path: root/APEX_1.4/shared/internal/include/authoring/ApexGSA.h

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#ifndef APEX_GSA_H
#define APEX_GSA_H


#include "ApexCSGMath.h"

#ifndef WITHOUT_APEX_AUTHORING

namespace ApexCSG
{
namespace GSA
{

// Utility vector format translation
inline physx::PxVec3 toPxVec3(const Vec4Real& p)
{
	return physx::PxVec3(static_cast<float>(p[0]), static_cast<float>(p[1]), static_cast<float>(p[2]));
}
	
	
/*** Compact implementation of the void simplex algorithm for D = 3 ***/

typedef physx::PxF32 real;

/*
	The implementation of farthest_halfspace should return the half-space "most below" the given point.  The point
	is represented by a vector in projective coordinates, and its last element (point[3]) will not necessarily equal 1.
	However, point[3] will be non-negative.  The plane returned is the boundary of the half-space found, and is also
	represented as a vector in projective coordinates (the coefficients of the plane equation).  The plane vector
	returned should have the greatest dot product with the input point.

	plane = the returned half-space boundary
	point = the input point
	returns the dot product of point and plane
*/
struct VS3D_Halfspace_Set
{
	virtual	real	farthest_halfspace(real plane[4], const real point[4]) = 0;
};


#define	VS3D_HIGH_ACCURACY	1
#define VS3D_UNNORMALIZED_PLANE_HANDLING	0	// 0 = planes must be normalized, 1 = planes must be near-normalized, 2 = planes may be arbitrary 
#define REAL_DOUBLE	0


#if VS3D_UNNORMALIZED_PLANE_HANDLING == 1
// Returns approximation to 1/sqrt(x)
inline real	vs3d_recip_sqrt(real x)
{
	real y = (real)1.5 - (real)0.5*x;
#if REAL_DOUBLE
	y *= (real)1.5 - (real)0.5*x*y*y;	// Perform another iteration for doubles, to handle the case where float-normalized normals are converted to double-precision
#endif
	return y;
}
#elif VS3D_UNNORMALIZED_PLANE_HANDLING == 2
#include <cmath>
inline real	vs3d_recip_sqrt(real x) { return 1/sqrt(x); }
#elif VS3D_UNNORMALIZED_PLANE_HANDLING != 0
#error Unrecognized value given for VS3D_UNNORMALIZED_PLANE_HANDLING.  Please set to 0, 1, or 2.
#endif


// Simple types and operations for internal calculations
struct Vec3 { real x, y, z; };	// 3-vector 
inline Vec3	vec3(real x, real y, real z)				{ Vec3 r; r.x = x; r.y = y; r.z = z; return r; }	// vector builder
inline Vec3	operator +	(const Vec3& a, const Vec3& b)	{ return vec3(a.x+b.x, a.y+b.y, a.z+b.z); }			// vector addition
inline Vec3	operator *	(real s, const Vec3& v)			{ return vec3(s*v.x, s*v.y, s*v.z); }				// scalar multiplication
inline real	operator |	(const Vec3& a, const Vec3& b)	{ return a.x*b.x + a.y*b.y + a.z*b.z; }				// dot product
inline Vec3 operator ^	(const Vec3& a, const Vec3& b)	{ return vec3(a.y*b.z - b.y*a.z, a.z*b.x - b.z*a.x, a.x*b.y - b.x*a.y); }	// cross product

struct Vec4 { Vec3 v; real w; };	// 4-vector split into 3-vector and scalar parts
inline Vec4	vec4(const Vec3& v, real w)					{ Vec4 r; r.v = v; r.w = w; return r; }	// vector builder
inline real	operator | (const Vec4& a, const Vec4& b)	{ return (a.v|b.v) + a.w*b.w; }			// dot product

// More accurate perpendicular
inline Vec3	perp(const Vec3& a, const Vec3& b)
{
	Vec3 c = a^b;	// Cross-product gives perpendicular
#if VS3D_HIGH_ACCURACY || REAL_DOUBLE
	const real c2 = c|c;
	if (c2 != 0) c = c + (1/c2)*((a|c)*(c^b) + (b|c)*(a^c));	// Improvement to (a b)^T(c) = (0)
#endif
	return c;
}

