Plane: TYPE = REF PlaneObj;
PlaneObj: TYPE = SVPolygon3d.PlaneObj;
Point3d: TYPE = SV3d.Point3d;
Poly3d: TYPE = REF Poly3dObj;
Poly3dObj: TYPE = SV3d.Poly3dObj;
Polygon: TYPE = REF PolygonObj;
PolygonObj: TYPE = SV2d.PolygonObj;
Vector: TYPE = SVVector3d.Vector;
GLOBAL VARIABLES
root3, root3Over3, twoRoot3Over3: REAL; -- constants computed at load time in Init (see below).
CreatePoly:
PUBLIC
PROC [len:
NAT]
RETURNS [poly3d: Poly3d] = {
poly3d ← NEW[Poly3dObj[len]];
poly3d.len ← 0;
};
CircumHexagon:
PUBLIC
PROC [y:
REAL, r:
REAL]
RETURNS [hex: Poly3d] = {
Creates a hexagon which circumscribes the circle (in the Y = y plane) whose center is at [0, y, 0] and whose radius is r.
hex ← CreatePoly[6];
hex ← AddPolyPoint[hex, [r*root3Over3, y, r]]; -- front right
hex ← AddPolyPoint[hex, [r*twoRoot3Over3, y, 0]];-- right
hex ← AddPolyPoint[hex, [r*root3Over3, y, -r]];-- back right
hex ← AddPolyPoint[hex, [-r*root3Over3, y, -r]];-- back left
hex ← AddPolyPoint[hex, [-r*twoRoot3Over3, y, 0]];-- left
hex ← AddPolyPoint[hex, [-r*root3Over3, y, r]];-- front left
};
ClearPoly:
PUBLIC
PROC [poly: Poly3d] = {
poly.len ← 0;
};
AddPolyPoint:
PUBLIC
PROC [poly: Poly3d, point: Point3d]
RETURNS [polyPlusPoint: Poly3d] = {
IF poly.len = poly.maxVerts
THEN {
polyPlusPoint ← CreatePoly[poly.maxVerts+1];
polyPlusPoint ← PartPolyGetsPartPoly[poly, 0, polyPlusPoint, 0, poly.maxVerts];
polyPlusPoint[poly.maxVerts] ← point;
polyPlusPoint.len ← poly.maxVerts + 1;
}
ELSE {
poly[poly.len] ← point;
poly.len ← poly.len + 1;
polyPlusPoint ← poly;
};
}; -- end of AddPolyPoint
PutPolyPoint:
PUBLIC
PROC [poly: Poly3d, index:
NAT, point: Point3d]
RETURNS [newPoly: Poly3d] = {
IF index+1 > poly.maxVerts
THEN {
newPoly ← CreatePoly[index+1];
newPoly ← PartPolyGetsPartPoly[poly, 0, newPoly, 0, poly.maxVerts];
newPoly[index] ← point;
newPoly.len ← index+1;
}
ELSE {
IF index+1 > poly.len THEN poly.len ← index+1;
poly[index] ← point;
newPoly ← poly};
};
PartPolyGetsPartPoly:
PUBLIC
PROC [fromPoly: Poly3d, fromStart:
NAT, toPoly: Poly3d, toStart:
NAT, duration:
NAT]
RETURNS [newPoly: Poly3d] = {
toPoly in range [toStart..toStart+duration-1] gets fromPoly in range [fromStart..fromStart+duration-1];
j: NAT ← fromStart;
IF toStart+duration > toPoly.maxVerts
THEN {
newPoly ← CreatePoly[toStart+duration];
newPoly ← PartPolyGetsPartPoly[toPoly, 0, newPoly, 0, toPoly.len];
newPoly.len ← toStart+duration}
ELSE {
newPoly ← toPoly;
IF toStart+duration > newPoly.len THEN newPoly.len ← toStart+duration};
IF fromStart+duration > fromPoly.len THEN ERROR; -- don't use bad data
FOR i:
NAT
IN[toStart..toStart+duration-1]
DO
newPoly[i] ← fromPoly[j];
j ← j + 1;
ENDLOOP;
};
PlaneFromPoly3d:
PUBLIC
PROC [poly: Poly3d]
RETURNS [plane: Plane] = {
Assumes that the normal of the plane should point in the direction such if the normal points toward a viewer, the viewer will perceive the points of poly to run clockwise.
