in commons-geometry-enclosing/src/main/java/org/apache/commons/geometry/enclosing/euclidean/threed/SphereGenerator.java [49:135]
public EnclosingBall<Vector3D> ballOnSupport(final List<Vector3D> support) {
if (support.isEmpty()) {
return new EnclosingBall<>(Vector3D.ZERO, Double.NEGATIVE_INFINITY, Collections.emptyList());
}
final Vector3D vA = support.get(0);
if (support.size() < 2) {
return new EnclosingBall<>(vA, 0, Collections.singletonList(vA));
}
final Vector3D vB = support.get(1);
if (support.size() < 3) {
return new EnclosingBall<>(vA.lerp(vB, 0.5),
0.5 * vA.distance(vB),
Arrays.asList(vA, vB));
}
final Vector3D vC = support.get(2);
if (support.size() < 4) {
final EmbeddingPlane p = Planes.fromPoints(vA, vB, vC, precision).getEmbedding();
final EnclosingBall<Vector2D> disk =
new DiskGenerator().ballOnSupport(Arrays.asList(p.toSubspace(vA),
p.toSubspace(vB),
p.toSubspace(vC)));
// convert back to 3D
return new EnclosingBall<>(p.toSpace(disk.getCenter()),
disk.getRadius(),
Arrays.asList(vA, vB, vC));
}
final Vector3D vD = support.get(3);
// a sphere is 3D can be defined as:
// (1) (x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = r^2
// which can be written:
// (2) (x^2 + y^2 + z^2) - 2 x_0 x - 2 y_0 y - 2 z_0 z + (x_0^2 + y_0^2 + z_0^2 - r^2) = 0
// or simply:
// (3) (x^2 + y^2 + z^2) + a x + b y + c z + d = 0
// with sphere center coordinates -a/2, -b/2, -c/2
// If the sphere exists, a b, c and d are a non zero solution to
// [ (x^2 + y^2 + z^2) x y z 1 ] [ 1 ] [ 0 ]
// [ (xA^2 + yA^2 + zA^2) xA yA zA 1 ] [ a ] [ 0 ]
// [ (xB^2 + yB^2 + zB^2) xB yB zB 1 ] * [ b ] = [ 0 ]
// [ (xC^2 + yC^2 + zC^2) xC yC zC 1 ] [ c ] [ 0 ]
// [ (xD^2 + yD^2 + zD^2) xD yD zD 1 ] [ d ] [ 0 ]
// So the determinant of the matrix is zero. Computing this determinant
// by expanding it using the minors m_ij of first row leads to
// (4) m_11 (x^2 + y^2 + z^2) - m_12 x + m_13 y - m_14 z + m_15 = 0
// So by identifying equations (2) and (4) we get the coordinates
// of center as:
// x_0 = +m_12 / (2 m_11)
// y_0 = -m_13 / (2 m_11)
// z_0 = +m_14 / (2 m_11)
// Note that the minors m_11, m_12, m_13 and m_14 all have the last column
// filled with 1.0, hence simplifying the computation
final BigFraction[] c2 = {
BigFraction.from(vA.getX()), BigFraction.from(vB.getX()),
BigFraction.from(vC.getX()), BigFraction.from(vD.getX())
};
final BigFraction[] c3 = {
BigFraction.from(vA.getY()), BigFraction.from(vB.getY()),
BigFraction.from(vC.getY()), BigFraction.from(vD.getY())
};
final BigFraction[] c4 = {
BigFraction.from(vA.getZ()), BigFraction.from(vB.getZ()),
BigFraction.from(vC.getZ()), BigFraction.from(vD.getZ())
};
final BigFraction[] c1 = {
c2[0].multiply(c2[0]).add(c3[0].multiply(c3[0])).add(c4[0].multiply(c4[0])),
c2[1].multiply(c2[1]).add(c3[1].multiply(c3[1])).add(c4[1].multiply(c4[1])),
c2[2].multiply(c2[2]).add(c3[2].multiply(c3[2])).add(c4[2].multiply(c4[2])),
c2[3].multiply(c2[3]).add(c3[3].multiply(c3[3])).add(c4[3].multiply(c4[3]))
};
final BigFraction twoM11 = minor(c2, c3, c4).multiply(2);
final BigFraction m12 = minor(c1, c3, c4);
final BigFraction m13 = minor(c1, c2, c4);
final BigFraction m14 = minor(c1, c2, c3);
final BigFraction centerX = m12.divide(twoM11);
final BigFraction centerY = m13.divide(twoM11).negate();
final BigFraction centerZ = m14.divide(twoM11);
final BigFraction dx = c2[0].subtract(centerX);
final BigFraction dy = c3[0].subtract(centerY);
final BigFraction dz = c4[0].subtract(centerZ);
final BigFraction r2 = dx.multiply(dx).add(dy.multiply(dy)).add(dz.multiply(dz));
return new EnclosingBall<>(Vector3D.of(centerX.doubleValue(),
centerY.doubleValue(),
centerZ.doubleValue()),
Math.sqrt(r2.doubleValue()),
Arrays.asList(vA, vB, vC, vD));
}