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@castano castano/Sphere.cpp
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// This code is in the public domain -- Ignacio Castaño <castano@gmail.com>
#include "Sphere.h"
#include "Vector.inl"
#include "Box.inl"
#include <float.h> // FLT_MAX
using namespace nv;
const float radiusEpsilon = 1e-4f;
Sphere::Sphere(Vector3::Arg p0, Vector3::Arg p1) {
if (p0 == p1) *this = Sphere(p0);
else {
center = (p0 + p1) * 0.5f;
radius = length(p0 - center) + radiusEpsilon;
float d0 = length(p0 - center);
float d1 = length(p1 - center);
nvDebugCheck(equal(d0, radius - radiusEpsilon));
nvDebugCheck(equal(d1, radius - radiusEpsilon));
}
}
Sphere::Sphere(Vector3::Arg p0, Vector3::Arg p1, Vector3::Arg p2) {
if (p0 == p1 || p0 == p2) *this = Sphere(p1, p2);
else if (p1 == p2) *this = Sphere(p0, p2);
else {
Vector3 a = p1 - p0;
Vector3 b = p2 - p0;
Vector3 c = cross(a, b);
float denominator = 2.0f * lengthSquared(c);
if (!isZero(denominator)) {
Vector3 d = (lengthSquared(b) * cross(c, a) + lengthSquared(a) * cross(b, c)) / denominator;
center = p0 + d;
radius = length(d) + radiusEpsilon;
float d0 = length(p0 - center);
float d1 = length(p1 - center);
float d2 = length(p2 - center);
nvDebugCheck(equal(d0, radius - radiusEpsilon));
nvDebugCheck(equal(d1, radius - radiusEpsilon));
nvDebugCheck(equal(d2, radius - radiusEpsilon));
}
else {
// @@ This is a specialization of the code below, but really, the only thing we need to do here is to find the two most distant points.
// Compute all possible spheres, invalidate those that do not contain the four points, keep the smallest.
Sphere s0(p1, p2);
float d0 = distanceSquared(s0, p0);
if (d0 > 0) s0.radius = NV_FLOAT_MAX;
Sphere s1(p0, p2);
float d1 = distanceSquared(s1, p1);
if (d1 > 0) s1.radius = NV_FLOAT_MAX;
Sphere s2(p0, p1);
float d2 = distanceSquared(s2, p2);
if (d2 > 0) s1.radius = NV_FLOAT_MAX;
if (s0.radius < s1.radius && s0.radius < s2.radius) {
center = s0.center;
radius = s0.radius;
}
else if (s1.radius < s2.radius) {
center = s1.center;
radius = s1.radius;
}
else {
center = s2.center;
radius = s2.radius;
}
}
}
}
Sphere::Sphere(Vector3::Arg p0, Vector3::Arg p1, Vector3::Arg p2, Vector3::Arg p3) {
if (p0 == p1 || p0 == p2 || p0 == p3) *this = Sphere(p1, p2, p3);
else if (p1 == p2 || p1 == p3) *this = Sphere(p0, p2, p3);
else if (p2 == p3) *this = Sphere(p0, p1, p2);
else {
// @@ This only works if the points are not coplanar!
Vector3 a = p1 - p0;
Vector3 b = p2 - p0;
Vector3 c = p3 - p0;
float denominator = 2.0f * dot(c, cross(a, b)); // triple product.
if (!isZero(denominator)) {
Vector3 d = (lengthSquared(c) * cross(a, b) + lengthSquared(b) * cross(c, a) + lengthSquared(a) * cross(b, c)) / denominator;
center = p0 + d;
radius = length(d) + radiusEpsilon;
float d0 = length(p0 - center);
float d1 = length(p1 - center);
float d2 = length(p2 - center);
float d3 = length(p3 - center);
nvDebugCheck(equal(d0, radius - radiusEpsilon));
nvDebugCheck(equal(d1, radius - radiusEpsilon));
nvDebugCheck(equal(d2, radius - radiusEpsilon));
nvDebugCheck(equal(d3, radius - radiusEpsilon));
}
else {
// Compute all possible spheres, invalidate those that do not contain the four points, keep the smallest.