This algorithm was taken from Foley and Van Dam "Fundamentals of Interactive Computer Graphics" p. 512-513, where the authors note that for the plane equation Ax + By + Cz + D = 0,
A, B, and C have the same relationship to each other as the areas of the polygon when it is projected onto the yz, xz, and xy planes respectively. Hence we can just set A, B, and C equal to these respective values and then plug in a point to solve for D. The areas are computed by the signed areas of the trapezoids formed by each each and its projection onto the y, z, and x axis respectively for A, B, and C.
iPlusOne: NAT;
iPoint, iPlusOnePoint: Point3d;
plane ← NEW[PlaneObj];
plane.A ← 0;
plane.B ← 0;
plane.C ← 0;
FOR i:
NAT
IN[1..poly.len]
DO
iPlusOne ← IF i = poly.len THEN 1 ELSE i + 1;
iPoint ← poly[i - 1];
iPlusOnePoint ← poly[iPlusOne - 1];
plane.A ← plane.A + (iPoint[3] + iPlusOnePoint[3])*(iPlusOnePoint[2] - iPoint[2]);
plane.B ← plane.B + (iPoint[1] + iPlusOnePoint[1])*(iPlusOnePoint[3] - iPoint[3]);
plane.C ← plane.C + (iPoint[2] + iPlusOnePoint[2])*(iPlusOnePoint[1] - iPoint[1]);
ENDLOOP;
Now find D. Use poly[0] as the point to substitute in.
iPoint ← poly[0];
plane.D ← -(iPoint[1]*plane.A + iPoint[2]*plane.B + iPoint[3]*plane.C);
}; -- end of PlaneFromPoly3d
PlaneFromPointAndNormal:
PUBLIC
PROC [point: Point3d, normal: Vector]
RETURNS [plane: Plane] = {
plane ← NEW[PlaneObj];
plane.A ← normal[1];
plane.B ← normal[2];
plane.C ← normal[3];
Ax + By + Cz + D = 0. D = -(Ax + By + Cz)
plane.D ← -(plane.A*point[1] + plane.B*point[2] + plane.C*point[3]);
}; -- end of PlaneFromPointAndNormal
PlaneFromCoefficients:
PUBLIC
PROC [
A,
B,
C,
D:
REAL]
RETURNS [plane: Plane] = {
plane ← NEW[PlaneObj ← [A, B, C, D]];
};
ClipPolyToPlane:
PUBLIC
PROC [poly: Poly3d, plane: Plane]
RETURNS [clippedPoly: Poly3d] = {
For each point in poly, plug the point into the plane equation to see which side of poly it is on. Whenever the side changes, compute the intersection with the plane, keeping the line segment on the opposite side of the plane from the side refered to be its surface normal [A, B, C]; Each iteration of the loop will output "thisPoint" if appropriate and/or any plane intersection of the line segment from lastPoint to thisPoint.
lastSide, thisSide: BOOL;
lastDotProduct, thisDotProduct: REAL;
lastPoint, thisPoint, intersectionPoint: Point3d;
clippedPoly ← CreatePoly[poly.len]; -- the clipped poly won't have more vertices than the old poly.
