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[vg.git] / vg_m.h

/* Copyright (C) 2021-2024 Harry Godden (hgn) - All Rights Reserved 
 *
 *  0. Misc
 *  1. Scalar operations
 *  2. Vectors
 *    2.a 2D Vectors
 *    2.b 3D Vectors
 *    2.c 4D Vectors
 *  3. Quaternions
 *  4. Matrices
 *    4.a 2x2 matrices
 *    4.b 3x3 matrices
 *    4.c 4x3 matrices
 *    4.d 4x4 matrices
 *  5. Geometry
 *    5.a Boxes
 *    5.b Planes
 *    5.c Closest points
 *    5.d Raycast & Spherecasts
 *    5.e Curves
 *    5.f Volumes
 *    5.g Inertia tensors
 *  6. Statistics
 *    6.a Random numbers
 */

#pragma once

#include "vg_platform.h"
#include <math.h>
#include <stdlib.h>

#define VG_PIf  3.14159265358979323846264338327950288f
#define VG_TAUf 6.28318530717958647692528676655900576f

/*
 * -----------------------------------------------------------------------------
 * Section 0.                    Misc Operations
 * -----------------------------------------------------------------------------
 */

/* get the f32 as the raw bits in a u32 without converting */
static u32 vg_ftu32( f32 a )
{
   u32 *ptr = (u32 *)(&a);
   return *ptr;
}

/* check if f32 is infinite */
static int vg_isinff( f32 a )
{
   return ((vg_ftu32(a)) & 0x7FFFFFFFU) == 0x7F800000U;
}

/* check if f32 is not a number */
static int vg_isnanf( f32 a )
{
   return !vg_isinff(a) && ((vg_ftu32(a)) & 0x7F800000U) == 0x7F800000U;
}

/* check if f32 is a number and is not infinite */
static int vg_validf( f32 a )
{
   return ((vg_ftu32(a)) & 0x7F800000U) != 0x7F800000U;
}

static int v3_valid( v3f a ){
   for( u32 i=0; i<3; i++ )
      if( !vg_validf(a[i]) ) return 0;
   return 1;
}

/*
 * -----------------------------------------------------------------------------
 * Section 1.                   Scalar Operations
 * -----------------------------------------------------------------------------
 */

static inline f32 vg_minf( f32 a, f32 b ){ return a < b? a: b; }
static inline f32 vg_maxf( f32 a, f32 b ){ return a > b? a: b; }

static inline int vg_min( int a, int b ){ return a < b? a: b; }
static inline int vg_max( int a, int b ){ return a > b? a: b; }

static inline f32 vg_clampf( f32 a, f32 min, f32 max )
{
   return vg_minf( max, vg_maxf( a, min ) );
}

static inline f32 vg_signf( f32 a )
{
   return a < 0.0f? -1.0f: 1.0f;
}

static inline f32 vg_fractf( f32 a )
{
   return a - floorf( a );
}

static inline f64 vg_fractf64( f64 a ){
   return a - floor( a );
}

static f32 vg_cfrictf( f32 velocity, f32 F )
{
   return -vg_signf(velocity) * vg_minf( F, fabsf(velocity) );
}

static inline f32 vg_rad( f32 deg )
{
   return deg * VG_PIf / 180.0f;
}

/* angle to reach b from a */
static f32 vg_angle_diff( f32 a, f32 b ){
   f32 d = fmod(b,VG_TAUf)-fmodf(a,VG_TAUf);
   if( fabsf(d) > VG_PIf )
      d = -vg_signf(d) * (VG_TAUf - fabsf(d));

   return d;
}

/*
 * quantize float to bit count
 */
static u32 vg_quantf( f32 a, u32 bits, f32 min, f32 max ){
   u32 mask = (0x1 << bits) - 1;
   return vg_clampf((a - min) * ((f32)mask/(max-min)), 0.0f, mask );
}

/*
 * un-quantize discreet to float
 */
static f32 vg_dequantf( u32 q, u32 bits, f32 min, f32 max ){
   u32 mask = (0x1 << bits) - 1;
   return min + (f32)q * ((max-min) / (f32)mask);
}

/* https://iquilezles.org/articles/functions/ 
 *
 * Use k to control the stretching of the function. Its maximum, which is 1, 
 * happens at exactly x = 1/k. 
 */
static f32 vg_exp_impulse( f32 x, f32 k ){
    f32 h = k*x;
    return h*expf(1.0f-h);
}

/*
 * -----------------------------------------------------------------------------
 * Section 2.a                   2D Vectors
 * -----------------------------------------------------------------------------
 */

static inline void v2_copy( v2f a, v2f d )
{
   d[0] = a[0]; d[1] = a[1];
}

static inline void v2_zero( v2f a )
{
   a[0] = 0.f; a[1] = 0.f;
}

static inline void v2_add( v2f a, v2f b, v2f d )
{
   d[0] = a[0]+b[0]; d[1] = a[1]+b[1];
}

static inline void v2_sub( v2f a, v2f b, v2f d )
{
   d[0] = a[0]-b[0]; d[1] = a[1]-b[1];
}

static inline void v2_minv( v2f a, v2f b, v2f dest ) 
{
   dest[0] = vg_minf(a[0], b[0]);
   dest[1] = vg_minf(a[1], b[1]);
}

static inline void v2_maxv( v2f a, v2f b, v2f dest ) 
{
   dest[0] = vg_maxf(a[0], b[0]);
   dest[1] = vg_maxf(a[1], b[1]);
}