Sphere s0(p1, p2, p3);
float d0 = distanceSquared(s0, p0);
if (d0 > 0) s0.radius = NV_FLOAT_MAX;
Sphere s1(p0, p2, p3);
float d1 = distanceSquared(s1, p1);
if (d1 > 0) s1.radius = NV_FLOAT_MAX;
Sphere s2(p0, p1, p3);
float d2 = distanceSquared(s2, p2);
if (d2 > 0) s2.radius = NV_FLOAT_MAX;
Sphere s3(p0, p1, p2);
float d3 = distanceSquared(s3, p3);
if (d3 > 0) s2.radius = NV_FLOAT_MAX;
if (s0.radius < s1.radius && s0.radius < s2.radius && s0.radius < s3.radius) {
center = s0.center;
radius = s0.radius;
}
else if (s1.radius < s2.radius && s1.radius < s3.radius) {
center = s1.center;
radius = s1.radius;
}
else if (s1.radius < s3.radius) {
center = s2.center;
radius = s2.radius;
}
else {
center = s3.center;
radius = s3.radius;
}
}
}
}
float nv::distanceSquared(const Sphere & sphere, const Vector3 & point) {
return lengthSquared(sphere.center - point) - square(sphere.radius);
}
// Implementation of "MiniBall" based on:
// http://www.flipcode.com/archives/Smallest_Enclosing_Spheres.shtml
static Sphere recurseMini(const Vector3 *P[], uint p, uint b = 0) {
Sphere MB;
switch(b) {
case 0:
MB = Sphere(*P[0]);
break;
case 1:
MB = Sphere(*P[-1]);
break;
case 2:
MB = Sphere(*P[-1], *P[-2]);
break;
case 3:
MB = Sphere(*P[-1], *P[-2], *P[-3]);
break;
case 4:
MB = Sphere(*P[-1], *P[-2], *P[-3], *P[-4]);
return MB;
}
for (uint i = 0; i < p; i++) {
if (distanceSquared(MB, *P[i]) > 0) { // Signed square distance to sphere
for (uint j = i; j > 0; j--) {
swap(P[j], P[j-1]);
}
MB = recurseMini(P + 1, i, b + 1);
}
}
return MB;
}
static bool allInside(const Sphere & sphere, const Vector3 * pointArray, const uint pointCount) {
for (uint i = 0; i < pointCount; i++) {
if (distanceSquared(sphere, pointArray[i]) >= NV_EPSILON) {
return false;
}
}
return true;
}
Sphere nv::miniBall(const Vector3 * pointArray, const uint pointCount) {
nvDebugCheck(pointArray != NULL);
nvDebugCheck(pointCount > 0);
const Vector3 **L = new const Vector3*[pointCount];
for (uint i = 0; i < pointCount; i++) {
L[i] = &pointArray[i];
}
Sphere sphere = recurseMini(L, pointCount);
delete [] L;
nvDebugCheck(allInside(sphere, pointArray, pointCount));
return sphere;
}
// Approximate bounding sphere, based on "An Efficient Bounding Sphere" by Jack Ritter, from "Graphics Gems"
Sphere nv::approximateSphere_Ritter(const Vector3 * pointArray, const uint pointCount) {
nvDebugCheck(pointArray != NULL);
nvDebugCheck(pointCount > 0);
Vector3 xmin, xmax, ymin, ymax, zmin, zmax;
xmin = xmax = ymin = ymax = zmin = zmax = pointArray[0];
// FIRST PASS: find 6 minima/maxima points
xmin.x = ymin.y = zmin.z = FLT_MAX;
xmax.x = ymax.y = zmax.z = -FLT_MAX;
for (uint i = 0; i < pointCount; i++) {
const Vector3 & p = pointArray[i];
if (p.x < xmin.x) xmin = p;
if (p.x > xmax.x) xmax = p;
if (p.y < ymin.y) ymin = p;
if (p.y > ymax.y) ymax = p;
if (p.z < zmin.z) zmin = p;
if (p.z > zmax.z) zmax = p;
}
float xspan = lengthSquared(xmax - xmin);
float yspan = lengthSquared(ymax - ymin);
float zspan = lengthSquared(zmax - zmin);
// Set points dia1 & dia2 to the maximally separated pair.