IF poly.len = 0 THEN RETURN;
lastPoint ← poly[poly.len-1]; -- the last point
lastDotProduct ← (plane.A*lastPoint[1] + plane.B*lastPoint[2] + plane.C*lastPoint[3] + plane.D);
lastSide ← lastDotProduct <= 0;
FOR i:
NAT
IN [0..poly.len)
DO
thisPoint ← poly[i];
thisDotProduct ← (plane.A*thisPoint[1] + plane.B*thisPoint[2] + plane.C*thisPoint[3] + plane.D);
thisSide ← thisDotProduct <= 0;
IF thisSide
THEN {
IF lastSide THEN clippedPoly ← AddPolyPoint[clippedPoly, thisPoint]
ELSE {
-- we have travelled from the badSide to the keepSide. Add the intersection point first.
intersectionPoint ← LineSegmentMeetsPlaneFast[lastPoint, thisPoint, lastDotProduct, thisDotProduct];
clippedPoly ← AddPolyPoint[clippedPoly, intersectionPoint];
clippedPoly ← AddPolyPoint[clippedPoly, thisPoint];
};
}
ELSE {
-- this point is on the bad side.
IF lastSide
THEN {
-- we have gone from good side to bad side
Output only the intersection point.
intersectionPoint ← LineSegmentMeetsPlaneFast[lastPoint, thisPoint, lastDotProduct, thisDotProduct];
clippedPoly ← AddPolyPoint[clippedPoly, intersectionPoint];
}
ELSE {}; -- bad side to bad side. Do nothing.
};
lastSide ← thisSide;
lastPoint ← thisPoint;
lastDotProduct ← thisDotProduct;
ENDLOOP;
};
ClipPolyToPlanes:
PUBLIC
PROC [poly: Poly3d, planes:
LIST
OF Plane]
RETURNS [clippedPoly: Poly3d] = {
clippedPoly ← poly;
FOR planeList:
LIST
OF Plane ← planes, planeList.rest
UNTIL planeList =
NIL
DO
clippedPoly ← ClipPolyToPlane[clippedPoly, planeList.first];
ENDLOOP;
};
LineSegmentMeetsPlaneFast:
PRIVATE
PROC [p1, p2: Point3d, d1, d2:
REAL]
RETURNS [intersection: Point3d] = {
This procedure assumes that p1 and p2 are on opposite sides of plane. Hence, we can use just the magnitude of d1 and d2.
total: REAL;
d1 ← IF d1 < 0 THEN -d1 ELSE d1;
d2 ← IF d2 < 0 THEN -d2 ELSE d2;
total ← d1+d2;
intersection[1] ← (d1/total)*(p2[1]-p1[1])+p1[1];
intersection[2] ← (d1/total)*(p2[2]-p1[2])+p1[2];
intersection[3] ← (d1/total)*(p2[3]-p1[3])+p1[3];
};
ClipLineSegmentToPlane:
PUBLIC
PROC [p1, p2: Point3d, plane: Plane]
RETURNS [newP1, newP2: Point3d, newP1isP1, newP2isP2, nullSegment:
BOOL] = {
dotProduct1, dotProduct2: REAL;
nullSegment ← FALSE;
dotProduct1 ← (plane.A*p1[1] + plane.B*p1[2] + plane.C*p1[3] + plane.D);
newP1isP1 ← dotProduct1 <= 0;
dotProduct2 ← (plane.A*p2[1] + plane.B*p2[2] + plane.C*p2[3] + plane.D);
newP2isP2 ← dotProduct2 <= 0;
IF newP1isP1
THEN {
newP1 ← p1;
IF newP2isP2 THEN newP2 ← p2
ELSE newP2 ← LineSegmentMeetsPlaneFast[p1, p2, dotProduct1, dotProduct2];
}
ELSE {
IF newP2isP2
THEN {
newP1 ← LineSegmentMeetsPlaneFast[p1, p2, dotProduct1, dotProduct2];
newP2 ← p2;
}
ELSE nullSegment ← TRUE;
};
}; -- end of ClipLineSegmentToPlane
ClipLineSegmentToPlanes:
PUBLIC
PROC [p1, p2: Point3d, planes:
LIST
OF Plane]
RETURNS [newP1, newP2: Point3d, newP1isP1, newP2isP2, nullSegment:
BOOL] = {
plane: Plane;
keepSide1, keepSide2: BOOL;
newP1isP1 ← newP2isP2 ← TRUE;
newP1 ← p1;
newP2 ← p2;
FOR planeList:
LIST
OF Plane ← planes, planeList.rest
UNTIL planeList =
NIL
DO
plane ← planeList.first;
[newP1, newP2, keepSide1, keepSide2, nullSegment] ← ClipLineSegmentToPlane[newP1, newP2, plane];
IF nullSegment THEN RETURN;
newP1isP1 ← newP1isP1 AND keepSide1;
newP2isP2 ← newP2isP2 AND keepSide2;
ENDLOOP;
}; -- end of ClipLineSegmentToPlanes
NormalOfPlane:
PUBLIC
PROC [plane: Plane]
RETURNS [normal: Vector] = {
normal[1] ← plane.A;
normal[2] ← plane.B;
normal[3] ← plane.C;
}; -- end of NormalFromPlane
MaximumDeviationWithinPoly3d:
PROC [poly: Poly3d]
RETURNS [distance:
REAL] = {
Finds the distance of each point in the poly from the plane of the poly and returns the maximum such distance.
