t3D geometry - plan9port - [fork] Plan 9 from user space
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---
DIR commit d1e9002f81f14fbfef1ebc4261edccd9eb97b72c
DIR parent 46f79934b79ef526ed42bbe5a565e6b5d884d24a
HTML Author: rsc <devnull@localhost>
Date: Tue, 4 Jan 2005 21:23:01 +0000
3D geometry
Diffstat:
A src/libgeometry/arith3.c | 215 +++++++++++++++++++++++++++++++
A src/libgeometry/matrix.c | 106 ++++++++++++++++++++++++++++++
A src/libgeometry/mkfile | 17 +++++++++++++++++
A src/libgeometry/qball.c | 66 +++++++++++++++++++++++++++++++
A src/libgeometry/quaternion.c | 242 +++++++++++++++++++++++++++++++
A src/libgeometry/transform.c | 75 +++++++++++++++++++++++++++++++
A src/libgeometry/tstack.c | 169 +++++++++++++++++++++++++++++++
7 files changed, 890 insertions(+), 0 deletions(-)
---
DIR diff --git a/src/libgeometry/arith3.c b/src/libgeometry/arith3.c
t@@ -0,0 +1,215 @@
+#include <u.h>
+#include <libc.h>
+#include <draw.h>
+#include <geometry.h>
+/*
+ * Routines whose names end in 3 work on points in Affine 3-space.
+ * They ignore w in all arguments and produce w=1 in all results.
+ * Routines whose names end in 4 work on points in Projective 3-space.
+ */
+Point3 add3(Point3 a, Point3 b){
+ a.x+=b.x;
+ a.y+=b.y;
+ a.z+=b.z;
+ a.w=1.;
+ return a;
+}
+Point3 sub3(Point3 a, Point3 b){
+ a.x-=b.x;
+ a.y-=b.y;
+ a.z-=b.z;
+ a.w=1.;
+ return a;
+}
+Point3 neg3(Point3 a){
+ a.x=-a.x;
+ a.y=-a.y;
+ a.z=-a.z;
+ a.w=1.;
+ return a;
+}
+Point3 div3(Point3 a, double b){
+ a.x/=b;
+ a.y/=b;
+ a.z/=b;
+ a.w=1.;
+ return a;
+}
+Point3 mul3(Point3 a, double b){
+ a.x*=b;
+ a.y*=b;
+ a.z*=b;
+ a.w=1.;
+ return a;
+}
+int eqpt3(Point3 p, Point3 q){
+ return p.x==q.x && p.y==q.y && p.z==q.z;
+}
+/*
+ * Are these points closer than eps, in a relative sense
+ */
+int closept3(Point3 p, Point3 q, double eps){
+ return 2.*dist3(p, q)<eps*(len3(p)+len3(q));
+}
+double dot3(Point3 p, Point3 q){
+ return p.x*q.x+p.y*q.y+p.z*q.z;
+}
+Point3 cross3(Point3 p, Point3 q){
+ Point3 r;
+ r.x=p.y*q.z-p.z*q.y;
+ r.y=p.z*q.x-p.x*q.z;
+ r.z=p.x*q.y-p.y*q.x;
+ r.w=1.;
+ return r;
+}
+double len3(Point3 p){
+ return sqrt(p.x*p.x+p.y*p.y+p.z*p.z);
+}
+double dist3(Point3 p, Point3 q){
+ p.x-=q.x;
+ p.y-=q.y;
+ p.z-=q.z;
+ return sqrt(p.x*p.x+p.y*p.y+p.z*p.z);
+}
+Point3 unit3(Point3 p){
+ double len=sqrt(p.x*p.x+p.y*p.y+p.z*p.z);
+ p.x/=len;
+ p.y/=len;
+ p.z/=len;
+ p.w=1.;
+ return p;
+}
+Point3 midpt3(Point3 p, Point3 q){
+ p.x=.5*(p.x+q.x);
+ p.y=.5*(p.y+q.y);
+ p.z=.5*(p.z+q.z);
+ p.w=1.;
+ return p;
+}
+Point3 lerp3(Point3 p, Point3 q, double alpha){
+ p.x+=(q.x-p.x)*alpha;
+ p.y+=(q.y-p.y)*alpha;
+ p.z+=(q.z-p.z)*alpha;
+ p.w=1.;
+ return p;
+}
+/*
+ * Reflect point p in the line joining p0 and p1
+ */
+Point3 reflect3(Point3 p, Point3 p0, Point3 p1){
+ Point3 a, b;
+ a=sub3(p, p0);
+ b=sub3(p1, p0);
+ return add3(a, mul3(b, 2*dot3(a, b)/dot3(b, b)));
+}
+/*
+ * Return the nearest point on segment [p0,p1] to point testp
+ */
+Point3 nearseg3(Point3 p0, Point3 p1, Point3 testp){
+ double num, den;
+ Point3 q, r;
+ q=sub3(p1, p0);
+ r=sub3(testp, p0);
+ num=dot3(q, r);;
+ if(num<=0) return p0;
+ den=dot3(q, q);
+ if(num>=den) return p1;
+ return add3(p0, mul3(q, num/den));
+}
+/*
+ * distance from point p to segment [p0,p1]
+ */
+#define SMALL 1e-8 /* what should this value be? */
+double pldist3(Point3 p, Point3 p0, Point3 p1){
+ Point3 d, e;
+ double dd, de, dsq;
+ d=sub3(p1, p0);
+ e=sub3(p, p0);
+ dd=dot3(d, d);
+ de=dot3(d, e);
+ if(dd<SMALL*SMALL) return len3(e);
+ dsq=dot3(e, e)-de*de/dd;
+ if(dsq<SMALL*SMALL) return 0;
+ return sqrt(dsq);
+}
+/*
+ * vdiv3(a, b) is the magnitude of the projection of a onto b
+ * measured in units of the length of b.
