will do
IF LinePointOnEdge[V, edge] THEN edgePoint _ V
ELSE edgePoint _ NearEndpointToLine[line, edge];
};

DistanceAndPointLineToEdge: PUBLIC PROC [line: Line3d, edge: Edge3d] RETURNS [distance: REAL, edgePoint: Point3d] = {
U, V: Point3d;
parallel: BOOL;
[U, V, parallel] _ NearLinePoints[line, edge.line];
IF parallel THEN {
distance _ DistancePointToLine[line.base, edge.line];
edgePoint _ edge.start;  -- any point on edge will do
}
ELSE {
IF LinePointOnEdge[V, edge] THEN {
distance _ SVVector3d.Distance[U,V];
edgePoint _ V;
}
ELSE {
[distance, edgePoint] _ DistanceAndEndpointLineToEdge[line, edge];
};
};
};

Between: PRIVATE PROC [test, a, b: REAL] RETURNS [BOOL] = {
SELECT a FROM
< b => RETURN [a <= test AND test <= b];
= b => RETURN [test = b];
> b => RETURN [b <= test AND test <= a];
ENDCASE => ERROR;
};
END.

	ÆSVLines3dImpl.mesa
Copyright c 1986 by Xerox Corporation.  All rights reserved.
Last edited by Bier on February 18, 1987
Contents:  Procedures for performing geometric operations on lines and points in 3-space.

Making Lines
line and pt determine a plane.  Within that plane, compute the line that is perpendicular to line and passes throught pt.

Our first job is to find the nearest point on the line to the origin.
base = v1 + [(O-v1)oN] N.
FillLineFromPoints: PUBLIC PROC [v1, v2: Point3d, line: Line3d] = {
Our first job is to find the nearest point on the line to the origin.
originInWorld: Point3d = [0,0,0];
originInLine: Point3d;
lineInWorld, worldInLine: Matrix4by4;
zInLine: REAL;
line.direction _ SVVector3d.Normalize[SVVector3d.Sub[v2, v1]];
lineInWorld _ Matrix3d.MakeHorizontalMatFromZAxis[line.direction, v1];
worldInLine _ Matrix3d.Inverse[lineInWorld];
originInLine _ Matrix3d.Update[originInWorld, worldInLine];
zInLine _ originInLine[3];
line.base _ SVVector3d.Add[v1, SVVector3d.Scale[line.direction, zInLine]];
line.mat _ Matrix3d.SetOrigin[lineInWorld, line.base];
};

line and pt determine a plane.  Within that plane, compute the line that is perpendicular to line and passes throught pt.
Making Edges
to.startIsFirst _ from.startIsFirst;
Direction and Distance for Lines
Returns the unit direction vector of line.
 [Artwork node; type 'Artwork on' to command tool] 
Dropping a perpendicular from V to line [A, N].
We drop a normal from the point onto the line and find where it hits.  The line equation of the normal we drop can be found using FillLineNormalToLineThruPoint above.
The projectedPt U is:
U = A + [(V-A)oN] N.
 [Artwork node; type 'Artwork on' to command tool] 
Find the points of closest approach of lines [A, N] and [B, M].
Method:  Let l1 be (A, N) and l2 be (B, M).  Let p1 be U and p2 be V.
Find U on (A, N) and V on (B, M) such that
(U-V)oM = 0
(U-V)oN = 0.
Let k = MoN.  If k S 1 then d = DistancePointToLine[A, (B, M)].
Otherwise,
U = A + [((B-A)oP)/(1-k2)] N, where P = N- kM.
V = B + [(U-B)oM] M.
d = Distance[U, V].
Distance for Edges
Comparing Edges to Points
Look for an obvious winner first.  If that fails, do math.

Perpendicular distance if possible, else distance to nearest endpoint.
perpendicular distance if possible, else distance to nearest endpoint.

Assumes pt is on edge.line.  Is it on edge?

Comparing Edges to Lines
distance _ DistanceLineToEndpoint[line, edge];
edgePoint _ NearEndpointToLine[line, edge];

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