// Square
inline real	sq(real x) { return x*x; }

// Returns index of the extremal element in a three-element set {e0, e1, e2} based upon comparisons c_ij. The extremal index m is such that c_mn is true, or e_m == e_n, for all n.
inline int	ext_index(int c_10, int c_21, int c_20)	{ return c_10<<c_21|(c_21&c_20)<<1; }

// Returns index (0, 1, or 2) of minimum argument
inline int	index_of_min(real x0, real x1, real x2)	{ return ext_index((int)(x1 < x0), (int)(x2 < x1), (int)(x2 < x0)); }

// Compare fractions with positive deominators.  Returns a_num*sqrt(a_rden2) > b_num*sqrt(b_rden2)
inline bool frac_gt(real a_num, real a_rden2, real b_num, real b_rden2)
{
	const bool a_num_neg = a_num < 0;
	const bool b_num_neg = b_num < 0;
	return a_num_neg != b_num_neg ? b_num_neg : ((a_num*a_num*a_rden2 > b_num*b_num*b_rden2) != a_num_neg);
}

// Returns index (0, 1, or 2) of maximum fraction with positive deominators
inline int	index_of_max_frac(real x0_num, real x0_rden2, real x1_num, real x1_rden2, real x2_num, real x2_rden2)
{
	return ext_index((int)frac_gt(x1_num, x1_rden2, x0_num, x0_rden2), (int)frac_gt(x2_num, x2_rden2, x1_num, x1_rden2), (int)frac_gt(x2_num, x2_rden2, x0_num, x0_rden2));
}

// Compare values given their signs and squares.  Returns a > b.  a2 and b2 may have any constant offset applied to them.
inline bool sgn_sq_gt(real sgn_a, real a2, real sgn_b, real b2) { return sgn_a*sgn_b < 0 ? (sgn_b < 0) : ((a2 > b2) != (sgn_a < 0)); }

// Returns index (0, 1, or 2) of maximum value given their signs and squares.  sq_x0, sq_x1, and sq_x2 may have any constant offset applied to them.
inline int	index_of_max_sgn_sq(real sgn_x0, real sq_x0, real sgn_x1, real sq_x1, real sgn_x2, real sq_x2)
{
	return ext_index((int)sgn_sq_gt(sgn_x1, sq_x1, sgn_x0, sq_x0), (int)sgn_sq_gt(sgn_x2, sq_x2, sgn_x1, sq_x1), (int)sgn_sq_gt(sgn_x2, sq_x2, sgn_x0, sq_x0));
}

// Project 2D (homogeneous) vector onto 2D half-space boundary
inline void project2D(Vec3& r, const Vec3& plane, real delta, real recip_n2, real eps2)
{
	r = r + (-delta*recip_n2)*vec3(plane.x, plane.y, 0);
	r = r + (-(r|plane)*recip_n2)*vec3(plane.x, plane.y, 0);	// Second projection for increased accuracy
	if ((r|r) > eps2) return;
	r = (-plane.z*recip_n2)*vec3(plane.x, plane.y, 0);
	r.z = 1;
}


// Update function for vs3d_test
static bool vs3d_update(Vec4& p, Vec4 S[4], int& plane_count, const Vec4& q, real eps2)
{
	// h plane is the last plane
	const Vec4& h = S[plane_count-1];

	// Handle plane_count == 1 specially (optimization; this could be commented out)
	if (plane_count == 1)
	{
		// Solution is objective projected onto h plane
		p = q;
		p.v = p.v + -(p|h)*h.v;
		if ((p|p) <= eps2) p = vec4(-h.w*h.v, 1);	// If p == 0 then q is a direction vector, any point in h is a support point
		return true;
	}

