commit d1e9002f81f14fbfef1ebc4261edccd9eb97b72c
from: rsc <devnull@localhost>
date: Tue Jan 04 21:23:01 2005 UTC

3D geometry

commit - 46f79934b79ef526ed42bbe5a565e6b5d884d24a
commit + d1e9002f81f14fbfef1ebc4261edccd9eb97b72c
blob - /dev/null
blob + 8ab1755e6d55e28cb30043a7039560b7bc18da55 (mode 644)
--- /dev/null
+++ src/libgeometry/arith3.c
@@ -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;
+}
blob - /dev/null
blob + 2c372ef664b613c273e63a093c3be1e744640938 (mode 644)
--- /dev/null
+++ src/libgeometry/matrix.c
@@ -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;
+}
blob - /dev/null
blob + 6510104e6755b9ae999998cd19ba1c7844dd5217 (mode 644)
--- /dev/null
+++ src/libgeometry/mkfile
@@ -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
blob - /dev/null
blob + b73ecc511c358f84a0a478c4e09e0ae4ebd0d7b7 (mode 644)
--- /dev/null
+++ src/libgeometry/qball.c
@@ -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)();
+	}
+}
blob - /dev/null
blob + 1f920f5a8cd4768fe9352b710aaa00087721e021 (mode 644)
--- /dev/null
+++ src/libgeometry/quaternion.c
@@ -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);
+}
blob - /dev/null
blob + a59248725cb84a65cd94786adc0ced4005340665 (mode 644)
--- /dev/null
+++ src/libgeometry/transform.c
@@ -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;
+}
blob - /dev/null
blob + bc41c4acfdf0fcfe3af7dd968e9dfd95fcde31d9 (mode 644)
--- /dev/null
+++ src/libgeometry/tstack.c
@@ -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);
+}