DIRECTORY
BasicObject3d,
CastRays,
CSG,
ObjectCast,
Polynomial,
RealFns,
SV2d,
SV3d,
SVRayTypes,
Roots;


ObjectCastImplC: PROGRAM
IMPORTS CastRays, CSG, Polynomial, RealFns
EXPORTS ObjectCast =

BEGIN

Classification: TYPE = SVRayTypes.Classification;
Point2d: TYPE = SV2d.Point2d;
Point3d: TYPE = SV3d.Point3d;
Primitive: TYPE = SVRayTypes.Primitive;
Ray: TYPE = SVRayTypes.Ray;
Surface: TYPE = SVRayTypes.Surface;
TorusRec: TYPE = BasicObject3d.TorusRec;
Vector: TYPE = SV3d.Vector;

ShortRealRootRec: TYPE = Polynomial.ShortRealRootRec;
Ref: TYPE = Polynomial.Ref;

globalPoly: Polynomial.Ref;
newWay: BOOL _ TRUE;


PositiveRoots: PROCEDURE [a4, a3, a2, a1, a0: REAL] RETURNS [rootArray: Polynomial.ShortRealRootRec] = {
i: NAT _ 1;
globalPoly[4] _ a4;  -- optional
globalPoly[3] _ a3;  -- optional
globalPoly[2] _ a2;  -- optional
globalPoly[1] _ a1;  -- optional
globalPoly[0] _ a0;  -- optional
rootArray _ CheapRealRoots[globalPoly]; -- optional

IF rootArray.nRoots = 0 THEN RETURN;
WHILE i <= rootArray.nRoots DO
IF rootArray.realRoot[i-1] <= 0 THEN {
FOR j: NAT IN [i..rootArray.nRoots-1] DO
rootArray.realRoot[j-1] _ rootArray.realRoot[j];
ENDLOOP;
rootArray.nRoots _ rootArray.nRoots - 1;
}
ELSE {i _ i + 1};
ENDLOOP;
};

ToroidCast: PUBLIC PROC [localRay: Ray,  prim: Primitive, torusRec: REF ANY, surface: Surface]
RETURNS [class: Classification] = {
IF newWay THEN RETURN[NewToroidCast[localRay, prim, torusRec, surface]]
ELSE RETURN[OldToroidCast[localRay, prim, torusRec, surface]];
};

OldToroidCast: PUBLIC PROC [localRay: Ray,  prim: Primitive, torusRec: REF ANY, surface: Surface]
RETURNS [class: Classification] = {
torusData: TorusRec;
realRoots: Polynomial.ShortRealRootRec;  -- optional
B,C: REAL;
n,N,q,Q,l,L: REAL;
BB, CC: REAL;
D,E,F: REAL;
nn, NN, qq, QQ, ll, LL, NNNN, QQQQ, LLLL, nN, qQ, lL: REAL;
a4,a3,a2,a1,a0: REAL;
p: Point3d;
d: Vector;
class _ CastRays.GetClassFromPool[];
torusData _ NARROW[torusRec];
[p, d] _ CSG.GetLocalRay[localRay];
B _ torusData.bigRadius;
C _ torusData.sectionRadius;
n _ p[1];	N _ d[1];
q _ p[2];	Q _ d[2];
l _ p[3];	L _ d[3];
BB _ B*B;
CC _ C*C;
D _ BB + CC;
E _ BB - CC;
F _ BB*CC;
nn _ n*n;  NN _ N*N;  qq _ q*q;  QQ _ Q*Q;  ll _ l*l;  LL _ L*L;
NNNN _ NN*NN;  QQQQ _ QQ*QQ;  LLLL _ LL*LL;
nN _ n*N;  qQ _ q*Q;  lL _ l*L;
a4 _ NNNN + QQQQ + LLLL + 2*(NN*(QQ + LL) + QQ*LL);
a3 _ 4*(nN*NN + qQ*QQ + lL*LL + nN*(QQ + LL) + NN*(qQ + lL) + lL*QQ + qQ*LL);
a2 _ 2*(3*(nn*NN + qq*QQ + ll*LL) + NN*(qq + ll) + nn*(QQ+LL) + 4*nN*(qQ+lL)
		- D*NN + E*QQ - D*LL + ll*QQ + LL*qq + 4*qQ*lL);
a1 _ 4*(nn*nN + qq*qQ + ll*lL + nN*(qq + ll) + (qQ + lL)*nn - nN*D + qQ*E - lL*D + qQ*ll + lL*qq);
a0 _ nn*nn + qq*qq + ll*ll + BB*BB + CC*CC +
	2*(nn*(qq + ll) - D*nn + E*qq - D*ll + qq*ll - F);