static inline f32 v2_dot( v2f a, v2f b )
{
   return a[0] * b[0] + a[1] * b[1];
}

static inline f32 v2_cross( v2f a, v2f b )
{
   return a[0]*b[1] - a[1]*b[0];
}

static inline void v2_abs( v2f a, v2f d )
{
   d[0] = fabsf( a[0] );
   d[1] = fabsf( a[1] );
}

static inline void v2_muls( v2f a, f32 s, v2f d )
{
   d[0] = a[0]*s; d[1] = a[1]*s;
}

static inline void v2_divs( v2f a, f32 s, v2f d )
{
   d[0] = a[0]/s; d[1] = a[1]/s;
}

static inline void v2_mul( v2f a, v2f b, v2f d )
{
   d[0] = a[0]*b[0]; 
   d[1] = a[1]*b[1];
}

static inline void v2_div( v2f a, v2f b, v2f d )
{
   d[0] = a[0]/b[0]; d[1] = a[1]/b[1];
}

static inline void v2_muladd( v2f a, v2f b, v2f s, v2f d )
{
   d[0] = a[0]+b[0]*s[0]; 
   d[1] = a[1]+b[1]*s[1];
}

static inline void v2_muladds( v2f a, v2f b, f32 s, v2f d )
{
   d[0] = a[0]+b[0]*s; 
   d[1] = a[1]+b[1]*s;
}

static inline f32 v2_length2( v2f a )
{
   return a[0]*a[0] + a[1]*a[1];
}

static inline f32 v2_length( v2f a )
{
   return sqrtf( v2_length2( a ) );
}

static inline f32 v2_dist2( v2f a, v2f b )
{
   v2f delta;
   v2_sub( a, b, delta );
   return v2_length2( delta );
}

static inline f32 v2_dist( v2f a, v2f b )
{
   return sqrtf( v2_dist2( a, b ) );
}

static inline void v2_lerp( v2f a, v2f b, f32 t, v2f d )
{
   d[0] = a[0] + t*(b[0]-a[0]);
   d[1] = a[1] + t*(b[1]-a[1]);
}

static inline void v2_normalize( v2f a )
{
   v2_muls( a, 1.0f / v2_length( a ), a );
}

static void v2_normalize_clamp( v2f a )
{
   f32 l2 = v2_length2( a );
   if( l2 > 1.0f )
      v2_muls( a, 1.0f/sqrtf(l2), a );
}

static inline void v2_floor( v2f a, v2f b )
{
   b[0] = floorf( a[0] );
   b[1] = floorf( a[1] );
}

static inline void v2_fill( v2f a, f32 v )
{
   a[0] = v;
   a[1] = v;
}

static inline void v2_copysign( v2f a, v2f b )
{
   a[0] = copysignf( a[0], b[0] );
   a[1] = copysignf( a[1], b[1] );
}

/* integer variants 
 * ---------------- */

static inline void v2i_copy( v2i a, v2i b )
{
   b[0] = a[0]; b[1] = a[1];
}

static inline int v2i_eq( v2i a, v2i b )
{
   return ((a[0] == b[0]) && (a[1] == b[1]));
}

static inline void v2i_add( v2i a, v2i b, v2i d )
{
   d[0] = a[0]+b[0]; d[1] = a[1]+b[1];
}

static inline void v2i_sub( v2i a, v2i b, v2i d )
{
   d[0] = a[0]-b[0]; d[1] = a[1]-b[1];
}

/*
 * -----------------------------------------------------------------------------
 * Section 2.b                   3D Vectors
 * -----------------------------------------------------------------------------
 */

static inline void v3_copy( v3f a, v3f b )
{
   b[0] = a[0]; b[1] = a[1]; b[2] = a[2];
}

static inline void v3_zero( v3f a )
{
   a[0] = 0.f; a[1] = 0.f; a[2] = 0.f;
}

static inline void v3_add( v3f a, v3f b, v3f d )
{
   d[0] = a[0]+b[0]; d[1] = a[1]+b[1]; d[2] = a[2]+b[2];
}

static inline void v3i_add( v3i a, v3i b, v3i d )
{
   d[0] = a[0]+b[0]; d[1] = a[1]+b[1]; d[2] = a[2]+b[2];
}

static inline void v3_sub( v3f a, v3f b, v3f d )
{
   d[0] = a[0]-b[0]; d[1] = a[1]-b[1]; d[2] = a[2]-b[2];
}

static inline void v3i_sub( v3i a, v3i b, v3i d )
{
   d[0] = a[0]-b[0]; d[1] = a[1]-b[1]; d[2] = a[2]-b[2];
}

static inline void v3_mul( v3f a, v3f b, v3f d )
{
   d[0] = a[0]*b[0]; d[1] = a[1]*b[1]; d[2] = a[2]*b[2];
}

static inline void v3_div( v3f a, v3f b, v3f d )
{
   d[0] = b[0]!=0.0f? a[0]/b[0]: INFINITY;
   d[1] = b[1]!=0.0f? a[1]/b[1]: INFINITY;
   d[2] = b[2]!=0.0f? a[2]/b[2]: INFINITY;
}

static inline void v3_muls( v3f a, f32 s, v3f d )
{
   d[0] = a[0]*s; d[1] = a[1]*s; d[2] = a[2]*s;
}

static inline void v3_fill( v3f a, f32 v )
{
   a[0] = v;
   a[1] = v;
   a[2] = v;
}