Vector3 dia1 = xmin;
Vector3 dia2 = xmax;
float maxspan = xspan;
if (yspan > maxspan) {
maxspan = yspan;
dia1 = ymin;
dia2 = ymax;
}
if (zspan > maxspan) {
dia1 = zmin;
dia2 = zmax;
}
// |dia1-dia2| is a diameter of initial sphere
// calc initial center
Sphere sphere;
sphere.center = (dia1 + dia2) / 2.0f;
// calculate initial radius**2 and radius
float rad_sq = lengthSquared(dia2 - sphere.center);
sphere.radius = sqrtf(rad_sq);
// SECOND PASS: increment current sphere
for (uint i = 0; i < pointCount; i++) {
const Vector3 & p = pointArray[i];
float old_to_p_sq = lengthSquared(p - sphere.center);
if (old_to_p_sq > rad_sq) { // do r**2 test first
// this point is outside of current sphere
float old_to_p = sqrtf(old_to_p_sq);
// calc radius of new sphere
sphere.radius = (sphere.radius + old_to_p) / 2.0f;
rad_sq = sphere.radius * sphere.radius; // for next r**2 compare
float old_to_new = old_to_p - sphere.radius;
// calc center of new sphere
sphere.center = (sphere.radius * sphere.center + old_to_new * p) / old_to_p;
}
}
nvDebugCheck(allInside(sphere, pointArray, pointCount));
return sphere;
}
static float computeSphereRadius(const Vector3 & center, const Vector3 * pointArray, const uint pointCount) {
float maxRadius2 = 0;
for (uint i = 0; i < pointCount; i++) {
const Vector3 & p = pointArray[i];
float r2 = lengthSquared(center - p);
if (r2 > maxRadius2) {
maxRadius2 = r2;
}
}
return sqrtf(maxRadius2) + radiusEpsilon;
}
Sphere nv::approximateSphere_AABB(const Vector3 * pointArray, const uint pointCount) {
nvDebugCheck(pointArray != NULL);
nvDebugCheck(pointCount > 0);
Box box;
box.clearBounds();
for (uint i = 0; i < pointCount; i++) {
box.addPointToBounds(pointArray[i]);
}
Sphere sphere;
sphere.center = box.center();
sphere.radius = computeSphereRadius(sphere.center, pointArray, pointCount);
nvDebugCheck(allInside(sphere, pointArray, pointCount));
return sphere;
}
static void computeExtremalPoints(const Vector3 & dir, const Vector3 * pointArray, uint pointCount, Vector3 * minPoint, Vector3 * maxPoint) {
nvDebugCheck(pointCount > 0);
uint mini = 0;
uint maxi = 0;
float minDist = FLT_MAX;
float maxDist = -FLT_MAX;
for (uint i = 0; i < pointCount; i++) {
float d = dot(dir, pointArray[i]);
if (d < minDist) {
minDist = d;
mini = i;
}
if (d > maxDist) {
maxDist = d;
maxi = i;
}
}
nvDebugCheck(minDist != FLT_MAX);
nvDebugCheck(maxDist != -FLT_MAX);
*minPoint = pointArray[mini];
*maxPoint = pointArray[maxi];
}
// EPOS algorithm based on:
// http://www.ep.liu.se/ecp/034/009/ecp083409.pdf
Sphere nv::approximateSphere_EPOS6(const Vector3 * pointArray, uint pointCount) {
nvDebugCheck(pointArray != NULL);
nvDebugCheck(pointCount > 0);
Vector3 extremalPoints[6];
// Compute 6 extremal points.