realRoots _ PositiveRoots[1.0, a3/a4, a2/a4, a1/a4, a0/a4];
SELECT realRoots.nRoots FROM
0 =>  {-- complete miss.
class.count _ 0;
class.classifs[1] _ FALSE;
};
1 => {-- a skim.  Count as a miss.
class.count _ 0;
class.classifs[1] _ FALSE;
};
2 => {-- cuts one cross section of the doughnut.  Sort roots by size.
x, y, z, r, OneMinusBoverR: REAL;
class.count _ 2;
IF realRoots.realRoot[0] < realRoots.realRoot[1] THEN
{class.params[1] _ realRoots.realRoot[0];
 class.params[2] _ realRoots.realRoot[1]}
ELSE
{class.params[2] _ realRoots.realRoot[0];
 class.params[1] _ realRoots.realRoot[1]};
class.surfaces[1] _ surface; class.surfaces[2] _ surface;
class.classifs[1] _ FALSE;  class.classifs[2] _ TRUE;  class.classifs[3] _ FALSE;
class.primitives[1] _ class.primitives[2] _ prim;
x _ n + class.params[1]*N;  y _ q + class.params[1]*Q;  z _ l + class.params[1]*L;
r _ RealFns.SqRt[x*x + z*z];
OneMinusBoverR _ 1 - (B/r);
class.normals[1] _ [x*OneMinusBoverR, y, z*OneMinusBoverR];
x _ n + class.params[2]*N;  y _ q + class.params[2]*Q;  z _ l + class.params[2]*L;
r _ RealFns.SqRt[x*x + z*z];
OneMinusBoverR _ 1 - (B/r);
class.normals[2] _ [x*OneMinusBoverR, y, z*OneMinusBoverR];
};
3 => {-- cuts one cross section.  Skims the other.  For now, count as a miss.
class.count _ 0;
class.classifs[1] _ FALSE;
};
4 => {-- cuts both cross sections.  Sort the roots.  Pair them in order.
x, y, z, r, OneMinusBoverR: REAL;
SortHitRec[realRoots];
class.count _ 4;
class.params[1] _ realRoots.realRoot[0];
class.params[2] _ realRoots.realRoot[1];
class.params[3] _ realRoots.realRoot[2];
class.params[4] _ realRoots.realRoot[3];
class.surfaces[1] _ class.surfaces[2] _ class.surfaces[3] _ class.surfaces[4] _ surface;
class.classifs[1] _ FALSE;  class.classifs[2] _ TRUE;  class.classifs[3] _ FALSE;
class.classifs[4] _ TRUE;  class.classifs[5] _ FALSE;
class.primitives[1] _ class.primitives[2] _ class.primitives[3] _ class.primitives[4] _ prim;
x _ n + class.params[1]*N;  y _ q + class.params[1]*Q;  z _ l + class.params[1]*L;
r _ RealFns.SqRt[x*x + z*z];
OneMinusBoverR _ 1 - (B/r);
class.normals[1] _ [x*OneMinusBoverR, y, z*OneMinusBoverR];
x _ n + class.params[2]*N;  y _ q + class.params[2]*Q;  z _ l + class.params[2]*L;
r _ RealFns.SqRt[x*x + z*z];
OneMinusBoverR _ 1 - (B/r);
class.normals[2] _ [x*OneMinusBoverR, y, z*OneMinusBoverR];
x _ n + class.params[3]*N;  y _ q + class.params[3]*Q;  z _ l + class.params[3]*L;
r _ RealFns.SqRt[x*x + z*z];
OneMinusBoverR _ 1 - (B/r);
class.normals[3] _ [x*OneMinusBoverR, y, z*OneMinusBoverR];
x _ n + class.params[4]*N;  y _ q + class.params[4]*Q;  z _ l + class.params[4]*L;
r _ RealFns.SqRt[x*x + z*z];
OneMinusBoverR _ 1 - (B/r);
class.normals[4] _ [x*OneMinusBoverR, y, z*OneMinusBoverR];
};
ENDCASE => ERROR;
};


SortHitRec: PROC [roots: Polynomial.ShortRealRootRec] = {
temp: REAL;
FOR i: NAT IN[0..roots.nRoots-1) DO
FOR j: NAT IN[0..roots.nRoots-i-1) DO
IF roots.realRoot[j] > roots.realRoot[j+1] THEN {
temp _ roots.realRoot[j];
roots.realRoot[j] _ roots.realRoot[j+1];
roots.realRoot[j+1] _ temp;};
ENDLOOP;
ENDLOOP;
};