+ * vrem3(a, b) is the component of a perpendicular to b.
+ */
+double vdiv3(Point3 a, Point3 b){
+ return (a.x*b.x+a.y*b.y+a.z*b.z)/(b.x*b.x+b.y*b.y+b.z*b.z);
+}
+Point3 vrem3(Point3 a, Point3 b){
+ double quo=(a.x*b.x+a.y*b.y+a.z*b.z)/(b.x*b.x+b.y*b.y+b.z*b.z);
+ a.x-=b.x*quo;
+ a.y-=b.y*quo;
+ a.z-=b.z*quo;
+ a.w=1.;
+ return a;
+}
+/*
+ * Compute face (plane) with given normal, containing a given point
+ */
+Point3 pn2f3(Point3 p, Point3 n){
+ n.w=-dot3(p, n);
+ return n;
+}
+/*
+ * Compute face containing three points
+ */
+Point3 ppp2f3(Point3 p0, Point3 p1, Point3 p2){
+ Point3 p01, p02;
+ p01=sub3(p1, p0);
+ p02=sub3(p2, p0);
+ return pn2f3(p0, cross3(p01, p02));
+}
+/*
+ * Compute point common to three faces.
+ * Cramer's rule, yuk.
+ */
+Point3 fff2p3(Point3 f0, Point3 f1, Point3 f2){
+ double det;
+ Point3 p;
+ det=dot3(f0, cross3(f1, f2));
+ if(fabs(det)<SMALL){ /* parallel planes, bogus answer */
+ p.x=0.;
+ p.y=0.;
+ p.z=0.;
+ p.w=0.;
+ return p;
+ }
+ p.x=(f0.w*(f2.y*f1.z-f1.y*f2.z)
+ +f1.w*(f0.y*f2.z-f2.y*f0.z)+f2.w*(f1.y*f0.z-f0.y*f1.z))/det;
+ p.y=(f0.w*(f2.z*f1.x-f1.z*f2.x)
+ +f1.w*(f0.z*f2.x-f2.z*f0.x)+f2.w*(f1.z*f0.x-f0.z*f1.x))/det;
+ p.z=(f0.w*(f2.x*f1.y-f1.x*f2.y)
+ +f1.w*(f0.x*f2.y-f2.x*f0.y)+f2.w*(f1.x*f0.y-f0.x*f1.y))/det;
+ p.w=1.;
+ return p;
+}
+/*
+ * pdiv4 does perspective division to convert a projective point to affine coordinates.
+ */
+Point3 pdiv4(Point3 a){
+ if(a.w==0) return a;
+ a.x/=a.w;
+ a.y/=a.w;
+ a.z/=a.w;
+ a.w=1.;
+ return a;
+}
+Point3 add4(Point3 a, Point3 b){
+ a.x+=b.x;
+ a.y+=b.y;
+ a.z+=b.z;
+ a.w+=b.w;
+ return a;
+}
+Point3 sub4(Point3 a, Point3 b){
+ a.x-=b.x;
+ a.y-=b.y;
+ a.z-=b.z;
+ a.w-=b.w;
+ return a;
+}
DIR diff --git a/src/libgeometry/matrix.c b/src/libgeometry/matrix.c
t@@ -0,0 +1,106 @@
+/*
+ * ident(m) store identity matrix in m
+ * matmul(a, b) matrix multiply a*=b
+ * matmulr(a, b) matrix multiply a=b*a
+ * determinant(m) returns det(m)
+ * adjoint(m, minv) minv=adj(m)
+ * invertmat(m, minv) invert matrix m, result in minv, returns det(m)
+ * if m is singular, minv=adj(m)
+ */
+#include <u.h>
+#include <libc.h>
+#include <draw.h>
+#include <geometry.h>
+void ident(Matrix m){
+ register double *s=&m[0][0];
+ *s++=1;*s++=0;*s++=0;*s++=0;
+ *s++=0;*s++=1;*s++=0;*s++=0;
+ *s++=0;*s++=0;*s++=1;*s++=0;
+ *s++=0;*s++=0;*s++=0;*s=1;
+}
+void matmul(Matrix a, Matrix b){
+ int i, j, k;
+ double sum;
+ Matrix tmp;
+ for(i=0;i!=4;i++) for(j=0;j!=4;j++){
+ sum=0;
+ for(k=0;k!=4;k++)
+ sum+=a[i][k]*b[k][j];
+ tmp[i][j]=sum;
+ }
+ for(i=0;i!=4;i++) for(j=0;j!=4;j++)
+ a[i][j]=tmp[i][j];
+}
+void matmulr(Matrix a, Matrix b){