static inline void v3_divs( v3f a, f32 s, v3f d )
{
   if( s == 0.0f )
      v3_fill( d, INFINITY );
   else
   {
      d[0] = a[0]/s; 
      d[1] = a[1]/s; 
      d[2] = a[2]/s;
   }
}

static inline void v3_muladds( v3f a, v3f b, f32 s, v3f d )
{
   d[0] = a[0]+b[0]*s; d[1] = a[1]+b[1]*s; d[2] = a[2]+b[2]*s;
}

static inline void v3_muladd( v2f a, v2f b, v2f s, v2f d )
{
   d[0] = a[0]+b[0]*s[0]; 
   d[1] = a[1]+b[1]*s[1];
   d[2] = a[2]+b[2]*s[2];
}

static inline f32 v3_dot( v3f a, v3f b )
{
   return a[0] * b[0] + a[1] * b[1] + a[2] * b[2];
}

static inline void v3_cross( v3f a, v3f b, v3f dest )
{
   v3f d;
   d[0] = a[1]*b[2] - a[2]*b[1];
   d[1] = a[2]*b[0] - a[0]*b[2];
   d[2] = a[0]*b[1] - a[1]*b[0];
   v3_copy( d, dest );
}

static inline f32 v3_length2( v3f a )
{
   return v3_dot( a, a );
}

static inline f32 v3_length( v3f a )
{
   return sqrtf( v3_length2( a ) );
}

static inline f32 v3_dist2( v3f a, v3f b )
{
   v3f delta;
   v3_sub( a, b, delta );
   return v3_length2( delta );
}

static inline f32 v3_dist( v3f a, v3f b )
{
   return sqrtf( v3_dist2( a, b ) );
}

static inline void v3_normalize( v3f a )
{
   v3_muls( a, 1.f / v3_length( a ), a );
}

static inline f32 vg_lerpf( f32 a, f32 b, f32 t ){
   return a + t*(b-a);
}

static inline f64 vg_lerp( f64 a, f64 b, f64 t )
{
   return a + t*(b-a);
}

static inline void vg_slewf( f32 *a, f32 b, f32 speed ){
   f32 d = vg_signf( b-*a ),
       c = *a + d*speed;
   *a = vg_minf( b*d, c*d ) * d;
}

static inline f32 vg_smoothstepf( f32 x ){
   return x*x*(3.0f - 2.0f*x);
}


/* correctly lerp around circular period -pi -> pi */
static f32 vg_alerpf( f32 a, f32 b, f32 t )
{
   f32 d = fmodf( b-a, VG_TAUf ),
         s = fmodf( 2.0f*d, VG_TAUf ) - d;
   return a + s*t;
}

static inline void v3_lerp( v3f a, v3f b, f32 t, v3f d )
{
   d[0] = a[0] + t*(b[0]-a[0]);
   d[1] = a[1] + t*(b[1]-a[1]);
   d[2] = a[2] + t*(b[2]-a[2]);
}

static inline void v3_minv( v3f a, v3f b, v3f dest ) 
{
   dest[0] = vg_minf(a[0], b[0]);
   dest[1] = vg_minf(a[1], b[1]);
   dest[2] = vg_minf(a[2], b[2]);
}

static inline void v3_maxv( v3f a, v3f b, v3f dest ) 
{
   dest[0] = vg_maxf(a[0], b[0]);
   dest[1] = vg_maxf(a[1], b[1]);
   dest[2] = vg_maxf(a[2], b[2]);
}

static inline f32 v3_minf( v3f a )
{
   return vg_minf( vg_minf( a[0], a[1] ), a[2] );
}

static inline f32 v3_maxf( v3f a )
{
   return vg_maxf( vg_maxf( a[0], a[1] ), a[2] );
}

static inline void v3_floor( v3f a, v3f b )
{
   b[0] = floorf( a[0] );
   b[1] = floorf( a[1] );
   b[2] = floorf( a[2] );
}

static inline void v3_ceil( v3f a, v3f b )
{
   b[0] = ceilf( a[0] );
   b[1] = ceilf( a[1] );
   b[2] = ceilf( a[2] );
}

static inline void v3_negate( v3f a, v3f b )
{
   b[0] = -a[0];
   b[1] = -a[1];
   b[2] = -a[2];
}

static inline void v3_rotate( v3f v, f32 angle, v3f axis, v3f d ) 
{
  v3f v1, v2, k;
  f32 c, s;

  c = cosf( angle );
  s = sinf( angle );

  v3_copy( axis, k );
  v3_normalize( k );
  v3_muls( v, c, v1 );
  v3_cross( k, v, v2 );
  v3_muls( v2, s, v2 );
  v3_add( v1, v2, v1 );
  v3_muls( k, v3_dot(k, v) * (1.0f - c), v2);
  v3_add( v1, v2, d );
}

static void v3_tangent_basis( v3f n, v3f tx, v3f ty ){
   /* Compute tangent basis (box2d) */
   if( fabsf( n[0] ) >= 0.57735027f ){
      tx[0] =  n[1];
      tx[1] = -n[0];
      tx[2] =  0.0f;
   }
   else{
      tx[0] =  0.0f;
      tx[1] =  n[2];
      tx[2] = -n[1];
   }

   v3_normalize( tx );
   v3_cross( n, tx, ty );
}

/*
 * Compute yaw and pitch based of a normalized vector representing forward
 * forward: -z
 * result -> (YAW,PITCH,0.0)
 */
static void v3_angles( v3f v, v3f out_angles ){
   float yaw = atan2f( v[0], -v[2] ),
       pitch = atan2f( 
                   -v[1], 
                   sqrtf(
                     v[0]*v[0] + v[2]*v[2]
                   )
               );

   out_angles[0] = yaw;
   out_angles[1] = pitch;
   out_angles[2] = 0.0f;
}

/*
 * Compute the forward vector from (YAW,PITCH,ROLL)
 * forward: -z
 */
static void v3_angles_vector( v3f angles, v3f out_v ){
   out_v[0] =  sinf( angles[0] ) * cosf( angles[1] );
   out_v[1] = -sinf( angles[1] );
   out_v[2] = -cosf( angles[0] ) * cosf( angles[1] );
}