EvalTorus: PRIVATE PROC [RR, RRminusSS: REAL, degree: NAT, t: REAL, localRay: Ray] RETURNS [f: REAL] = {
HH, x, y, z: REAL;
p1,p2,p3,d1,d2,d3: REAL;
basePt: Point3d;
direction: Vector;
[basePt, direction] _ CSG.GetLocalRay[localRay];
p1 _ basePt[1];
p2 _ basePt[2];
p3 _ basePt[3];
d1 _ direction[1];
d2 _ direction[2];
d3 _ direction[3];
x _ p1 + t*d1;
y _ p2 + t*d2;
z _ p3 + t*d3;
SELECT degree FROM
4=>{
xxPLUSzz: REAL;
xxPLUSzz _ x*x + z*z;
HH _ xxPLUSzz + y*y;
f _ RRminusSS*RRminusSS + HH*(HH+2.0*RRminusSS) - 4.0*RR*xxPLUSzz;
};
3=>{
TwoHHdot: REAL;
HH _ x*x + y*y + z*z;
TwoHHdot _ 2.0*(x*d1 + y*d2 + z*d3);
f _ TwoHHdot*(HH + 2.0*RRminusSS) + HH*TwoHHdot - 8.0*RR*(x*d1 + z*d3);
};
2=>{
HHdot, HHdotPrime: REAL;
HH _ x*x + y*y + z*z;
HHdot _ (x*d1 + y*d2 + z*d3);
HHdotPrime _ (d1*d1+d2*d2+d3*d3);
f _ 4.0*HHdotPrime*(HH + RRminusSS) + 8.0*(HHdot*HHdot - RR*(d1*d1+d3*d3));
};
1=>{
f _ 24.0*(x*d1 + y*d2 + z*d3)*(d1*d1+d2*d2+d3*d3);
};
ENDCASE => ERROR;
};

TorusRoots: PRIVATE PROC [RR, RRminusSS: REAL, localRay: Ray, degree: NAT, bound: REAL] RETURNS [roots: Polynomial.ShortRealRootRec] =
BEGIN
roots.nRoots _ 0;
IF degree=1 THEN
BEGIN
p: Point3d;
d: Vector;
[p,d] _ CSG.GetLocalRay[localRay];
roots.nRoots _ 1;
roots.realRoot[0] _ -(p[1]*d[1] + p[2]*d[2] + p[3]*d[3])/(d[1]*d[1] + d[2]*d[2] + d[3]*d[3])
END
ELSE
BEGIN
BEGIN
extrema: Polynomial.ShortRealRootRec _ TorusRoots[RR, RRminusSS, localRay, degree-1, bound];
x: REAL;
IF extrema.nRoots>0 THEN
BEGIN
x _ TRootBetween[RR, RRminusSS, degree, -bound, extrema.realRoot[0], localRay];
IF x<=extrema.realRoot[0] THEN {roots.nRoots _ 1; roots.realRoot[0] _ x};
FOR i:NAT IN [0..extrema.nRoots-1) DO
x _ TRootBetween[RR, RRminusSS, degree, extrema.realRoot[i], extrema.realRoot[i+1], localRay];
IF x<=extrema.realRoot[i+1] THEN {roots.realRoot[roots.nRoots] _ x; roots.nRoots _ roots.nRoots + 1};
ENDLOOP;
x _ TRootBetween[RR, RRminusSS, degree, extrema.realRoot[extrema.nRoots-1],bound, localRay];
IF x<=bound THEN {roots.realRoot[roots.nRoots] _ x; roots.nRoots _ roots.nRoots + 1}
END
ELSE
BEGIN
x _ TRootBetween[RR, RRminusSS, degree, -bound, bound, localRay];
IF x<=bound THEN {roots.realRoot[0] _ x; roots.nRoots _ 1}
END
END
END
END;


TRootBound: PROC [R, S: REAL, p: Point3d, d: Vector] RETURNS [REAL] =
BEGIN -- finds a bound for the absolute values of the roots
p1, dSup: REAL;
p1 _ ABS[p[1]]+ABS[p[2]]+ABS[p[3]];
dSup _ MAX[ABS[d[1]], MAX[ABS[d[2]], ABS[d[3]]]];
RETURN[(p1 + R + S)/dSup];
END;