+ int i, j, k;
+ double sum;
+ Matrix tmp;
+ for(i=0;i!=4;i++) for(j=0;j!=4;j++){
+ sum=0;
+ for(k=0;k!=4;k++)
+ sum+=b[i][k]*a[k][j];
+ tmp[i][j]=sum;
+ }
+ for(i=0;i!=4;i++) for(j=0;j!=4;j++)
+ a[i][j]=tmp[i][j];
+}
+/*
+ * Return det(m)
+ */
+double determinant(Matrix m){
+ return m[0][0]*(m[1][1]*(m[2][2]*m[3][3]-m[2][3]*m[3][2])+
+ m[1][2]*(m[2][3]*m[3][1]-m[2][1]*m[3][3])+
+ m[1][3]*(m[2][1]*m[3][2]-m[2][2]*m[3][1]))
+ -m[0][1]*(m[1][0]*(m[2][2]*m[3][3]-m[2][3]*m[3][2])+
+ m[1][2]*(m[2][3]*m[3][0]-m[2][0]*m[3][3])+
+ m[1][3]*(m[2][0]*m[3][2]-m[2][2]*m[3][0]))
+ +m[0][2]*(m[1][0]*(m[2][1]*m[3][3]-m[2][3]*m[3][1])+
+ m[1][1]*(m[2][3]*m[3][0]-m[2][0]*m[3][3])+
+ m[1][3]*(m[2][0]*m[3][1]-m[2][1]*m[3][0]))
+ -m[0][3]*(m[1][0]*(m[2][1]*m[3][2]-m[2][2]*m[3][1])+
+ m[1][1]*(m[2][2]*m[3][0]-m[2][0]*m[3][2])+
+ m[1][2]*(m[2][0]*m[3][1]-m[2][1]*m[3][0]));
+}
+/*
+ * Store the adjoint (matrix of cofactors) of m in madj.
+ * Works fine even if m and madj are the same matrix.
+ */
+void adjoint(Matrix m, Matrix madj){
+ double m00=m[0][0], m01=m[0][1], m02=m[0][2], m03=m[0][3];
+ double m10=m[1][0], m11=m[1][1], m12=m[1][2], m13=m[1][3];
+ double m20=m[2][0], m21=m[2][1], m22=m[2][2], m23=m[2][3];
+ double m30=m[3][0], m31=m[3][1], m32=m[3][2], m33=m[3][3];
+ madj[0][0]=m11*(m22*m33-m23*m32)+m21*(m13*m32-m12*m33)+m31*(m12*m23-m13*m22);
+ madj[0][1]=m01*(m23*m32-m22*m33)+m21*(m02*m33-m03*m32)+m31*(m03*m22-m02*m23);
+ madj[0][2]=m01*(m12*m33-m13*m32)+m11*(m03*m32-m02*m33)+m31*(m02*m13-m03*m12);
+ madj[0][3]=m01*(m13*m22-m12*m23)+m11*(m02*m23-m03*m22)+m21*(m03*m12-m02*m13);
+ madj[1][0]=m10*(m23*m32-m22*m33)+m20*(m12*m33-m13*m32)+m30*(m13*m22-m12*m23);
+ madj[1][1]=m00*(m22*m33-m23*m32)+m20*(m03*m32-m02*m33)+m30*(m02*m23-m03*m22);
+ madj[1][2]=m00*(m13*m32-m12*m33)+m10*(m02*m33-m03*m32)+m30*(m03*m12-m02*m13);
+ madj[1][3]=m00*(m12*m23-m13*m22)+m10*(m03*m22-m02*m23)+m20*(m02*m13-m03*m12);
+ madj[2][0]=m10*(m21*m33-m23*m31)+m20*(m13*m31-m11*m33)+m30*(m11*m23-m13*m21);
+ madj[2][1]=m00*(m23*m31-m21*m33)+m20*(m01*m33-m03*m31)+m30*(m03*m21-m01*m23);
+ madj[2][2]=m00*(m11*m33-m13*m31)+m10*(m03*m31-m01*m33)+m30*(m01*m13-m03*m11);
+ madj[2][3]=m00*(m13*m21-m11*m23)+m10*(m01*m23-m03*m21)+m20*(m03*m11-m01*m13);
+ madj[3][0]=m10*(m22*m31-m21*m32)+m20*(m11*m32-m12*m31)+m30*(m12*m21-m11*m22);
+ madj[3][1]=m00*(m21*m32-m22*m31)+m20*(m02*m31-m01*m32)+m30*(m01*m22-m02*m21);
+ madj[3][2]=m00*(m12*m31-m11*m32)+m10*(m01*m32-m02*m31)+m30*(m02*m11-m01*m12);
+ madj[3][3]=m00*(m11*m22-m12*m21)+m10*(m02*m21-m01*m22)+m20*(m01*m12-m02*m11);
+}
+/*
+ * Store the inverse of m in minv.
