1 /*
2  * Quaternion arithmetic:
3  *	qadd(q, r)	returns q+r
4  *	qsub(q, r)	returns q-r
5  *	qneg(q)		returns -q
6  *	qmul(q, r)	returns q*r
7  *	qdiv(q, r)	returns q/r, can divide check.
8  *	qinv(q)		returns 1/q, can divide check.
9  *	double qlen(p)	returns modulus of p
10  *	qunit(q)	returns a unit quaternion parallel to q
11  * The following only work on unit quaternions and rotation matrices:
12  *	slerp(q, r, a)	returns q*(r*q^-1)^a
13  *	qmid(q, r)	slerp(q, r, .5)
14  *	qsqrt(q)	qmid(q, (Quaternion){1,0,0,0})
15  *	qtom(m, q)	converts a unit quaternion q into a rotation matrix m
16  *	mtoq(m)		returns a quaternion equivalent to a rotation matrix m
17  */
18 #include <u.h>
19 #include <libc.h>
20 #include <draw.h>
21 #include <geometry.h>
22 void qtom(Matrix m, Quaternion q){
23 #ifndef new
24 	m[0][0]=1-2*(q.j*q.j+q.k*q.k);
25 	m[0][1]=2*(q.i*q.j+q.r*q.k);
26 	m[0][2]=2*(q.i*q.k-q.r*q.j);
27 	m[0][3]=0;
28 	m[1][0]=2*(q.i*q.j-q.r*q.k);
29 	m[1][1]=1-2*(q.i*q.i+q.k*q.k);
30 	m[1][2]=2*(q.j*q.k+q.r*q.i);
31 	m[1][3]=0;
32 	m[2][0]=2*(q.i*q.k+q.r*q.j);
33 	m[2][1]=2*(q.j*q.k-q.r*q.i);
34 	m[2][2]=1-2*(q.i*q.i+q.j*q.j);
35 	m[2][3]=0;
36 	m[3][0]=0;
37 	m[3][1]=0;
38 	m[3][2]=0;
39 	m[3][3]=1;
40 #else
41 	/*
42 	 * Transcribed from Ken Shoemake's new code -- not known to work
43 	 */
44 	double Nq = q.r*q.r+q.i*q.i+q.j*q.j+q.k*q.k;
45 	double s = (Nq > 0.0) ? (2.0 / Nq) : 0.0;
46 	double xs = q.i*s,		ys = q.j*s,		zs = q.k*s;
47 	double wx = q.r*xs,		wy = q.r*ys,		wz = q.r*zs;
48 	double xx = q.i*xs,		xy = q.i*ys,		xz = q.i*zs;
49 	double yy = q.j*ys,		yz = q.j*zs,		zz = q.k*zs;
50 	m[0][0] = 1.0 - (yy + zz); m[1][0] = xy + wz;         m[2][0] = xz - wy;
51 	m[0][1] = xy - wz;         m[1][1] = 1.0 - (xx + zz); m[2][1] = yz + wx;
52 	m[0][2] = xz + wy;         m[1][2] = yz - wx;         m[2][2] = 1.0 - (xx + yy);
53 	m[0][3] = m[1][3] = m[2][3] = m[3][0] = m[3][1] = m[3][2] = 0.0;
54 	m[3][3] = 1.0;
55 #endif
56 }
57 Quaternion mtoq(Matrix mat){
58 #ifndef new
59 #define	EPS	1.387778780781445675529539585113525e-17	/* 2^-56 */
60 	double t;
61 	Quaternion q;
62 	q.r=0.;
63 	q.i=0.;
64 	q.j=0.;
65 	q.k=1.;
66 	if((t=.25*(1+mat[0][0]+mat[1][1]+mat[2][2]))>EPS){
67 		q.r=sqrt(t);
68 		t=4*q.r;
69 		q.i=(mat[1][2]-mat[2][1])/t;
70 		q.j=(mat[2][0]-mat[0][2])/t;
71 		q.k=(mat[0][1]-mat[1][0])/t;
72 	}
73 	else if((t=-.5*(mat[1][1]+mat[2][2]))>EPS){
74 		q.i=sqrt(t);
75 		t=2*q.i;
76 		q.j=mat[0][1]/t;
77 		q.k=mat[0][2]/t;
78 	}
79 	else if((t=.5*(1-mat[2][2]))>EPS){
80 		q.j=sqrt(t);
81 		q.k=mat[1][2]/(2*q.j);
82 	}
83 	return q;
84 #else
85 	/*
86 	 * Transcribed from Ken Shoemake's new code -- not known to work
87 	 */
88 	/* This algorithm avoids near-zero divides by looking for a large
89 	 * component -- first r, then i, j, or k.  When the trace is greater than zero,
90 	 * |r| is greater than 1/2, which is as small as a largest component can be.
91 	 * Otherwise, the largest diagonal entry corresponds to the largest of |i|,
92 	 * |j|, or |k|, one of which must be larger than |r|, and at least 1/2.