TRootBetween: PROC [RR, RRminusSS: REAL, degree: NAT, x0, x1: REAL, localRay: Ray]
RETURNS [x: REAL] =
BEGIN
y0: REAL _ EvalTorus[RR, RRminusSS, degree, x0, localRay];
y1: REAL _ EvalTorus[RR, RRminusSS, degree, x1, localRay];
xx: REAL;
xx0: REAL _ IF y0<y1 THEN x0 ELSE x1;
xx1: REAL _ IF y0<y1 THEN x1 ELSE x0;
yy0: REAL _ MIN[y0,y1];
yy1: REAL _ MAX[y0,y1];
NewtonStep: PROC [x: REAL] RETURNS [newx:REAL] =
BEGIN
y: REAL _ EvalTorus[RR, RRminusSS, degree, x, localRay];
deriv: REAL _ EvalTorus[RR, RRminusSS, degree-1, x, localRay];
IF y<0 THEN {IF y>yy0 THEN {xx0 _ x; yy0 _ y}};
IF y>0 THEN {IF y<yy1 THEN {xx1 _ x; yy1 _ y}};
newx _ IF deriv=0 THEN x+y ELSE x-y/deriv;
END;
IF y0=0 THEN RETURN[x0];
IF y1=0 THEN RETURN[x1];
IF (y0 > 0.0 AND y1 > 0.0) OR (y0 < 0.0 AND y1 < 0.0) THEN RETURN[x1+100]; -- return a non-sense value which the caller will check for (there is no root in this interval).
xx _ x _ (x0+x1)/2;
WHILE x IN [x0..x1] DO
newx:REAL _ NewtonStep[x];
IF x=newx THEN {
RETURN;
};
x _ NewtonStep[newx];
xx _ NewtonStep[xx];
IF xx=x THEN EXIT;
ENDLOOP;
IF xx0 IN [x0..x1] AND xx1 IN [x0..x1] THEN
BEGIN
x0 _ MIN[xx0,xx1];
x1 _ MAX[xx0,xx1];
END;
y0 _ EvalTorus[RR, RRminusSS, degree, x0, localRay];
y1 _ EvalTorus[RR, RRminusSS, degree, x1, localRay];
THROUGH [0..500) DO
y: REAL _ EvalTorus[RR, RRminusSS, degree, x _ (x0+x1)/2, localRay];
IF x=x0 OR x=x1 THEN
{IF ABS[y0] < ABS[y1] THEN RETURN[x0] ELSE RETURN[x1]};
IF y*y0 < 0 THEN {x1 _ x; y1 _ y}
ELSE {x0 _ x; y0 _ y};
IF y0*y1>=0 THEN {IF ABS[y0] < ABS[y1] THEN RETURN[x0] ELSE RETURN[x1]}
ENDLOOP;
ERROR;
END;


PositiveTorusRoots: PRIVATE PROC [RR, RRminusSS: REAL, localRay: Ray, bound: REAL] RETURNS [roots: Polynomial.ShortRealRootRec] = {
i: NAT _ 1;
roots _ TorusRoots[RR, RRminusSS, localRay, 4, bound];
IF roots.nRoots = 0 THEN RETURN;
WHILE i <= roots.nRoots DO
IF roots.realRoot[i-1] <= 0 THEN {
FOR j: NAT IN [i..roots.nRoots-1] DO
roots.realRoot[j-1] _ roots.realRoot[j];
ENDLOOP;
roots.nRoots _ roots.nRoots - 1;
}
ELSE {i _ i + 1};
ENDLOOP;
};

InsideTorus: PRIVATE PROC [R, S: REAL, testPt: Point3d] RETURNS [BOOL] = {
r, rMinusR: REAL;
r _ RealFns.SqRt[testPt[1]*testPt[1] + testPt[3]*testPt[3]];
rMinusR _ r-R;
RETURN[rMinusR*rMinusR + testPt[2]*testPt[2] <= S*S];
};

NewToroidCast: PUBLIC PROC [localRay: Ray, prim: Primitive, torusRec: REF ANY, surface: Surface]
RETURNS [class: Classification] = {
torusData: TorusRec;
realRoots: Polynomial.ShortRealRootRec;  -- optional
B, R, S, RR, RRminusSS: REAL; -- big and little torus radii
p: Point3d;
d: Vector;
bound: REAL;
class _ CastRays.GetClassFromPool[];
torusData _ NARROW[torusRec];
R _ B _ torusData.bigRadius;
S _ torusData.sectionRadius;
RR _ R*R;
RRminusSS _ RR - S*S;
[p, d] _ CSG.GetLocalRay[localRay];

bound _ TRootBound[R, S, p, d];
realRoots _ PositiveTorusRoots[RR, RRminusSS, localRay, bound];