+ * If m is singular, minv is instead its adjoint.
+ * Returns det(m).
+ * Works fine even if m and minv are the same matrix.
+ */
+double invertmat(Matrix m, Matrix minv){
+ double d, dinv;
+ int i, j;
+ d=determinant(m);
+ adjoint(m, minv);
+ if(d!=0.){
+ dinv=1./d;
+ for(i=0;i!=4;i++) for(j=0;j!=4;j++) minv[i][j]*=dinv;
+ }
+ return d;
+}
DIR diff --git a/src/libgeometry/mkfile b/src/libgeometry/mkfile
t@@ -0,0 +1,17 @@
+<$PLAN9/src/mkhdr
+
+LIB=libgeometry.a
+OFILES=\
+ arith3.$O\
+ matrix.$O\
+ qball.$O\
+ quaternion.$O\
+ transform.$O\
+ tstack.$O\
+
+HFILES=$PLAN9/include/geometry.h
+
+<$PLAN9/src/mksyslib
+
+listing:V:
+ pr mkfile $HFILES $CFILES|lp -du
DIR diff --git a/src/libgeometry/qball.c b/src/libgeometry/qball.c
t@@ -0,0 +1,66 @@
+/*
+ * Ken Shoemake's Quaternion rotation controller
+ */
+#include <u.h>
+#include <libc.h>
+#include <draw.h>
+#include <stdio.h>
+#include <event.h>
+#include <geometry.h>
+#define BORDER 4
+static Point ctlcen; /* center of qball */
+static int ctlrad; /* radius of qball */
+static Quaternion *axis; /* constraint plane orientation, 0 if none */
+/*
+ * Convert a mouse point into a unit quaternion, flattening if
+ * constrained to a particular plane.
+ */
+static Quaternion mouseq(Point p){
+ double qx=(double)(p.x-ctlcen.x)/ctlrad;
+ double qy=(double)(p.y-ctlcen.y)/ctlrad;
+ double rsq=qx*qx+qy*qy;
+ double l;
+ Quaternion q;
+ if(rsq>1){
+ rsq=sqrt(rsq);
+ q.r=0.;
+ q.i=qx/rsq;
+ q.j=qy/rsq;
+ q.k=0.;
+ }
+ else{
+ q.r=0.;
+ q.i=qx;
+ q.j=qy;
+ q.k=sqrt(1.-rsq);
+ }
+ if(axis){
+ l=q.i*axis->i+q.j*axis->j+q.k*axis->k;
+ q.i-=l*axis->i;
+ q.j-=l*axis->j;
+ q.k-=l*axis->k;
+ l=sqrt(q.i*q.i+q.j*q.j+q.k*q.k);
+ if(l!=0.){
+ q.i/=l;
+ q.j/=l;
+ q.k/=l;
+ }
+ }
+ return q;
+}
+void qball(Rectangle r, Mouse *m, Quaternion *result, void (*redraw)(void), Quaternion *ap){
+ Quaternion q, down;
+ Point rad;
+ axis=ap;
+ ctlcen=divpt(addpt(r.min, r.max), 2);
+ rad=divpt(subpt(r.max, r.min), 2);
+ ctlrad=(rad.x<rad.y?rad.x:rad.y)-BORDER;
+ down=qinv(mouseq(m->xy));
+ q=*result;
+ for(;;){
+ *m=emouse();
+ if(!m->buttons) break;
+ *result=qmul(q, qmul(down, mouseq(m->xy)));
+ (*redraw)();
+ }
+}
DIR diff --git a/src/libgeometry/quaternion.c b/src/libgeometry/quaternion.c
t@@ -0,0 +1,242 @@
+/*
+ * Quaternion arithmetic:
+ * qadd(q, r) returns q+r
+ * qsub(q, r) returns q-r
+ * qneg(q) returns -q
+ * qmul(q, r) returns q*r
+ * qdiv(q, r) returns q/r, can divide check.
+ * qinv(q) returns 1/q, can divide check.