/*
 * -----------------------------------------------------------------------------
 * Section 2.c                   4D Vectors
 * -----------------------------------------------------------------------------
 */

static inline void v4_copy( v4f a, v4f b )
{
   b[0] = a[0]; b[1] = a[1]; b[2] = a[2]; b[3] = a[3];
}

static inline void v4_add( v4f a, v4f b, v4f d )
{
   d[0] = a[0]+b[0]; 
   d[1] = a[1]+b[1];
   d[2] = a[2]+b[2];
   d[3] = a[3]+b[3];
}

static inline void v4_zero( v4f a )
{
   a[0] = 0.f; a[1] = 0.f; a[2] = 0.f; a[3] = 0.f;
}

static inline void v4_muls( v4f a, f32 s, v4f d )
{
   d[0] = a[0]*s; 
   d[1] = a[1]*s;
   d[2] = a[2]*s;
   d[3] = a[3]*s;
}

static inline void v4_muladds( v4f a, v4f b, f32 s, v4f d )
{
   d[0] = a[0]+b[0]*s; 
   d[1] = a[1]+b[1]*s;
   d[2] = a[2]+b[2]*s;
   d[3] = a[3]+b[3]*s;
}

static inline void v4_lerp( v4f a, v4f b, f32 t, v4f d )
{
   d[0] = a[0] + t*(b[0]-a[0]);
   d[1] = a[1] + t*(b[1]-a[1]);
   d[2] = a[2] + t*(b[2]-a[2]);
   d[3] = a[3] + t*(b[3]-a[3]);
}

static inline f32 v4_dot( v4f a, v4f b )
{
   return a[0]*b[0] + a[1]*b[1] + a[2]*b[2] + a[3]*b[3];
}

static inline f32 v4_length( v4f a )
{
   return sqrtf( v4_dot(a,a) );
}

/*
 * -----------------------------------------------------------------------------
 * Section 3                   Quaternions
 * -----------------------------------------------------------------------------
 */

static inline void q_identity( v4f q )
{
   q[0] = 0.0f; q[1] = 0.0f; q[2] = 0.0f; q[3] = 1.0f;
}

static inline void q_axis_angle( v4f q, v3f axis, f32 angle )
{
   f32 a = angle*0.5f,
         c = cosf(a),
         s = sinf(a);

   q[0] = s*axis[0];
   q[1] = s*axis[1];
   q[2] = s*axis[2];
   q[3] = c;
}

static inline void q_mul( v4f q, v4f q1, v4f d )
{
   v4f t;
   t[0] = q[3]*q1[0] + q[0]*q1[3] + q[1]*q1[2] - q[2]*q1[1];
   t[1] = q[3]*q1[1] - q[0]*q1[2] + q[1]*q1[3] + q[2]*q1[0];
   t[2] = q[3]*q1[2] + q[0]*q1[1] - q[1]*q1[0] + q[2]*q1[3];
   t[3] = q[3]*q1[3] - q[0]*q1[0] - q[1]*q1[1] - q[2]*q1[2];
   v4_copy( t, d );
}

static inline void q_normalize( v4f q )
{
   f32 l2 = v4_dot(q,q);
   if( l2 < 0.00001f ) q_identity( q );
   else {
      f32 s = 1.0f/sqrtf(l2);
      q[0] *= s;
      q[1] *= s;
      q[2] *= s;
      q[3] *= s;
   }
}

static inline void q_inv( v4f q, v4f d )
{
   f32 s = 1.0f / v4_dot(q,q);
   d[0] = -q[0]*s;
   d[1] = -q[1]*s;
   d[2] = -q[2]*s;
   d[3] =  q[3]*s;
}

static inline void q_nlerp( v4f a, v4f b, f32 t, v4f d ){
   if( v4_dot(a,b) < 0.0f ){
      v4f c;
      v4_muls( b, -1.0f, c );
      v4_lerp( a, c, t, d );
   }
   else
      v4_lerp( a, b, t, d );

   q_normalize( d );
}

static inline void q_m3x3( v4f q, m3x3f d )
{
   f32
      l = v4_length(q),
      s = l > 0.0f? 2.0f/l: 0.0f,

      xx = s*q[0]*q[0], xy = s*q[0]*q[1], wx = s*q[3]*q[0],
      yy = s*q[1]*q[1], yz = s*q[1]*q[2], wy = s*q[3]*q[1],
      zz = s*q[2]*q[2], xz = s*q[0]*q[2], wz = s*q[3]*q[2];

   d[0][0] = 1.0f - yy - zz;
   d[1][1] = 1.0f - xx - zz;
   d[2][2] = 1.0f - xx - yy;
   d[0][1] = xy + wz;
   d[1][2] = yz + wx;
   d[2][0] = xz + wy;
   d[1][0] = xy - wz;
   d[2][1] = yz - wx;
   d[0][2] = xz - wy;
}

static void q_mulv( v4f q, v3f v, v3f d )
{
   v3f v1, v2;

   v3_muls( q, 2.0f*v3_dot(q,v), v1 );
   v3_muls( v, q[3]*q[3] - v3_dot(q,q), v2 );
   v3_add( v1, v2, v1 );
   v3_cross( q, v, v2 );
   v3_muls( v2, 2.0f*q[3], v2 );
   v3_add( v1, v2, d );
}

static f32 q_dist( v4f q0, v4f q1 ){
   return acosf( 2.0f * v4_dot(q0,q1) -1.0f );
}