SELECT realRoots.nRoots FROM
0 =>  {-- complete miss.
class.count _ 0;
class.classifs[1] _ FALSE;
};
1 => {-- a skim.  Count as a miss.
IF InsideTorus[R, S, p] THEN {
x, y, z, r, OneMinusBoverR: REAL;
class.count _ 1;
class.classifs[1] _ TRUE; class.classifs[2] _ FALSE;
class.params[1] _ realRoots.realRoot[0];
[x, y, z] _ CSG.EvaluateLocalRay2[localRay, class.params[1]];
r _ RealFns.SqRt[x*x + z*z];
OneMinusBoverR _ 1 - (B/r);
class.normals[1] _ [x*OneMinusBoverR, y, z*OneMinusBoverR];
class.surfaces[1] _ surface;
class.primitives[1] _ prim;
}
ELSE CastRays.MakeClassAMiss[class];
};
2 => {-- cuts one cross section of the doughnut.  Sort roots by size.
x, y, z, r, OneMinusBoverR: REAL;
class.count _ 2;
IF realRoots.realRoot[0] < realRoots.realRoot[1] THEN
{class.params[1] _ realRoots.realRoot[0];
 class.params[2] _ realRoots.realRoot[1]}
ELSE
{class.params[2] _ realRoots.realRoot[0];
 class.params[1] _ realRoots.realRoot[1]};
class.surfaces[1] _ surface; class.surfaces[2] _ surface;
class.classifs[1] _ FALSE;  class.classifs[2] _ TRUE;  class.classifs[3] _ FALSE;
class.primitives[1] _ class.primitives[2] _ prim;
[x, y, z] _ CSG.EvaluateLocalRay2[localRay, class.params[1]];
r _ RealFns.SqRt[x*x + z*z];
OneMinusBoverR _ 1 - (B/r);
class.normals[1] _ [x*OneMinusBoverR, y, z*OneMinusBoverR];
[x, y, z] _ CSG.EvaluateLocalRay2[localRay, class.params[2]];
r _ RealFns.SqRt[x*x + z*z];
OneMinusBoverR _ 1 - (B/r);
class.normals[2] _ [x*OneMinusBoverR, y, z*OneMinusBoverR];
};
3 => {-- cuts one cross section.  Skims the other (or starts inside of the torus).  For now, count as a miss.
class.count _ 0;
class.classifs[1] _ FALSE;
};
4 => {-- cuts both cross sections.  Sort the roots.  Pair them in order.
x, y, z, r, OneMinusBoverR: REAL;
SortHitRec[realRoots];
class.count _ 4;
class.params[1] _ realRoots.realRoot[0];
class.params[2] _ realRoots.realRoot[1];
class.params[3] _ realRoots.realRoot[2];
class.params[4] _ realRoots.realRoot[3];
class.surfaces[1] _ class.surfaces[2] _ class.surfaces[3] _ class.surfaces[4] _ surface;
class.classifs[1] _ FALSE;  class.classifs[2] _ TRUE;  class.classifs[3] _ FALSE;
class.classifs[4] _ TRUE;  class.classifs[5] _ FALSE;
class.primitives[1] _ class.primitives[2] _ class.primitives[3] _ class.primitives[4] _ prim;
[x, y, z] _ CSG.EvaluateLocalRay2[localRay, class.params[1]];
r _ RealFns.SqRt[x*x + z*z];
OneMinusBoverR _ 1 - (B/r);
class.normals[1] _ [x*OneMinusBoverR, y, z*OneMinusBoverR];
[x, y, z] _ CSG.EvaluateLocalRay2[localRay, class.params[2]];
r _ RealFns.SqRt[x*x + z*z];
OneMinusBoverR _ 1 - (B/r);
class.normals[2] _ [x*OneMinusBoverR, y, z*OneMinusBoverR];
[x, y, z] _ CSG.EvaluateLocalRay2[localRay, class.params[3]];
r _ RealFns.SqRt[x*x + z*z];
OneMinusBoverR _ 1 - (B/r);
class.normals[3] _ [x*OneMinusBoverR, y, z*OneMinusBoverR];
[x, y, z] _ CSG.EvaluateLocalRay2[localRay, class.params[4]];
r _ RealFns.SqRt[x*x + z*z];
OneMinusBoverR _ 1 - (B/r);
class.normals[4] _ [x*OneMinusBoverR, y, z*OneMinusBoverR];
};
ENDCASE => ERROR;
};

RootBound: PROC [poly: Ref] RETURNS [REAL] =
BEGIN -- finds a bound for the absolute values of the roots
d: NAT _ Degree[poly];
b: REAL _ 0;
FOR i:NAT IN [0..d) DO b _ b + ABS[poly[i]] ENDLOOP;
RETURN [b*1.01/ABS[poly[d]]+1];
END;