+ * double qlen(p) returns modulus of p
+ * qunit(q) returns a unit quaternion parallel to q
+ * The following only work on unit quaternions and rotation matrices:
+ * slerp(q, r, a) returns q*(r*q^-1)^a
+ * qmid(q, r) slerp(q, r, .5)
+ * qsqrt(q) qmid(q, (Quaternion){1,0,0,0})
+ * qtom(m, q) converts a unit quaternion q into a rotation matrix m
+ * mtoq(m) returns a quaternion equivalent to a rotation matrix m
+ */
+#include <u.h>
+#include <libc.h>
+#include <draw.h>
+#include <geometry.h>
+void qtom(Matrix m, Quaternion q){
+#ifndef new
+ m[0][0]=1-2*(q.j*q.j+q.k*q.k);
+ m[0][1]=2*(q.i*q.j+q.r*q.k);
+ m[0][2]=2*(q.i*q.k-q.r*q.j);
+ m[0][3]=0;
+ m[1][0]=2*(q.i*q.j-q.r*q.k);
+ m[1][1]=1-2*(q.i*q.i+q.k*q.k);
+ m[1][2]=2*(q.j*q.k+q.r*q.i);
+ m[1][3]=0;
+ m[2][0]=2*(q.i*q.k+q.r*q.j);
+ m[2][1]=2*(q.j*q.k-q.r*q.i);
+ m[2][2]=1-2*(q.i*q.i+q.j*q.j);
+ m[2][3]=0;
+ m[3][0]=0;
+ m[3][1]=0;
+ m[3][2]=0;
+ m[3][3]=1;
+#else
+ /*
+ * Transcribed from Ken Shoemake's new code -- not known to work
+ */
+ double Nq = q.r*q.r+q.i*q.i+q.j*q.j+q.k*q.k;
+ double s = (Nq > 0.0) ? (2.0 / Nq) : 0.0;
+ double xs = q.i*s, ys = q.j*s, zs = q.k*s;
+ double wx = q.r*xs, wy = q.r*ys, wz = q.r*zs;
+ double xx = q.i*xs, xy = q.i*ys, xz = q.i*zs;
+ double yy = q.j*ys, yz = q.j*zs, zz = q.k*zs;
+ m[0][0] = 1.0 - (yy + zz); m[1][0] = xy + wz; m[2][0] = xz - wy;
+ m[0][1] = xy - wz; m[1][1] = 1.0 - (xx + zz); m[2][1] = yz + wx;
+ m[0][2] = xz + wy; m[1][2] = yz - wx; m[2][2] = 1.0 - (xx + yy);
+ m[0][3] = m[1][3] = m[2][3] = m[3][0] = m[3][1] = m[3][2] = 0.0;
+ m[3][3] = 1.0;
+#endif
+}
+Quaternion mtoq(Matrix mat){
+#ifndef new
+#define EPS 1.387778780781445675529539585113525e-17 /* 2^-56 */
+ double t;
+ Quaternion q;
+ q.r=0.;
+ q.i=0.;
+ q.j=0.;
+ q.k=1.;
+ if((t=.25*(1+mat[0][0]+mat[1][1]+mat[2][2]))>EPS){
+ q.r=sqrt(t);
+ t=4*q.r;
+ q.i=(mat[1][2]-mat[2][1])/t;
+ q.j=(mat[2][0]-mat[0][2])/t;
+ q.k=(mat[0][1]-mat[1][0])/t;
+ }
+ else if((t=-.5*(mat[1][1]+mat[2][2]))>EPS){
+ q.i=sqrt(t);
+ t=2*q.i;
+ q.j=mat[0][1]/t;
+ q.k=mat[0][2]/t;
+ }
+ else if((t=.5*(1-mat[2][2]))>EPS){
+ q.j=sqrt(t);
+ q.k=mat[1][2]/(2*q.j);
+ }
+ return q;
+#else
+ /*
+ * Transcribed from Ken Shoemake's new code -- not known to work
+ */
+ /* This algorithm avoids near-zero divides by looking for a large
+ * component -- first r, then i, j, or k. When the trace is greater than zero,
+ * |r| is greater than 1/2, which is as small as a largest component can be.
+ * Otherwise, the largest diagonal entry corresponds to the largest of |i|,
+ * |j|, or |k|, one of which must be larger than |r|, and at least 1/2.