/*
 * -----------------------------------------------------------------------------
 * Section 4.a                  2x2 matrices
 * -----------------------------------------------------------------------------
 */

#define M2X2_INDENTIY {{1.0f, 0.0f, }, \
                       {0.0f, 1.0f, }}
                          
#define M2X2_ZERO     {{0.0f, 0.0f, }, \
                       {0.0f, 0.0f, }}

static inline void m2x2_copy( m2x2f a, m2x2f b )
{
   v2_copy( a[0], b[0] );
   v2_copy( a[1], b[1] );
}

static inline void m2x2_identity( m2x2f a )
{
   m2x2f id = M2X2_INDENTIY;
   m2x2_copy( id, a );
}

static inline void m2x2_create_rotation( m2x2f a, f32 theta )
{
   f32 s, c;
   
   s = sinf( theta );
   c = cosf( theta );
   
   a[0][0] =  c;
   a[0][1] = -s;
   a[1][0] =  s;
   a[1][1] =  c;
}

static inline void m2x2_mulv( m2x2f m, v2f v, v2f d )
{
   v2f res;
   
   res[0] = m[0][0]*v[0] + m[1][0]*v[1];
   res[1] = m[0][1]*v[0] + m[1][1]*v[1];
   
   v2_copy( res, d );
}

/*
 * -----------------------------------------------------------------------------
 * Section 4.b                  3x3 matrices
 * -----------------------------------------------------------------------------
 */

#define M3X3_IDENTITY   {{1.0f, 0.0f, 0.0f, },\
                        { 0.0f, 1.0f, 0.0f, },\
                        { 0.0f, 0.0f, 1.0f, }}
                        
#define M3X3_ZERO       {{0.0f, 0.0f, 0.0f, },\
                        { 0.0f, 0.0f, 0.0f, },\
                        { 0.0f, 0.0f, 0.0f, }}


static void euler_m3x3( v3f angles, m3x3f d )
{
   f32 cosY = cosf( angles[0] ),
       sinY = sinf( angles[0] ),
       cosP = cosf( angles[1] ),
       sinP = sinf( angles[1] ),
       cosR = cosf( angles[2] ),
       sinR = sinf( angles[2] );

   d[2][0] = -sinY * cosP;
   d[2][1] =  sinP;
   d[2][2] =  cosY * cosP;

   d[0][0] =  cosY * cosR;
   d[0][1] =  sinR;
   d[0][2] =  sinY * cosR;

   v3_cross( d[0], d[2], d[1] );
}

static void m3x3_q( m3x3f m, v4f q )
{
   f32 diag, r, rinv;

   diag = m[0][0] + m[1][1] + m[2][2];
   if( diag >= 0.0f )
   {
      r    = sqrtf( 1.0f + diag );
      rinv = 0.5f / r;
      q[0] = rinv * (m[1][2] - m[2][1]);
      q[1] = rinv * (m[2][0] - m[0][2]);
      q[2] = rinv * (m[0][1] - m[1][0]);
      q[3] = r    * 0.5f;
   } 
   else if( m[0][0] >= m[1][1] && m[0][0] >= m[2][2] )
   {
      r    = sqrtf( 1.0f - m[1][1] - m[2][2] + m[0][0] );
      rinv = 0.5f / r;
      q[0] = r    * 0.5f;
      q[1] = rinv * (m[0][1] + m[1][0]);
      q[2] = rinv * (m[0][2] + m[2][0]);
      q[3] = rinv * (m[1][2] - m[2][1]);
   } 
   else if( m[1][1] >= m[2][2] )
   {
      r    = sqrtf( 1.0f - m[0][0] - m[2][2] + m[1][1] );
      rinv = 0.5f / r;
      q[0] = rinv * (m[0][1] + m[1][0]);
      q[1] = r    * 0.5f;
      q[2] = rinv * (m[1][2] + m[2][1]);
      q[3] = rinv * (m[2][0] - m[0][2]);
   } 
   else 
   {
      r    = sqrtf( 1.0f - m[0][0] - m[1][1] + m[2][2] );
      rinv = 0.5f / r;
      q[0] = rinv * (m[0][2] + m[2][0]);
      q[1] = rinv * (m[1][2] + m[2][1]);
      q[2] = r    * 0.5f;
      q[3] = rinv * (m[0][1] - m[1][0]);
   }
}

/* a X b == [b]T a == ...*/
static void m3x3_skew_symetric( m3x3f a, v3f v )
{
   a[0][0] =  0.0f;
   a[0][1] =  v[2];
   a[0][2] = -v[1];
   a[1][0] = -v[2];
   a[1][1] =  0.0f;
   a[1][2] =  v[0];
   a[2][0] =  v[1];
   a[2][1] = -v[0];
   a[2][2] =  0.0f;
}