computeExtremalPoints(Vector3(1, 0, 0), pointArray, pointCount, extremalPoints+0, extremalPoints+1);
computeExtremalPoints(Vector3(0, 1, 0), pointArray, pointCount, extremalPoints+2, extremalPoints+3);
computeExtremalPoints(Vector3(0, 0, 1), pointArray, pointCount, extremalPoints+4, extremalPoints+5);
Sphere sphere = miniBall(extremalPoints, 6);
sphere.radius = computeSphereRadius(sphere.center, pointArray, pointCount);
nvDebugCheck(allInside(sphere, pointArray, pointCount));
return sphere;
}
Sphere nv::approximateSphere_EPOS14(const Vector3 * pointArray, uint pointCount) {
nvDebugCheck(pointArray != NULL);
nvDebugCheck(pointCount > 0);
Vector3 extremalPoints[14];
// Compute 14 extremal points.
computeExtremalPoints(Vector3(1, 0, 0), pointArray, pointCount, extremalPoints+0, extremalPoints+1);
computeExtremalPoints(Vector3(0, 1, 0), pointArray, pointCount, extremalPoints+2, extremalPoints+3);
computeExtremalPoints(Vector3(0, 0, 1), pointArray, pointCount, extremalPoints+4, extremalPoints+5);
float d = sqrtf(1.0f/3.0f);
computeExtremalPoints(Vector3(d, d, d), pointArray, pointCount, extremalPoints+6, extremalPoints+7);
computeExtremalPoints(Vector3(-d, d, d), pointArray, pointCount, extremalPoints+8, extremalPoints+9);
computeExtremalPoints(Vector3(-d, -d, d), pointArray, pointCount, extremalPoints+10, extremalPoints+11);
computeExtremalPoints(Vector3(d, -d, d), pointArray, pointCount, extremalPoints+12, extremalPoints+13);
Sphere sphere = miniBall(extremalPoints, 14);
sphere.radius = computeSphereRadius(sphere.center, pointArray, pointCount);
nvDebugCheck(allInside(sphere, pointArray, pointCount));
return sphere;
}
// This code is in the public domain -- Ignacio Castaño <castano@gmail.com>
#pragma once
#ifndef NV_MATH_SPHERE_H
#define NV_MATH_SPHERE_H
#include "Vector.h"
namespace nv
{
class Sphere
{
public:
Sphere() {}
Sphere(Vector3::Arg center, float radius) : center(center), radius(radius) {}
Sphere(Vector3::Arg center) : center(center), radius(0.0f) {}
Sphere(Vector3::Arg p0, Vector3::Arg p1);
Sphere(Vector3::Arg p0, Vector3::Arg p1, Vector3::Arg p2);
Sphere(Vector3::Arg p0, Vector3::Arg p1, Vector3::Arg p2, Vector3::Arg p3);
Vector3 center;
float radius;
};
// Returns negative values if point is inside.
float distanceSquared(const Sphere & sphere, const Vector3 &point);
// Welz's algorithm. Fairly slow, recursive implementation uses large stack.
Sphere miniBall(const Vector3 * pointArray, uint pointCount);
Sphere approximateSphere_Ritter(const Vector3 * pointArray, uint pointCount);
Sphere approximateSphere_AABB(const Vector3 * pointArray, uint pointCount);
Sphere approximateSphere_EPOS6(const Vector3 * pointArray, uint pointCount);
Sphere approximateSphere_EPOS14(const Vector3 * pointArray, uint pointCount);
} // nv namespace
#endif // NV_MATH_SPHERE_H
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