+ */
+ Quaternion qu;
+ double tr, s;
+
+ tr = mat[0][0] + mat[1][1] + mat[2][2];
+ if (tr >= 0.0) {
+ s = sqrt(tr + mat[3][3]);
+ qu.r = s*0.5;
+ s = 0.5 / s;
+ qu.i = (mat[2][1] - mat[1][2]) * s;
+ qu.j = (mat[0][2] - mat[2][0]) * s;
+ qu.k = (mat[1][0] - mat[0][1]) * s;
+ }
+ else {
+ int i = 0;
+ if (mat[1][1] > mat[0][0]) i = 1;
+ if (mat[2][2] > mat[i][i]) i = 2;
+ switch(i){
+ case 0:
+ s = sqrt( (mat[0][0] - (mat[1][1]+mat[2][2])) + mat[3][3] );
+ qu.i = s*0.5;
+ s = 0.5 / s;
+ qu.j = (mat[0][1] + mat[1][0]) * s;
+ qu.k = (mat[2][0] + mat[0][2]) * s;
+ qu.r = (mat[2][1] - mat[1][2]) * s;
+ break;
+ case 1:
+ s = sqrt( (mat[1][1] - (mat[2][2]+mat[0][0])) + mat[3][3] );
+ qu.j = s*0.5;
+ s = 0.5 / s;
+ qu.k = (mat[1][2] + mat[2][1]) * s;
+ qu.i = (mat[0][1] + mat[1][0]) * s;
+ qu.r = (mat[0][2] - mat[2][0]) * s;
+ break;
+ case 2:
+ s = sqrt( (mat[2][2] - (mat[0][0]+mat[1][1])) + mat[3][3] );
+ qu.k = s*0.5;
+ s = 0.5 / s;
+ qu.i = (mat[2][0] + mat[0][2]) * s;
+ qu.j = (mat[1][2] + mat[2][1]) * s;
+ qu.r = (mat[1][0] - mat[0][1]) * s;
+ break;
+ }
+ }
+ if (mat[3][3] != 1.0){
+ s=1/sqrt(mat[3][3]);
+ qu.r*=s;
+ qu.i*=s;
+ qu.j*=s;
+ qu.k*=s;
+ }
+ return (qu);
+#endif
+}
+Quaternion qadd(Quaternion q, Quaternion r){
+ q.r+=r.r;
+ q.i+=r.i;
+ q.j+=r.j;
+ q.k+=r.k;
+ return q;
+}
+Quaternion qsub(Quaternion q, Quaternion r){
+ q.r-=r.r;
+ q.i-=r.i;
+ q.j-=r.j;
+ q.k-=r.k;
+ return q;
+}
+Quaternion qneg(Quaternion q){
+ q.r=-q.r;
+ q.i=-q.i;
+ q.j=-q.j;
+ q.k=-q.k;
+ return q;
+}
+Quaternion qmul(Quaternion q, Quaternion r){
+ Quaternion s;
+ s.r=q.r*r.r-q.i*r.i-q.j*r.j-q.k*r.k;
+ s.i=q.r*r.i+r.r*q.i+q.j*r.k-q.k*r.j;
+ s.j=q.r*r.j+r.r*q.j+q.k*r.i-q.i*r.k;
+ s.k=q.r*r.k+r.r*q.k+q.i*r.j-q.j*r.i;
+ return s;
+}
+Quaternion qdiv(Quaternion q, Quaternion r){
+ return qmul(q, qinv(r));
+}
+Quaternion qunit(Quaternion q){
+ double l=qlen(q);
+ q.r/=l;
+ q.i/=l;
+ q.j/=l;
+ q.k/=l;
+ return q;
+}
+/*
+ * Bug?: takes no action on divide check
+ */
+Quaternion qinv(Quaternion q){
+ double l=q.r*q.r+q.i*q.i+q.j*q.j+q.k*q.k;
+ q.r/=l;
+ q.i=-q.i/l;
+ q.j=-q.j/l;
+ q.k=-q.k/l;
+ return q;
+}
+double qlen(Quaternion p){
+ return sqrt(p.r*p.r+p.i*p.i+p.j*p.j+p.k*p.k);
+}
+Quaternion slerp(Quaternion q, Quaternion r, double a){
+ double u, v, ang, s;
+ double dot=q.r*r.r+q.i*r.i+q.j*r.j+q.k*r.k;
+ ang=dot<-1?PI:dot>1?0:acos(dot); /* acos gives NaN for dot slightly out of range */
+ s=sin(ang);
+ if(s==0) return ang<PI/2?q:r;
+ u=sin((1-a)*ang)/s;
+ v=sin(a*ang)/s;
+ q.r=u*q.r+v*r.r;
+ q.i=u*q.i+v*r.i;
+ q.j=u*q.j+v*r.j;
+ q.k=u*q.k+v*r.k;
+ return q;
+}
+/*
+ * Only works if qlen(q)==qlen(r)==1
+ */
+Quaternion qmid(Quaternion q, Quaternion r){
+ double l;
+ q=qadd(q, r);
+ l=qlen(q);
+ if(l<1e-12){
+ q.r=r.i;
+ q.i=-r.r;
+ q.j=r.k;
+ q.k=-r.j;
+ }
+ else{
+ q.r/=l;
+ q.i/=l;
+ q.j/=l;
+ q.k/=l;
+ }
+ return q;
+}
+/*
+ * Only works if qlen(q)==1
+ */
+static Quaternion qident={1,0,0,0};
+Quaternion qsqrt(Quaternion q){
+ return qmid(q, qident);
+}
DIR diff --git a/src/libgeometry/transform.c b/src/libgeometry/transform.c
t@@ -0,0 +1,75 @@
+/*
+ * The following routines transform points and planes from one space
+ * to another. Points and planes are represented by their
+ * homogeneous coordinates, stored in variables of type Point3.
+ */
+#include <u.h>
+#include <libc.h>
+#include <draw.h>
+#include <geometry.h>
+/*
+ * Transform point p.