/* aka kronecker product */
static void m3x3_outer_product( m3x3f out_m, v3f a, v3f b )
{
   out_m[0][0] = a[0]*b[0];
   out_m[0][1] = a[0]*b[1];
   out_m[0][2] = a[0]*b[2];
   out_m[1][0] = a[1]*b[0];
   out_m[1][1] = a[1]*b[1];
   out_m[1][2] = a[1]*b[2];
   out_m[2][0] = a[2]*b[0];
   out_m[2][1] = a[2]*b[1];
   out_m[2][2] = a[2]*b[2];
}

static void m3x3_add( m3x3f a, m3x3f b, m3x3f d )
{
   v3_add( a[0], b[0], d[0] );
   v3_add( a[1], b[1], d[1] );
   v3_add( a[2], b[2], d[2] );
}

static void m3x3_sub( m3x3f a, m3x3f b, m3x3f d )
{
   v3_sub( a[0], b[0], d[0] );
   v3_sub( a[1], b[1], d[1] );
   v3_sub( a[2], b[2], d[2] );
}

static inline void m3x3_copy( m3x3f a, m3x3f b )
{
   v3_copy( a[0], b[0] );
   v3_copy( a[1], b[1] );
   v3_copy( a[2], b[2] );
}

static inline void m3x3_identity( m3x3f a )
{
   m3x3f id = M3X3_IDENTITY;
   m3x3_copy( id, a );
}

static void m3x3_diagonal( m3x3f out_a, f32 v )
{
   m3x3_identity( out_a );
   out_a[0][0] = v;
   out_a[1][1] = v;
   out_a[2][2] = v;
}

static void m3x3_setdiagonalv3( m3x3f a, v3f v )
{
   a[0][0] = v[0];
   a[1][1] = v[1];
   a[2][2] = v[2];
}

static inline void m3x3_zero( m3x3f a )
{
   m3x3f z = M3X3_ZERO;
   m3x3_copy( z, a );
}

static inline void m3x3_inv( m3x3f src, m3x3f dest )
{
   f32 a = src[0][0], b = src[0][1], c = src[0][2],
         d = src[1][0], e = src[1][1], f = src[1][2],
         g = src[2][0], h = src[2][1], i = src[2][2];

   f32 det =    1.f / 
               (+a*(e*i-h*f)
                -b*(d*i-f*g)
                +c*(d*h-e*g));
                
   dest[0][0] =  (e*i-h*f)*det;
   dest[0][1] = -(b*i-c*h)*det;
   dest[0][2] =  (b*f-c*e)*det;
   dest[1][0] = -(d*i-f*g)*det;
   dest[1][1] =  (a*i-c*g)*det;
   dest[1][2] = -(a*f-d*c)*det;
   dest[2][0] =  (d*h-g*e)*det;
   dest[2][1] = -(a*h-g*b)*det;
   dest[2][2] =  (a*e-d*b)*det;
}

static f32 m3x3_det( m3x3f m )
{
   return   m[0][0] * (m[1][1] * m[2][2] - m[2][1] * m[1][2])
          - m[0][1] * (m[1][0] * m[2][2] - m[1][2] * m[2][0])
          + m[0][2] * (m[1][0] * m[2][1] - m[1][1] * m[2][0]);
}

static inline void m3x3_transpose( m3x3f src, m3x3f dest )
{
   f32 a = src[0][0], b = src[0][1], c = src[0][2],
         d = src[1][0], e = src[1][1], f = src[1][2],
         g = src[2][0], h = src[2][1], i = src[2][2];
         
   dest[0][0] = a;
   dest[0][1] = d;
   dest[0][2] = g;
   dest[1][0] = b;
   dest[1][1] = e;
   dest[1][2] = h;
   dest[2][0] = c;
   dest[2][1] = f;
   dest[2][2] = i;
}

static inline void m3x3_mul( m3x3f a, m3x3f b, m3x3f d )
{
   f32 a00 = a[0][0], a01 = a[0][1], a02 = a[0][2],
         a10 = a[1][0], a11 = a[1][1], a12 = a[1][2],
         a20 = a[2][0], a21 = a[2][1], a22 = a[2][2],

         b00 = b[0][0], b01 = b[0][1], b02 = b[0][2],
         b10 = b[1][0], b11 = b[1][1], b12 = b[1][2],
         b20 = b[2][0], b21 = b[2][1], b22 = b[2][2];

   d[0][0] = a00*b00 + a10*b01 + a20*b02;
   d[0][1] = a01*b00 + a11*b01 + a21*b02;
   d[0][2] = a02*b00 + a12*b01 + a22*b02;
   d[1][0] = a00*b10 + a10*b11 + a20*b12;
   d[1][1] = a01*b10 + a11*b11 + a21*b12;
   d[1][2] = a02*b10 + a12*b11 + a22*b12;
   d[2][0] = a00*b20 + a10*b21 + a20*b22;
   d[2][1] = a01*b20 + a11*b21 + a21*b22;
   d[2][2] = a02*b20 + a12*b21 + a22*b22;
}

static inline void m3x3_mulv( m3x3f m, v3f v, v3f d )
{
   v3f res;
   
   res[0] = m[0][0]*v[0] + m[1][0]*v[1] + m[2][0]*v[2];
   res[1] = m[0][1]*v[0] + m[1][1]*v[1] + m[2][1]*v[2];
   res[2] = m[0][2]*v[0] + m[1][2]*v[1] + m[2][2]*v[2];
   
   v3_copy( res, d );
}

static inline void m3x3_projection( m3x3f dst, 
      f32 const left, f32 const right, f32 const bottom, f32 const top )
{
   f32 rl, tb;
   