93 	 */
94 	Quaternion qu;
95 	double tr, s;
96 
97 	tr = mat[0][0] + mat[1][1] + mat[2][2];
98 	if (tr >= 0.0) {
99 		s = sqrt(tr + mat[3][3]);
100 		qu.r = s*0.5;
101 		s = 0.5 / s;
102 		qu.i = (mat[2][1] - mat[1][2]) * s;
103 		qu.j = (mat[0][2] - mat[2][0]) * s;
104 		qu.k = (mat[1][0] - mat[0][1]) * s;
105 	}
106 	else {
107 		int i = 0;
108 		if (mat[1][1] > mat[0][0]) i = 1;
109 		if (mat[2][2] > mat[i][i]) i = 2;
110 		switch(i){
111 		case 0:
112 			s = sqrt( (mat[0][0] - (mat[1][1]+mat[2][2])) + mat[3][3] );
113 			qu.i = s*0.5;
114 			s = 0.5 / s;
115 			qu.j = (mat[0][1] + mat[1][0]) * s;
116 			qu.k = (mat[2][0] + mat[0][2]) * s;
117 			qu.r = (mat[2][1] - mat[1][2]) * s;
118 			break;
119 		case 1:
120 			s = sqrt( (mat[1][1] - (mat[2][2]+mat[0][0])) + mat[3][3] );
121 			qu.j = s*0.5;
122 			s = 0.5 / s;
123 			qu.k = (mat[1][2] + mat[2][1]) * s;
124 			qu.i = (mat[0][1] + mat[1][0]) * s;
125 			qu.r = (mat[0][2] - mat[2][0]) * s;
126 			break;
127 		case 2:
128 			s = sqrt( (mat[2][2] - (mat[0][0]+mat[1][1])) + mat[3][3] );
129 			qu.k = s*0.5;
130 			s = 0.5 / s;
131 			qu.i = (mat[2][0] + mat[0][2]) * s;
132 			qu.j = (mat[1][2] + mat[2][1]) * s;
133 			qu.r = (mat[1][0] - mat[0][1]) * s;
134 			break;
135 		}
136 	}
137 	if (mat[3][3] != 1.0){
138 		s=1/sqrt(mat[3][3]);
139 		qu.r*=s;
140 		qu.i*=s;
141 		qu.j*=s;
142 		qu.k*=s;
143 	}
144 	return (qu);
145 #endif
146 }
147 Quaternion qadd(Quaternion q, Quaternion r){
148 	q.r+=r.r;
149 	q.i+=r.i;
150 	q.j+=r.j;
151 	q.k+=r.k;
152 	return q;
153 }
154 Quaternion qsub(Quaternion q, Quaternion r){
155 	q.r-=r.r;
156 	q.i-=r.i;
157 	q.j-=r.j;
158 	q.k-=r.k;
159 	return q;
160 }
161 Quaternion qneg(Quaternion q){
162 	q.r=-q.r;
163 	q.i=-q.i;
164 	q.j=-q.j;
165 	q.k=-q.k;
166 	return q;
167 }
168 Quaternion qmul(Quaternion q, Quaternion r){
169 	Quaternion s;
170 	s.r=q.r*r.r-q.i*r.i-q.j*r.j-q.k*r.k;
171 	s.i=q.r*r.i+r.r*q.i+q.j*r.k-q.k*r.j;
172 	s.j=q.r*r.j+r.r*q.j+q.k*r.i-q.i*r.k;
173 	s.k=q.r*r.k+r.r*q.k+q.i*r.j-q.j*r.i;
174 	return s;
175 }
176 Quaternion qdiv(Quaternion q, Quaternion r){
177 	return qmul(q, qinv(r));
178 }
179 Quaternion qunit(Quaternion q){
180 	double l=qlen(q);
181 	q.r/=l;
182 	q.i/=l;
183 	q.j/=l;
184 	q.k/=l;
185 	return q;
186 }
187 /*
188  * Bug?: takes no action on divide check
189  */
190 Quaternion qinv(Quaternion q){
191 	double l=q.r*q.r+q.i*q.i+q.j*q.j+q.k*q.k;
192 	q.r/=l;
193 	q.i=-q.i/l;
194 	q.j=-q.j/l;
195 	q.k=-q.k/l;
196 	return q;
197 }
198 double qlen(Quaternion p){
199 	return sqrt(p.r*p.r+p.i*p.i+p.j*p.j+p.k*p.k);
200 }
201 Quaternion slerp(Quaternion q, Quaternion r, double a){
202 	double u, v, ang, s;
203 	double dot=q.r*r.r+q.i*r.i+q.j*r.j+q.k*r.k;
204 	ang=dot<-1?PI:dot>1?0:acos(dot); /* acos gives NaN for dot slightly out of range */
205 	s=sin(ang);
206 	if(s==0) return ang<PI/2?q:r;
207 	u=sin((1-a)*ang)/s;
208 	v=sin(a*ang)/s;
209 	q.r=u*q.r+v*r.r;
210 	q.i=u*q.i+v*r.i;
211 	q.j=u*q.j+v*r.j;
212 	q.k=u*q.k+v*r.k;
213 	return q;
214 }
215 /*
216  * Only works if qlen(q)==qlen(r)==1
217  */
218 Quaternion qmid(Quaternion q, Quaternion r){
219 	double l;
220 	q=qadd(q, r);
221 	l=qlen(q);
222 	if(l<1e-12){
223 		q.r=r.i;
224 		q.i=-r.r;
225 		q.j=r.k;
226 		q.k=-r.j;
227 	}
228 	else{
229 		q.r/=l;
230 		q.i/=l;
231 		q.j/=l;
232 		q.k/=l;
233 	}
234 	return q;
235 }
236 /*
237  * Only works if qlen(q)==1
238  */
239 static Quaternion qident={1,0,0,0};
240 Quaternion qsqrt(Quaternion q){
241 	return qmid(q, qident);
242 }