+ */
+Point3 xformpoint(Point3 p, Space *to, Space *from){
+ Point3 q, r;
+ register double *m;
+ if(from){
+ m=&from->t[0][0];
+ q.x=*m++*p.x; q.x+=*m++*p.y; q.x+=*m++*p.z; q.x+=*m++*p.w;
+ q.y=*m++*p.x; q.y+=*m++*p.y; q.y+=*m++*p.z; q.y+=*m++*p.w;
+ q.z=*m++*p.x; q.z+=*m++*p.y; q.z+=*m++*p.z; q.z+=*m++*p.w;
+ q.w=*m++*p.x; q.w+=*m++*p.y; q.w+=*m++*p.z; q.w+=*m *p.w;
+ }
+ else
+ q=p;
+ if(to){
+ m=&to->tinv[0][0];
+ r.x=*m++*q.x; r.x+=*m++*q.y; r.x+=*m++*q.z; r.x+=*m++*q.w;
+ r.y=*m++*q.x; r.y+=*m++*q.y; r.y+=*m++*q.z; r.y+=*m++*q.w;
+ r.z=*m++*q.x; r.z+=*m++*q.y; r.z+=*m++*q.z; r.z+=*m++*q.w;
+ r.w=*m++*q.x; r.w+=*m++*q.y; r.w+=*m++*q.z; r.w+=*m *q.w;
+ }
+ else
+ r=q;
+ return r;
+}
+/*
+ * Transform point p with perspective division.
+ */
+Point3 xformpointd(Point3 p, Space *to, Space *from){
+ p=xformpoint(p, to, from);
+ if(p.w!=0){
+ p.x/=p.w;
+ p.y/=p.w;
+ p.z/=p.w;
+ p.w=1;
+ }
+ return p;
+}
+/*
+ * Transform plane p -- same as xformpoint, except multiply on the
+ * other side by the inverse matrix.
+ */
+Point3 xformplane(Point3 p, Space *to, Space *from){
+ Point3 q, r;
+ register double *m;
+ if(from){
+ m=&from->tinv[0][0];
+ q.x =*m++*p.x; q.y =*m++*p.x; q.z =*m++*p.x; q.w =*m++*p.x;
+ q.x+=*m++*p.y; q.y+=*m++*p.y; q.z+=*m++*p.y; q.w+=*m++*p.y;
+ q.x+=*m++*p.z; q.y+=*m++*p.z; q.z+=*m++*p.z; q.w+=*m++*p.z;
+ q.x+=*m++*p.w; q.y+=*m++*p.w; q.z+=*m++*p.w; q.w+=*m *p.w;
+ }
+ else
+ q=p;
+ if(to){
+ m=&to->t[0][0];
+ r.x =*m++*q.x; r.y =*m++*q.x; r.z =*m++*q.x; r.w =*m++*q.x;
+ r.x+=*m++*q.y; r.y+=*m++*q.y; r.z+=*m++*q.y; r.w+=*m++*q.y;
+ r.x+=*m++*q.z; r.y+=*m++*q.z; r.z+=*m++*q.z; r.w+=*m++*q.z;
+ r.x+=*m++*q.w; r.y+=*m++*q.w; r.z+=*m++*q.w; r.w+=*m *q.w;
+ }
+ else
+ r=q;
+ return r;
+}
DIR diff --git a/src/libgeometry/tstack.c b/src/libgeometry/tstack.c
t@@ -0,0 +1,169 @@
+/*% cc -gpc %
+ * These transformation routines maintain stacks of transformations
+ * and their inverses.
+ * t=pushmat(t) push matrix stack
+ * t=popmat(t) pop matrix stack
+ * rot(t, a, axis) multiply stack top by rotation
+ * qrot(t, q) multiply stack top by rotation, q is unit quaternion
+ * scale(t, x, y, z) multiply stack top by scale
+ * move(t, x, y, z) multiply stack top by translation
+ * xform(t, m) multiply stack top by m
+ * ixform(t, m, inv) multiply stack top by m. inv is the inverse of m.
+ * look(t, e, l, u) multiply stack top by viewing transformation
+ * persp(t, fov, n, f) multiply stack top by perspective transformation
+ * viewport(t, r, aspect)
+ * multiply stack top by window->viewport transformation.
