static OrderedTriple[] sectPlanes(
OrderedTriple p11,
OrderedTriple p12,
OrderedTriple p13,
OrderedTriple p21,
OrderedTriple p22,
OrderedTriple p23) {
OrderedTriple n1 = p11.minus(p12).cross(p12.minus(p13));
OrderedTriple n2 = p21.minus(p22).cross(p22.minus(p23));
OrderedTriple n3 = n1.cross(n2);
double[] m1 = {n1.x, n1.y, n1.z};
double[] m2 = {n2.x, n2.y, n2.z};
double[] m3 = {n3.x, n3.y, n3.z};
double[][] m = {m1, m2, m3};
Matrix M = new Matrix(m);
double[] b1 = {n1.dot(p11)};
double[] b2 = {n2.dot(p21)};
double[] b3 = {0};
double[][] b = {b1, b2, b3};
Matrix B = new Matrix(b);
Matrix S = M.inverse().times(B);
// OrderedTriple s1 = new OrderedTriple( S.mat[0][0], S.mat[1][0],
// S.mat[2][0] );
OrderedTriple s1 = new OrderedTriple(S.mat[0][0], S.mat[1][0], S.mat[2][0]);
OrderedTriple s2 = s1.plus(n3);
OrderedTriple[] answer = {s1, s2};
return answer;
}
public static OrderedTriple sectPlaneLine(
OrderedTriple P1, OrderedTriple P2, OrderedTriple P3, OrderedTriple L1, OrderedTriple L2) {
// Find the 3D coordinates of the intersection of the plane containing
// points P1, P2, P3 and the line containing L1, L2
OrderedTriple n = P2.minus(P1).cross(P3.minus(P2));
OrderedTriple l = L2.minus(L1);
double A = n.dot(l);
if (A == 0) return null;
double B = n.dot(P1);
double C = n.dot(L1);
double K = (B - C) / A;
return l.times(K).plus(L1);
}
static OrderedTriple[] sectSphereLine(double r, OrderedTriple a, OrderedTriple b) {
// Find the intersection(s) of the sphere with radius length r and center at
// the origin and the line containing points a and b
OrderedTriple v = b.minus(a);
try {
double[] k = solveQuadratic(v.lengthSquared(), 2 * a.dot(v), a.lengthSquared() - r * r);
OrderedTriple[] answer = {v.times(k[0]).plus(a), v.times(k[1]).plus(a)};
return answer;
} catch (Exception e) {
return null;
}
}
static double[] planeFromNormalAndPoint(OrderedTriple n, OrderedTriple p) {
double[] answer = {n.x, n.y, n.z, n.dot(p)};
return answer;
}
public static OrderedTriple findThirdVector(
OrderedTriple v1, OrderedTriple v2, double a1, double a2, OrderedTriple testv)
throws Exception {
// v1 and v2 must be equal lengths N. returns the vector a1 rad from v1, a2
// rad from v2 and N length. if there are two such vectors
// it returns the one with positive component of testv
// System.out.println( "angle between v1 & v2: " + v1.degBetween( v2 ) );
double epsilon = 0.0001;
if (isApprox(a1, 0, epsilon)) return new OrderedTriple(v1);
if (isApprox(a1, Math.PI, epsilon)) return v1.negative();
if (isApprox(a2, 0, epsilon)) return new OrderedTriple(v2);
if (isApprox(a2, Math.PI, epsilon)) return v2.negative();
OrderedTriple v3 = null;
double a = v1.x;
double b = v1.y;
double c = v1.z;
double d = v2.x;
double e = v2.y;
double f = v2.z;
double g, h, i;
double roots[] = null;
double N = v1.length();
double A = N * N * Math.cos(a1);
double B = N * N * Math.cos(a2);
double x12 = a * e - b * d;
double x23 = b * f - c * e;
double x31 = c * d - a * f;
double P, Q, R, S;
P = Q = R = S = 0;
if (!isApprox(x12, 0, epsilon)) {
// System.out.println( "route 1" );
P = x23 / x12;
Q = (e * A - b * B) / x12;
R = x31 / x12;
S = (a * B - d * A) / x12;
roots = solveQuadratic(1 + P * P + R * R, 2 * P * Q + 2 * R * S, Q * Q + S * S - N * N);
i = roots[0];
v3 = new OrderedTriple(P * i + Q, R * i + S, i);
if (testv != null && v3.dot(testv) < 0) {
i = roots[1];
v3.become(P * i + Q, R * i + S, i);
}
} else if (!isApprox(x23, 0, epsilon)) {
// System.out.println( "route 2" );
P = x31 / x23;
Q = (f * A - c * B) / x23;
R = x12 / x23;
S = (b * B - e * A) / x23;
roots = solveQuadratic(1 + P * P + R * R, 2 * P * Q + 2 * R * S, Q * Q + S * S - N * N);
g = roots[0];
v3 = new OrderedTriple(g, P * g + Q, R * g + S);
if (testv != null && v3.dot(testv) < 0) {
g = roots[1];
v3.become(g, P * g + Q, R * g + S);
}
} else if (!isApprox(x31, 0, epsilon)) {
// System.out.println( "route 3" );
P = x23 / x31;
Q = (c * B - f * A) / x31;
R = x12 / x31;
S = (d * A - a * B) / x31;
roots = solveQuadratic(1 + P * P + R * R, 2 * P * Q + 2 * R * S, Q * Q + S * S - N * N);
h = roots[0];
v3 = new OrderedTriple(P * h + Q, h, R * h + S);
if (testv != null && v3.dot(testv) < 0) {
h = roots[1];
v3.become(P * h + Q, h, R * h + S);
}
} else {
throw new Exception("EXCEPTION: no route found in findThirdVector");
}
// if( ! isApprox( v3.radBetween( v1 ), a1, 0.1 ) ) System.out.println(
// "error v1" );
// if( ! isApprox( v3.radBetween( v2 ), a2, 0.1 ) ) System.out.println(
// "error v2" );
return v3;
}