   m3x3_zero( dst );

   rl = 1.0f / (right - left);
   tb = 1.0f / (top   - bottom);

   dst[0][0] = 2.0f * rl;
   dst[1][1] = 2.0f * tb;
   dst[2][2] = 1.0f;
}

static inline void m3x3_translate( m3x3f m, v3f v )
{
   m[2][0] = m[0][0] * v[0] + m[1][0] * v[1] + m[2][0];
   m[2][1] = m[0][1] * v[0] + m[1][1] * v[1] + m[2][1];
   m[2][2] = m[0][2] * v[0] + m[1][2] * v[1] + m[2][2];
}

static inline void m3x3_scale( m3x3f m, v3f v )
{
   v3_muls( m[0], v[0], m[0] );
   v3_muls( m[1], v[1], m[1] );
   v3_muls( m[2], v[2], m[2] );
}

static inline void m3x3_scalef( m3x3f m, f32 f )
{
   v3f v;
   v3_fill( v, f );
   m3x3_scale( m, v );
}

static inline void m3x3_rotate( m3x3f m, f32 angle )
{
   f32 m00 = m[0][0], m10 = m[1][0],
         m01 = m[0][1], m11 = m[1][1],
         m02 = m[0][2], m12 = m[1][2];
   f32 c, s;

   s = sinf( angle );
   c = cosf( angle );

   m[0][0] = m00 * c + m10 * s;
   m[0][1] = m01 * c + m11 * s;
   m[0][2] = m02 * c + m12 * s;

   m[1][0] = m00 * -s + m10 * c;
   m[1][1] = m01 * -s + m11 * c;
   m[1][2] = m02 * -s + m12 * c;
}

/*
 * -----------------------------------------------------------------------------
 * Section 4.c                  4x3 matrices
 * -----------------------------------------------------------------------------
 */

#define M4X3_IDENTITY   {{1.0f, 0.0f, 0.0f, },\
                        { 0.0f, 1.0f, 0.0f, },\
                        { 0.0f, 0.0f, 1.0f, },\
                        { 0.0f, 0.0f, 0.0f }}

static inline void m4x3_to_3x3( m4x3f a, m3x3f b )
{
   v3_copy( a[0], b[0] );
   v3_copy( a[1], b[1] );
   v3_copy( a[2], b[2] );
}

static inline void m4x3_invert_affine( m4x3f a, m4x3f b )
{
   m3x3_transpose( a, b );
   m3x3_mulv( b, a[3], b[3] );
   v3_negate( b[3], b[3] );
}

static void m4x3_invert_full( m4x3f src, m4x3f dst )
{
  f32 t2, t4, t5,
        det,
        a = src[0][0], b = src[0][1], c = src[0][2],
        e = src[1][0], f = src[1][1], g = src[1][2],
        i = src[2][0], j = src[2][1], k = src[2][2],
        m = src[3][0], n = src[3][1], o = src[3][2];

   t2 = j*o - n*k;
   t4 = i*o - m*k;
   t5 = i*n - m*j;
   
   dst[0][0] =  f*k - g*j;
   dst[1][0] =-(e*k - g*i);
   dst[2][0] =  e*j - f*i;
   dst[3][0] =-(e*t2 - f*t4 + g*t5);
   
   dst[0][1] =-(b*k - c*j);
   dst[1][1] =  a*k - c*i;
   dst[2][1] =-(a*j - b*i);
   dst[3][1] =  a*t2 - b*t4 + c*t5;
   
   t2 = f*o - n*g;
   t4 = e*o - m*g; 
   t5 = e*n - m*f;
   
   dst[0][2] =  b*g - c*f ;
   dst[1][2] =-(a*g - c*e );
   dst[2][2] =  a*f - b*e ;
   dst[3][2] =-(a*t2 - b*t4 + c * t5);

   det = 1.0f / (a * dst[0][0] + b * dst[1][0] + c * dst[2][0]);
   v3_muls( dst[0], det, dst[0] );
   v3_muls( dst[1], det, dst[1] );
   v3_muls( dst[2], det, dst[2] );
   v3_muls( dst[3], det, dst[3] );
}

static inline void m4x3_copy( m4x3f a, m4x3f b )
{
   v3_copy( a[0], b[0] );
   v3_copy( a[1], b[1] );
   v3_copy( a[2], b[2] );
   v3_copy( a[3], b[3] );
}

static inline void m4x3_identity( m4x3f a )
{
   m4x3f id = M4X3_IDENTITY;
   m4x3_copy( id, a );
}

static void m4x3_mul( m4x3f a, m4x3f b, m4x3f d ) 
{
   f32 
   a00 = a[0][0], a01 = a[0][1], a02 = a[0][2],
   a10 = a[1][0], a11 = a[1][1], a12 = a[1][2],
   a20 = a[2][0], a21 = a[2][1], a22 = a[2][2],
   a30 = a[3][0], a31 = a[3][1], a32 = a[3][2],
   b00 = b[0][0], b01 = b[0][1], b02 = b[0][2],
   b10 = b[1][0], b11 = b[1][1], b12 = b[1][2],
   b20 = b[2][0], b21 = b[2][1], b22 = b[2][2],
   b30 = b[3][0], b31 = b[3][1], b32 = b[3][2];
   
   d[0][0] = a00*b00 + a10*b01 + a20*b02;
   d[0][1] = a01*b00 + a11*b01 + a21*b02;
   d[0][2] = a02*b00 + a12*b01 + a22*b02;
   d[1][0] = a00*b10 + a10*b11 + a20*b12;
   d[1][1] = a01*b10 + a11*b11 + a21*b12;
   d[1][2] = a02*b10 + a12*b11 + a22*b12;
   d[2][0] = a00*b20 + a10*b21 + a20*b22;
   d[2][1] = a01*b20 + a11*b21 + a21*b22;
   d[2][2] = a02*b20 + a12*b21 + a22*b22;
   d[3][0] = a00*b30 + a10*b31 + a20*b32 + a30;
   d[3][1] = a01*b30 + a11*b31 + a21*b32 + a31;
   d[3][2] = a02*b30 + a12*b31 + a22*b32 + a32;
}