+ */
+#include <u.h>
+#include <libc.h>
+#include <draw.h>
+#include <geometry.h>
+Space *pushmat(Space *t){
+ Space *v;
+ v=malloc(sizeof(Space));
+ if(t==0){
+ ident(v->t);
+ ident(v->tinv);
+ }
+ else
+ *v=*t;
+ v->next=t;
+ return v;
+}
+Space *popmat(Space *t){
+ Space *v;
+ if(t==0) return 0;
+ v=t->next;
+ free(t);
+ return v;
+}
+void rot(Space *t, double theta, int axis){
+ double s=sin(radians(theta)), c=cos(radians(theta));
+ Matrix m, inv;
+ int i=(axis+1)%3, j=(axis+2)%3;
+ ident(m);
+ m[i][i] = c;
+ m[i][j] = -s;
+ m[j][i] = s;
+ m[j][j] = c;
+ ident(inv);
+ inv[i][i] = c;
+ inv[i][j] = s;
+ inv[j][i] = -s;
+ inv[j][j] = c;
+ ixform(t, m, inv);
+}
+void qrot(Space *t, Quaternion q){
+ Matrix m, inv;
+ int i, j;
+ qtom(m, q);
+ for(i=0;i!=4;i++) for(j=0;j!=4;j++) inv[i][j]=m[j][i];
+ ixform(t, m, inv);
+}
+void scale(Space *t, double x, double y, double z){
+ Matrix m, inv;
+ ident(m);
+ m[0][0]=x;
+ m[1][1]=y;
+ m[2][2]=z;
+ ident(inv);
+ inv[0][0]=1/x;
+ inv[1][1]=1/y;
+ inv[2][2]=1/z;
+ ixform(t, m, inv);
+}
+void move(Space *t, double x, double y, double z){
+ Matrix m, inv;
+ ident(m);
+ m[0][3]=x;
+ m[1][3]=y;
+ m[2][3]=z;
+ ident(inv);
+ inv[0][3]=-x;
+ inv[1][3]=-y;
+ inv[2][3]=-z;
+ ixform(t, m, inv);
+}
+void xform(Space *t, Matrix m){
+ Matrix inv;
+ if(invertmat(m, inv)==0) return;
+ ixform(t, m, inv);
+}
+void ixform(Space *t, Matrix m, Matrix inv){
+ matmul(t->t, m);
+ matmulr(t->tinv, inv);
+}
+/*
+ * multiply the top of the matrix stack by a view-pointing transformation
+ * with the eyepoint at e, looking at point l, with u at the top of the screen.
+ * The coordinate system is deemed to be right-handed.
+ * The generated transformation transforms this view into a view from
+ * the origin, looking in the positive y direction, with the z axis pointing up,
+ * and x to the right.
+ */
+void look(Space *t, Point3 e, Point3 l, Point3 u){
+ Matrix m, inv;
+ Point3 r;
+ l=unit3(sub3(l, e));
+ u=unit3(vrem3(sub3(u, e), l));
+ r=cross3(l, u);
+ /* make the matrix to transform from (rlu) space to (xyz) space */
+ ident(m);
+ m[0][0]=r.x; m[0][1]=r.y; m[0][2]=r.z;
+ m[1][0]=l.x; m[1][1]=l.y; m[1][2]=l.z;
+ m[2][0]=u.x; m[2][1]=u.y; m[2][2]=u.z;
+ ident(inv);
+ inv[0][0]=r.x; inv[0][1]=l.x; inv[0][2]=u.x;
+ inv[1][0]=r.y; inv[1][1]=l.y; inv[1][2]=u.y;
+ inv[2][0]=r.z; inv[2][1]=l.z; inv[2][2]=u.z;
+ ixform(t, m, inv);
+ move(t, -e.x, -e.y, -e.z);
+}
+/*
+ * generate a transformation that maps the frustum with apex at the origin,
+ * apex angle=fov and clipping planes y=n and y=f into the double-unit cube.
+ * plane y=n maps to y'=-1, y=f maps to y'=1
+ */
+int persp(Space *t, double fov, double n, double f){
+ Matrix m;
+ double z;
+ if(n<=0 || f<=n || fov<=0 || 180<=fov) /* really need f!=n && sin(v)!=0 */
+ return -1;
+ z=1/tan(radians(fov)/2);
+ m[0][0]=z; m[0][1]=0; m[0][2]=0; m[0][3]=0;
+ m[1][0]=0; m[1][1]=(f+n)/(f-n); m[1][2]=0; m[1][3]=f*(1-m[1][1]);
+ m[2][0]=0; m[2][1]=0; m[2][2]=z; m[2][3]=0;
+ m[3][0]=0; m[3][1]=1; m[3][2]=0; m[3][3]=0;
+ xform(t, m);
+ return 0;
+}
+/*
+ * Map the unit-cube window into the given screen viewport.
+ * r has min at the top left, max just outside the lower right. Aspect is the
+ * aspect ratio (dx/dy) of the viewport's pixels (not of the whole viewport!)
+ * The whole window is transformed to fit centered inside the viewport with equal
+ * slop on either top and bottom or left and right, depending on the viewport's
+ * aspect ratio.
+ * The window is viewed down the y axis, with x to the left and z up. The viewport
+ * has x increasing to the right and y increasing down. The window's y coordinates
+ * are mapped, unchanged, into the viewport's z coordinates.
+ */
+void viewport(Space *t, Rectangle r, double aspect){
+ Matrix m;
+ double xc, yc, wid, hgt, scale;
+ xc=.5*(r.min.x+r.max.x);
+ yc=.5*(r.min.y+r.max.y);
+ wid=(r.max.x-r.min.x)*aspect;
+ hgt=r.max.y-r.min.y;
+ scale=.5*(wid<hgt?wid:hgt);
+ ident(m);
+ m[0][0]=scale;
+ m[0][3]=xc;
+ m[1][1]=0;
+ m[1][2]=-scale;
+ m[1][3]=yc;
+ m[2][1]=1;
+ m[2][2]=0;
+ /* should get inverse by hand */
+ xform(t, m);
+}