RootBetween: PROC [poly: Ref, x0, x1: REAL]
RETURNS [x: REAL] =
BEGIN
y0: REAL _ Eval[poly,x0];
y1: REAL _ Eval[poly,x1];
xx: REAL;
xx0: REAL _ IF y0<y1 THEN x0 ELSE x1;
xx1: REAL _ IF y0<y1 THEN x1 ELSE x0;
yy0: REAL _ MIN[y0,y1];
yy1: REAL _ MAX[y0,y1];
NewtonStep: PROC [x: REAL] RETURNS [newx:REAL] =
BEGIN
y: REAL _ Eval[poly, x];
deriv: REAL _ EvalDeriv[poly, x];
IF y<0 THEN {IF y>yy0 THEN {xx0 _ x; yy0 _ y}};
IF y>0 THEN {IF y<yy1 THEN {xx1 _ x; yy1 _ y}};
newx _ IF deriv=0 THEN x+y ELSE x-y/deriv;
END;
IF y0=0 THEN RETURN[x0];
IF y1=0 THEN RETURN[x1];
IF (y0 > 0.0 AND y1 > 0.0) OR (y0 < 0.0 AND y1 < 0.0) THEN RETURN[x1+100]; -- return a non-sense value which the caller will check for (there is no root in this interval).
xx _ x _ (x0+x1)/2;
WHILE x IN [x0..x1] DO
newx:REAL _ NewtonStep[x];
IF x=newx THEN {
RETURN;
};
x _ NewtonStep[newx];
xx _ NewtonStep[xx];
IF xx=x THEN EXIT;
ENDLOOP;
IF xx0 IN [x0..x1] AND xx1 IN [x0..x1] THEN
BEGIN
x0 _ MIN[xx0,xx1];
x1 _ MAX[xx0,xx1];
END;
y0 _ Eval[poly,x0];
y1 _ Eval[poly,x1];
THROUGH [0..500) DO
y: REAL _ Eval[poly, x _ (x0+x1)/2];
IF x=x0 OR x=x1 THEN
{IF ABS[y0] < ABS[y1] THEN RETURN[x0] ELSE RETURN[x1]};
IF y*y0 < 0 THEN {x1 _ x; y1 _ y}
ELSE {x0 _ x; y0 _ y};
IF y0*y1>=0 THEN {IF ABS[y0] < ABS[y1] THEN RETURN[x0] ELSE RETURN[x1]}
ENDLOOP;
ERROR;
END;

Degree: PUBLIC PROC [poly: Ref] RETURNS [NAT] =
BEGIN
FOR i:NAT DECREASING IN [0..poly.maxDegreePlusOne) DO
IF poly[i] # 0 THEN RETURN[i];
ENDLOOP;
RETURN[0];
END;

Differentiate: PUBLIC PROC [poly: Ref] =
BEGIN
d:NAT _ Degree[poly];
FOR i: NAT IN (0..d] DO poly[i-1] _ i*poly[i] ENDLOOP;
poly[d] _ 0.0;
END;

CheapRealRoots: PUBLIC PROC [poly: Ref] RETURNS [roots: ShortRealRootRec] =
BEGIN
d: NAT _ Degree[poly];
roots.nRoots _ 0;
IF d<=1 THEN
BEGIN
IF d=1 THEN {roots.nRoots _ 1; roots.realRoot[0] _ -poly[0]/poly[1]}
END
ELSE
BEGIN
bound: REAL _ RootBound[poly];
savedCoeff: ARRAY [0..5] OF REAL;
FOR i: NAT IN [0..d] DO savedCoeff[i] _ poly[i] ENDLOOP;
Differentiate[poly];
BEGIN
extrema: ShortRealRootRec _ CheapRealRoots[poly];
x: REAL;
FOR i: NAT IN [0..d] DO poly[i] _ savedCoeff[i] ENDLOOP;
IF extrema.nRoots>0 THEN
BEGIN
x _ RootBetween[poly,-bound,extrema.realRoot[0]];
IF x<=extrema.realRoot[0] THEN {roots.nRoots _ 1; roots.realRoot[0] _ x};
FOR i:NAT IN [0..extrema.nRoots-1) DO
x _ RootBetween[poly,extrema.realRoot[i],extrema.realRoot[i+1]];
IF x<=extrema.realRoot[i+1] THEN {roots.realRoot[roots.nRoots] _ x; roots.nRoots _ roots.nRoots + 1};
ENDLOOP;
x _ RootBetween[poly,extrema.realRoot[extrema.nRoots-1],bound];
IF x<=bound THEN {roots.realRoot[roots.nRoots] _ x; roots.nRoots _ roots.nRoots + 1}
END
ELSE
BEGIN
x _ RootBetween[poly,-bound,bound];
IF x<=bound THEN {roots.realRoot[0] _ x; roots.nRoots _ 1}
END
END
END
END;

EvalDeriv: PUBLIC PROC [f: Ref, x: REAL] RETURNS [y: REAL] =
BEGIN
y _ 0.0;
FOR i:NAT DECREASING IN (0..Degree[f]] DO
y _ i*f[i] + y*x
ENDLOOP
END;

Eval: PUBLIC PROC [f: Ref, x: REAL] RETURNS [y: REAL] =
BEGIN
y _ 0.0;
FOR i:NAT DECREASING IN [0..Degree[f]] DO
y _ y*x + f[i]
ENDLOOP
END;


Init: PROC = {
globalPoly _ Polynomial.Quartic[[0,0,0,0,0]];
};

Init[];

END.
File: ObjectCastImplC.mesa
Last edited by Bier on January 2, 1983 8:31 pm
Copyright c 1984 by Xerox Corporation.  All rights reserved.
Author: Bier on January 2, 1985 1:13:56 pm PST