	// Create basis in the h plane
	const int min_i = index_of_min(h.v.x*h.v.x, h.v.y*h.v.y, h.v.z*h.v.z);
	const Vec3 y = h.v^vec3((real)(min_i == 0), (real)(min_i == 1), (real)(min_i == 2));
	const Vec3 x = y^h.v;

	// Use reduced vector r instead of p
	Vec3 r = {x|q.v, y|q.v, q.w*(y|y)};	// (x|x) = (y|y) = square of plane basis scale

	// If r == 0 (within epsilon), then it is a direction vector, and we have a bounded solution
	if ((r|r) <= eps2) r.z = 1;

	// Create plane equations in the h plane.  These will not be normalized in general.
	int N = 0;			// Plane count in h subspace
	Vec3 R[3];			// Planes in h subspace
	real recip_n2[3];	// Plane normal vector reciprocal lengths squared
	real delta[3];		// Signed distance of objective to the planes
	int index[3];		// Keep track of original plane indices
	for (int i = 0; i < plane_count-1; ++i)
	{
		const Vec3& vi = S[i].v;
		const real cos_theta = h.v|vi;
		R[N] = vec3(x|vi, y|vi, S[i].w - h.w*cos_theta);
		index[N] = i;
		const real n2 = R[N].x*R[N].x + R[N].y*R[N].y;
		if (n2 >= eps2)
		{
			const real lin_norm = (real)1.5-(real)0.5*n2;	// 1st-order approximation to 1/sqrt(n2) expanded about n2 = 1
			R[N] = lin_norm*R[N];	// We don't need normalized plane equations, but rescaling (even with an approximate normalization) gives better numerical behavior
			recip_n2[N] = 1/(R[N].x*R[N].x + R[N].y*R[N].y);
			delta[N] = r|R[N];
			++N;	// Keep this plane
		}
		else if (cos_theta < 0) return false;	// Parallel cases are redundant and rejected, anti-parallel cases are 1D voids
	}

	// Now work with the N-sized R array of half-spaces in the h plane
	switch (N)
	{
	case 1: one_plane:
		if (delta[0] < 0) N = 0;	// S[0] is redundant, eliminate it
		else project2D(r, R[0], delta[0], recip_n2[0], eps2);
		break;
	case 2: two_planes:
		if (delta[0] < 0 && delta[1] < 0) N = 0;	// S[0] and S[1] are redundant, eliminate them
		else
		{
			const int max_d_index = (int)frac_gt(delta[1], recip_n2[1], delta[0], recip_n2[0]);
			project2D(r, R[max_d_index], delta[max_d_index], recip_n2[max_d_index], eps2);
			const int min_d_index = max_d_index^1;
			const real new_delta_min = r|R[min_d_index];
			if (new_delta_min < 0)
			{
				index[0] = index[max_d_index];
				N = 1;	// S[min_d_index] is redundant, eliminate it
			}
			else
			{
				// Set r to the intersection of R[0] and R[1] and keep both
				r = perp(R[0], R[1]);
				if (r.z*r.z*recip_n2[0]*recip_n2[1] < eps2)
				{
					if (R[0].x*R[1].x + R[0].y*R[1].y < 0) return false;	// 2D void found
					goto one_plane;
				}
				r = (1/r.z)*r;	// We could just as well multiply r by sgn(r.z); we just need to ensure r.z > 0
			}
		}
		break;
	case 3:
		if (delta[0] < 0 && delta[1] < 0 && delta[2] < 0) N = 0;	// S[0], S[1], and S[2] are redundant, eliminate them
		else
		{
			const Vec3 row_x = {R[0].x, R[1].x, R[2].x};
			const Vec3 row_y = {R[0].y, R[1].y, R[2].y};
			const Vec3 row_w = {R[0].z, R[1].z, R[2].z};
			const Vec3 cof_w = perp(row_x, row_y);
			const bool detR_pos = (row_w|cof_w) > 0;
			const int nrw_sgn0 = cof_w.x*cof_w.x*recip_n2[1]*recip_n2[2] < eps2 ? 0 : (((int)((cof_w.x > 0) == detR_pos)<<1)-1);
			const int nrw_sgn1 = cof_w.y*cof_w.y*recip_n2[2]*recip_n2[0] < eps2 ? 0 : (((int)((cof_w.y > 0) == detR_pos)<<1)-1);
			const int nrw_sgn2 = cof_w.z*cof_w.z*recip_n2[0]*recip_n2[1] < eps2 ? 0 : (((int)((cof_w.z > 0) == detR_pos)<<1)-1);

