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+/* arcdot.c
+ *
+ * Angle between two vectors
+ *
+ *
+ *
+ *
+ * SYNOPSIS:
+ *
+ * double p[3], q[3], arcdot();
+ *
+ * y = arcdot( p, q );
+ *
+ *
+ *
+ * DESCRIPTION:
+ *
+ * For two vectors p, q, the angle A between them is given by
+ *
+ * p.q / (|p| |q|) = cos A .
+ *
+ * where "." represents inner product, "|x|" the length of vector x.
+ * If the angle is small, an expression in sin A is preferred.
+ * Set r = q - p. Then
+ *
+ * p.q = p.p + p.r ,
+ *
+ * |p|^2 = p.p ,
+ *
+ * |q|^2 = p.p + 2 p.r + r.r ,
+ *
+ * p.p^2 + 2 p.p p.r + p.r^2
+ * cos^2 A = ----------------------------
+ * p.p (p.p + 2 p.r + r.r)
+ *
+ * p.p + 2 p.r + p.r^2 / p.p
+ * = --------------------------- ,
+ * p.p + 2 p.r + r.r
+ *
+ * sin^2 A = 1 - cos^2 A
+ *
+ * r.r - p.r^2 / p.p
+ * = --------------------
+ * p.p + 2 p.r + r.r
+ *
+ * = (r.r - p.r^2 / p.p) / q.q .
+ *
+ * ACCURACY:
+ *
+ * Relative error:
+ * arithmetic domain # trials peak rms
+ * IEEE -1, 1 10^6 1.7e-16 4.2e-17
+ *
+ */
+
+/*
+Cephes Math Library Release 2.3: November, 1995
+Copyright 1995 by Stephen L. Moshier
+*/
+
+#include <math.h>
+#ifdef ANSIPROT
+extern double sqrt ( double );
+extern double acos ( double );
+extern double asin ( double );
+extern double atan ( double );
+#else
+double sqrt(), acos(), asin(), atan();
+#endif
+extern double PI;
+
+double arcdot(p,q)
+double p[], q[];
+{
+double pp, pr, qq, rr, rt, pt, qt, pq;
+int i;
+
+pq = 0.0;
+qq = 0.0;
+pp = 0.0;
+pr = 0.0;
+rr = 0.0;
+for (i=0; i<3; i++)
+ {
+ pt = p[i];
+ qt = q[i];
+ pq += pt * qt;
+ qq += qt * qt;
+ pp += pt * pt;
+ rt = qt - pt;
+ pr += pt * rt;
+ rr += rt * rt;
+ }
+if (rr == 0.0 || pp == 0.0 || qq == 0.0)
+ return 0.0;
+rt = (rr - (pr * pr) / pp) / qq;
+if (rt <= 0.75)
+ {
+ rt = sqrt(rt);
+ qt = asin(rt);
+ if (pq < 0.0)
+ qt = PI - qt;
+ }
+else
+ {
+ pt = pq / sqrt(pp*qq);
+ qt = acos(pt);
+ }
+return qt;
+}