Debugging
A torus has a single surface.  Classifying a ray with respect to this surface involves solving a single quadric equation.
Let the radius from the center of the doughnut hole to the center of the doughnut be C.
Let the cross-sectional radius be B.
Let the doughnut be a circle revolved around the y axis.
Consider a cross section of the doughnut.  Call the line parallel to the y axis and running through the center of the cross section L.  Let r(y) be the distance perpendicular to L from L to the circular boundary of the cross section.  Then:
(1)  r^2 + y^2 = C^2
From which we find:
(2)  r = +- SqRt[C^2 - y^2]
Thus the two concentric circles cut by the plane y = y0 have equations:
x^2 + z^2  = (B +- SqRt[C^2 - y^2])^2 or compactly:
(3) x^2 + z^2  = (B +- r)^2 = B^2 +- 2Br + r^2
We can get rid of the square root by regrouping:
(4) x^2 + z^2 - B^2 - r^2 = +- 2Br
and squaring:
(5) (x^2 + z^2 - B^2 - r^2)^2 = 4B^2r^2
Expanding, we have:
x^4 + x^2z^2 - x^2*B^2 - x^2*r^2 +
z^2*x^2 + z^4 - z^2*B^2 - z^2*r^2
-B^2*x^2 -B^2*z^2 + B^4 + B^2*r^2
-r^2*x^2 - r^2*z^2 + B^2*r^2 + r^4	=	4*B^2*r^2;		OR

x^4  + z^4 + B^4 + r^4 +
2*x^2z^2 - 2*x^2*B^2 - 2*x^2*r^2 - 2*z^2*B^2 - 2*z^2*r^2 + 2*B^2*r^2
	= 4*B^2*r^2
	
Plugging in r^2 = (C^2 - y^2), r^4 = (C^4 - 2*C^2*y^2 + y^4):

x^4  + z^4 + B^4 + C^4 - 2*C^2*y^2 + y^4 +
2*x^2z^2 - 2*x^2*B^2 - 2*x^2*(C^2 - y^2) - 2*z^2*B^2 - 2*z^2*(C^2 - y^2) + 2*B^2*(C^2 - y^2)
	= 4*B^2*(C^2 - y^2) = 4*B^2*C^2 - 4*B^2*y^2	OR
	
 x^4  + y^4 + z^4 + B^4 + C^4 +
2*x^2z^2 + 2*x^2*y^2 + 2*z^2*y^2
x^2 (- 2*B^2 - 2*C^2) +
y^2 (- 2*B^2- 2*C^2) +
z^2 (- 2*B^2 - 2*C^2) +
- 2*C^2 + 2*B^2*C^2	=	4*B^2*C^2 - 4*B^2*y^2	OR

x^4  + y^4 + z^4 + B^4 + C^4 +
2*x^2z^2 + 2*x^2*y^2 + 2*z^2*y^2
x^2 (- 2*B^2 - 2*C^2) +
y^2 (2*B^2 - 2*C^2) +
z^2 (- 2*B^2 - 2*C^2) +
- 2*B^2*C^2	= 0

Let D = B^2 + C^2,  E = B^2 - C^2, F = C^2B^2  Then:

(6)  x^4  + y^4 + z^4 + B^4 + C^4 +
		+ 2(x^2*y^2 + x^2z^2 + y^2z^2
		  - Dx^2 + Ey^2 - Dz^2 - F) = 0

Now recall the ray equations (using single letters for all variables for compactness):
(7)  x = n + t*N
(8)  y = q + t*Q
(9)  z = l + t*L
We wish to solve for t by plugging (7) - (9) into (6).  We get:

(n + t*N)^4 + (q + t*Q)^4 + (l + t*L)^4 + B^4 + C^4
+ 2( (n + t*N)^2*(q + t*Q)^2 + (n + t*N)^2*(l + t*L)^2 + (q + t*Q)^2*(l + t*L)^2
	-D(n + t*N)^2 + E(q + t*Q)^2 - D(l + t*L)^2 - F) = 0
	
If I've done my algebra correctly, this boils down to:
a4*t^4 + a3*t^3 + a2*t^2 + a1*t + a0
Where:
a4 = N^4 + Q^4 + L^4 + 2(N^2(Q^2+L^2) + Q^2L^2);
a3 = 4(nN^3 + qQ^3 + lL^3 + nN(Q^2 + L^2) + N^2(qQ + lL) + lLQ^2 + qQL^2)
a2 = 2(3n^2N^2 + 3q^2Q^2 + 3l^2L^2 + N^2(q^2 + l^2) + n^2(Q^2+L^2) + 4nN(qQ + lL)
- DN^2 + EQ^2 - DL^2 + l^2Q^2 + L^2q^2 + 4qQlL)
a1 = 4(n^3N + q^3Q + l^3L + nN(q^2 + l^2) + (qQ + lL)n^2 - nND + qQE - lLD + qQl^2 + lLq^2)
a0 = n^4 + q^4 + l^4 + B^4 + C^4 +
2(n^2(q^2 + l^2) - Dn^2 + Eq^2 - Dl^2+ q^2l^2  - F)
Here goes:

Use only the positive roots.
[realRootsOneOrigin, rootCount] _ Roots.RealQuartic[a4, a3, a2, a1, a0];
realRoots[0] _ realRootsOneOrigin[1]; realRoots[1] _ realRootsOneOrigin[2];
realRoots[2] _ realRootsOneOrigin[3]; realRoots[3] _ realRootsOneOrigin[4];

realRoots: ARRAY[0..3] OF REAL;
realRootsOneOrigin: ARRAY[1..4] OF REAL;
rootCount: NAT;

SELECT rootCount FROM
Calculate the surface normals at the two hits points.
To normalize, divide by C.  (I won't do this since the lighting model normalizes anyway)
SortHitRec[realRoots, 4];
Calculate the surface normals at the four hits points.
To normalize, divide by C.  (I won't do this since the lighting model normalizes anyway)
A simple bubble sort. Sorts in increasing order (depth).
SortHitRec: PROC [roots: ARRAY[0..3] OF REAL, rootCount: NAT] = {
temp: REAL;
FOR i: NAT IN[0..rootCount-1) DO
FOR j: NAT IN[0..rootCount-i-1) DO
IF roots[j] > roots[j+1] THEN {
temp _ roots[j];
roots[j] _ roots[j+1];
roots[j+1] _ temp;};
ENDLOOP;
ENDLOOP;
};
	
debug SVError.Append[IO.PutFR["%d degree, %d roots: ", IO.int[degree - 1], IO.int[extrema.nRoots]], TRUE, FALSE];
debug FOR i: NAT IN [0..extrema.nRoots) DO
debug SVError.Append[IO.PutFR["%g ", IO.real[extrema.realRoot[i]]], TRUE, FALSE];
debug ENDLOOP;
debug SVError.AppendTypescript[""];
We do this for the torus by reasoning geometrically.  In local coordinates, the torus is
centered on [0,0,0].  The ray base-point is at point p.  Hence (|p| + R + S)/|d| will bound
the value of d.  We can use the 1-norm to overestimate |p| and the sup-norm to underestimate |d| to get a conservative bound on t.
debug  xi: ARRAY [0..50) OF REAL;
debug  i: NAT _ 0;
IF y0*y1 > 0 THEN RETURN[x1+100]; -- return a non-sense value which the caller will check for (there is no root in this interval).
first try Newton's method
debug  SVError.Append[IO.PutFR["TNewton: %d steps: ", IO.int[i]], TRUE,FALSE];
debug  FOR j: NAT IN [0..i) DO
debug  SVError.Append[IO.PutFR["%g ", IO.real[xi[j]]], TRUE, FALSE];
debug  ENDLOOP;
debug  SVError.AppendTypescript[""];
debug  xi[i] _ xx; i _ i+1;
If it oscillated or went out of range, try bisection
debug  SIGNAL Bisection;
debug  SVError.Append[IO.PutFR["%d degree, %d roots: ", IO.int[4], IO.int[roots.nRoots]], TRUE, FALSE];
debug  FOR i: NAT IN [0..roots.nRoots) DO
debug  SVError.Append[IO.PutFR["%g ", IO.real[roots.realRoot[i]]], TRUE, FALSE];
debug  ENDLOOP;
debug  SVError.AppendTypescript[""];
SELECT rootCount FROM
Calculate the surface normals at the two hits points.
Calculate the surface normals at the two hits points.
To normalize, divide by C.  (I won't do this since the lighting model normalizes anyway)
SortHitRec[realRoots, 4];
Calculate the surface normals at the four hits points.
To normalize, divide by C.  (I won't do this since the lighting model normalizes anyway)
debug  xi: ARRAY [0..50) OF REAL;
debug  i: NAT _ 0;
IF y0*y1 > 0 THEN RETURN[x1+100];
first try Newton's method
debug  SVError.Append[IO.PutFR["Newton: %d steps: ", IO.int[i]], TRUE,FALSE];
debug  FOR j: NAT IN [0..i) DO
debug  SVError.Append[IO.PutFR["%g ", IO.real[xi[j]]], TRUE, FALSE];
debug  ENDLOOP;
debug  SVError.AppendTypescript[""];
debug  xi[i] _ xx; i _ i+1;
If it oscillated or went out of range, try bisection
debug  SIGNAL Bisection;
debug  SVError.Append[IO.PutFR["%g bound, %d degree %d roots: ", IO.real[bound], IO.int[d-1], IO.int[extrema.nRoots]], TRUE, FALSE];
debug  FOR i: NAT IN [0..extrema.nRoots) DO
debug  SVError.Append[IO.PutFR["%g ", IO.real[extrema.realRoot[i]]], TRUE, FALSE];
debug  ENDLOOP;
debug  SVError.AppendTypescript[""];
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