			if ((nrw_sgn0|nrw_sgn1|nrw_sgn2) >= 0) return false;	// 3D void found

			const int positive_width_count = ((nrw_sgn0>>1)&1) + ((nrw_sgn1>>1)&1) + ((nrw_sgn2>>1)&1);
			if (positive_width_count == 1)
			{
				// A single positive width results from a redundant plane.  Eliminate it and peform N = 2 calculation.
				const int pos_width_index = ((nrw_sgn1>>1)&1)|(nrw_sgn2&2);	// Calculates which index corresponds to the positive-width side
				R[pos_width_index] = R[2];
				recip_n2[pos_width_index] = recip_n2[2];
				delta[pos_width_index] = delta[2];
				index[pos_width_index] = index[2];
				N = 2;
				goto two_planes;
			}

			// Find the max dot product of r and R[i]/|R_normal[i]|.  For numerical accuracy when the angle between r and the i^{th} plane normal is small, we take some care below:
			const int max_d_index = r.z != 0
				? index_of_max_frac(delta[0], recip_n2[0], delta[1], recip_n2[1], delta[2], recip_n2[2])	// displacement term resolves small-angle ambiguity, just use dot product
				: index_of_max_sgn_sq(delta[0], -sq(r.x*R[0].y - r.y*R[0].x)*recip_n2[0], delta[1], -sq(r.x*R[1].y - r.y*R[1].x)*recip_n2[1], delta[2], -sq(r.x*R[2].y - r.y*R[2].x)*recip_n2[2]);	// No displacement term.  Use wedge product to find the sine of the angle.

			// Project r onto max-d plane
			project2D(r, R[max_d_index], delta[max_d_index], recip_n2[max_d_index], eps2);
			N = 1;	// Unless we use a vertex in the loop below
			const int index_max = index[max_d_index];

			// The number of finite widths should be >= 2.  If not, it should be 0, but in any case it implies three parallel lines in the plane, which we should not have here.
			// If we do have three parallel lines (# of finite widths < 2), we've picked the line corresponding to the half-plane farthest from r, which is correct.
			const int finite_width_count = (nrw_sgn0&1) + (nrw_sgn1&1) + (nrw_sgn2&1);
			if (finite_width_count >= 2)
			{
				const int i_remaining[2] = {(1<<max_d_index)&3, (3>>max_d_index)^1};	// = {(max_d_index+1)%3, (max_d_index+2)%3}
				const int i_select = (int)frac_gt(delta[i_remaining[1]], recip_n2[i_remaining[1]], delta[i_remaining[0]], recip_n2[i_remaining[0]]);	// Select the greater of the remaining dot products
				for (int i = 0; i < 2; ++i)
				{
					const int j = i_remaining[i_select^i];	// i = 0 => the next-greatest, i = 1 => the least
					if ((r|R[j]) >= 0)
					{
						r = perp(R[max_d_index], R[j]);
						r = (1/r.z)*r;	// We could just as well multiply r by sgn(r.z); we just need to ensure r.z > 0
						index[1] = index[j];
						N = 2;
						break;
					}
				}
			}

			index[0] = index_max;
		}
		break;
	}

	// Transform r back to 3D space
	p = vec4(r.x*x + r.y*y + (-r.z*h.w)*h.v, r.z);