plane: Plane ← PlaneFromPoly3d[poly];
SIGNAL NotYetImplemented;-- **** not yet implemented
};
NotYetImplemented: SIGNAL = CODE;
SignedPointToPlaneDistance:
PUBLIC
PROC [point: Point3d, plane: Plane]
RETURNS [distance:
REAL] = {
distance ← (plane.A*point[1] + plane.B*point[2] + plane.C*point[3] + plane.D)/
RealFns.SqRt[plane.A*plane.A + plane.B*plane.B + plane.C*plane.C];
}; -- end of SignedPointToPlaneDistance
PointOnNormalSideOfPlane:
PUBLIC
PROC [point: Point3d, plane: Plane]
RETURNS [
BOOL] = {
plugInPoint: REAL ← (plane.A*point[1] + plane.B*point[2] + plane.C*point[3] + plane.D);
RETURN [plugInPoint > 0];
};
Quadrant: TYPE = SVPolygon3d.Quadrant;-- {inin, inout, outin, outout};
QuadrantOfPoint:
PUBLIC
PROC [point: Point3d, plane1, plane2: Plane]
RETURNS [Quadrant] = {
Classifies point with respect to two planes. outin means the point is on the normal side of the first plane and the non-normal side of the other plane, etc.
IF PointOnNormalSideOfPlane[point, plane1] THEN {
IF PointOnNormalSideOfPlane[point, plane2] THEN RETURN[outout]
ELSE RETURN[outin]}
ELSE {
IF PointOnNormalSideOfPlane[point, plane2] THEN RETURN[inout]
ELSE RETURN[inin]};
}; -- QuadrantOfPoint
ProjectPolyToXYPlane:
PUBLIC
PROC [poly3d: Poly3d]
RETURNS [polygon: Polygon] = {
polygon ← SVPolygon2d.CreatePoly[poly3d.len];
FOR i:
NAT
IN[0..poly3d.len)
DO
polygon[i][1] ← poly3d[i][1];
polygon[i][2] ← poly3d[i][2];
ENDLOOP;
polygon.len ← poly3d.len;
};
ProjectPolyToYZPlane:
PUBLIC
PROC [poly3d: Poly3d]
RETURNS [polygon: Polygon] = {
FOR i:
NAT
IN[0..poly3d.len)
DO
polygon[i][1] ← poly3d[i][2];
polygon[i][2] ← poly3d[i][3];
ENDLOOP;
polygon.len ← poly3d.len;
};
ProjectPolyToXZPlane:
PUBLIC
PROC [poly3d: Poly3d]
RETURNS [polygon: Polygon] = {
FOR i:
NAT
IN[0..poly3d.len)
DO
polygon[i][1] ← poly3d[i][1];
polygon[i][2] ← poly3d[i][3];
ENDLOOP;
polygon.len ← poly3d.len;
};
Init:
PROC = {
root3 ← RealFns.SqRt[3];
root3Over3 ← root3/3;
twoRoot3Over3 ← 2*root3Over3;
};
Init[];