	// Pack S array with kept planes
	if (N < 2 || index[1] != 0) { for (int i = 0; i < N; ++i) S[i] = S[index[i]]; }	// Safe to copy columns in order
	else { const Vec4 temp = S[0]; S[0] = S[index[0]]; S[1] = temp; }	// Otherwise use temp storage to avoid overwrite
	S[N] = h;
	plane_count = N+1;

	return true;
}


// Performs the VS algorithm for D = 3
inline int vs3d_test(VS3D_Halfspace_Set& halfspace_set, real* q = NULL)
{
	// Objective = q if it is not NULL, otherwise it is the origin represented in homogeneous coordinates
	const Vec4 objective = q ? (q[3] != 0 ? vec4((1/q[3])*vec3(q[0], q[1], q[2]), 1) : *(Vec4*)q) : vec4(vec3(0, 0, 0), 1);

	// Tolerance for 3D void simplex algorithm
	const real eps_f = (real)1/(sizeof(real) == 4 ? (1L<<23) : (1LL<<52));	// Floating-point epsilon
#if VS3D_HIGH_ACCURACY || REAL_DOUBLE
	const real eps = 8*eps_f;
#else
	const real eps = 80*eps_f;
#endif
	const real eps2 = eps*eps;	// Using epsilon squared

	// Maximum allowed iterations of main loop.  If exceeded, error code is returned
	const int max_iteration_count = 50;

	// State
	Vec4 S[4];				// Up to 4 planes
	int plane_count = 0;	// Number of valid planes
	Vec4 p = objective;		// Test point, initialized to objective

	// Default result, changed to valid result if found in loop below
	int result = -1;

	// Iterate until a stopping condition is met or the maximum number of iterations is reached
	for (int i = 0; result < 0 && i < max_iteration_count; ++i)
	{
		Vec4& plane = S[plane_count++];
		real delta = halfspace_set.farthest_halfspace(&plane.v.x, &p.v.x);
#if VS3D_UNNORMALIZED_PLANE_HANDLING != 0
		const real recip_norm = vs3d_recip_sqrt(plane.v|plane.v);
		plane = vec4(recip_norm*plane.v, recip_norm*plane.w);
		delta *= recip_norm;
#endif
		if (delta <= 0 || delta*delta <= eps2*(p|p)) result = 1;	// Intersection found
		else if (!vs3d_update(p, S, plane_count, objective, eps2)) result = 0;	// Void simplex found
	}

	// If q is given, fill it with the solution (normalize p.w if it is not zero)
	if (q) *(Vec4*)q = (p.w != 0) ? vec4((1/p.w)*p.v, 1) : p;

	PX_ASSERT(result >= 0);

	return result;
}


/*
	Utility class derived from GSA::ConvexShape, to handle common implementations

	PlaneIterator must have:
		1) a constructor which takes an object of type IteratorInitValues (either by value or refrence) in its constructor,
		2) a valid() method which returns a bool (true iff the plane() function can return a valid plane, see below),
		3) an inc() method to advance to the next plane, and
		4) a plane() method which returns a plane of type ApexCSG::Plane, either by value or reference (the plane will be copied).
*/
template<class PlaneIterator, class IteratorInitValues>
class StaticConvexPolyhedron : public VS3D_Halfspace_Set
{
public:
	virtual	GSA::real farthest_halfspace(GSA::real plane[4], const GSA::real point[4])
	{
		plane[0] = plane[1] = plane[2] = 0.0f;
		plane[3] = 1.0f;
		Real greatest_s = -MAX_REAL;

		for (PlaneIterator it(m_initValues); it.valid(); it.inc())
		{
			const Plane test = it.plane();
			const Real s = point[0]*test[0] + point[1]*test[1] + point[2]*test[2] + point[3]*test[3];
			if (s > greatest_s)
			{
				greatest_s = s;
				for (int i = 0; i < 4; ++i)
				{
					plane[i] = (GSA::real)test[i];
				}
			}
		}

		// Return results
		return (GSA::real)greatest_s;
	}

protected:
	IteratorInitValues	m_initValues;
};

};	// namespace GSA
};	// namespace ApexCSG

#endif	// #ifndef WITHOUT_APEX_AUTHORING

#endif // #ifndef APEX_GSA_H