#if 0 /* shat appf mingw wstringop-overflow */
inline
#endif
static void m4x3_mulv( m4x3f m, v3f v, v3f d ) 
{
   v3f res;
  
   res[0] = m[0][0]*v[0] + m[1][0]*v[1] + m[2][0]*v[2] + m[3][0];
   res[1] = m[0][1]*v[0] + m[1][1]*v[1] + m[2][1]*v[2] + m[3][1];
   res[2] = m[0][2]*v[0] + m[1][2]*v[1] + m[2][2]*v[2] + m[3][2];

   v3_copy( res, d );
}

/* 
 * Transform plane ( xyz, distance )
 */
static void m4x3_mulp( m4x3f m, v4f p, v4f d )
{
   v3f o;

   v3_muls( p, p[3], o );
   m4x3_mulv( m, o, o );
   m3x3_mulv( m, p, d );
   
   d[3] = v3_dot( o, d );
}

/*
 * Affine transforms
 */

static void m4x3_translate( m4x3f m, v3f v )
{
   v3_muladds( m[3], m[0], v[0], m[3] );
   v3_muladds( m[3], m[1], v[1], m[3] );
   v3_muladds( m[3], m[2], v[2], m[3] );
}

static void m4x3_rotate_x( m4x3f m, f32 angle )
{
   m4x3f t = M4X3_IDENTITY;
   f32 c, s;

   c = cosf( angle );
   s = sinf( angle );

   t[1][1] =  c;
   t[1][2] =  s;
   t[2][1] = -s;
   t[2][2] =  c;

   m4x3_mul( m, t, m );
}

static void m4x3_rotate_y( m4x3f m, f32 angle )
{
   m4x3f t = M4X3_IDENTITY;
   f32 c, s;

   c = cosf( angle );
   s = sinf( angle );

   t[0][0] =  c;
   t[0][2] = -s;
   t[2][0] =  s;
   t[2][2] =  c;

   m4x3_mul( m, t, m );
}

static void m4x3_rotate_z( m4x3f m, f32 angle )
{
   m4x3f t = M4X3_IDENTITY;
   f32 c, s;

   c = cosf( angle );
   s = sinf( angle );

   t[0][0] =  c;
   t[0][1] =  s;
   t[1][0] = -s;
   t[1][1] =  c;

   m4x3_mul( m, t, m );
}

static void m4x3_expand( m4x3f m, m4x4f d )
{
   v3_copy( m[0], d[0] );
   v3_copy( m[1], d[1] );
   v3_copy( m[2], d[2] );
   v3_copy( m[3], d[3] );
   d[0][3] = 0.0f;
   d[1][3] = 0.0f;
   d[2][3] = 0.0f;
   d[3][3] = 1.0f;
}

static void m4x3_decompose( m4x3f m, v3f co, v4f q, v3f s )
{
   v3_copy( m[3], co );
   s[0] = v3_length(m[0]);
   s[1] = v3_length(m[1]);
   s[2] = v3_length(m[2]);

   m3x3f rot;
   v3_divs( m[0], s[0], rot[0] );
   v3_divs( m[1], s[1], rot[1] );
   v3_divs( m[2], s[2], rot[2] );

   m3x3_q( rot, q );
}

static void m4x3_expand_aabb_point( m4x3f m, boxf box, v3f point ){
   v3f v;
   m4x3_mulv( m, point, v );

   v3_minv( box[0], v, box[0] );
   v3_maxv( box[1], v, box[1] );
}

static void m4x3_expand_aabb_aabb( m4x3f m, boxf boxa, boxf boxb ){
   v3f a; v3f b;
   v3_copy( boxb[0], a );
   v3_copy( boxb[1], b );
   m4x3_expand_aabb_point( m, boxa, (v3f){ a[0], a[1], a[2] } );
   m4x3_expand_aabb_point( m, boxa, (v3f){ a[0], b[1], a[2] } );
   m4x3_expand_aabb_point( m, boxa, (v3f){ b[0], b[1], a[2] } );
   m4x3_expand_aabb_point( m, boxa, (v3f){ b[0], a[1], a[2] } );
   m4x3_expand_aabb_point( m, boxa, (v3f){ a[0], a[1], b[2] } );
   m4x3_expand_aabb_point( m, boxa, (v3f){ a[0], b[1], b[2] } );
   m4x3_expand_aabb_point( m, boxa, (v3f){ b[0], b[1], b[2] } );
   m4x3_expand_aabb_point( m, boxa, (v3f){ b[0], a[1], b[2] } );
}
static inline void m4x3_lookat( m4x3f m, v3f pos, v3f target, v3f up )
{
   v3f dir;
   v3_sub( target, pos, dir );
   v3_normalize( dir );

   v3_copy( dir, m[2] );

   v3_cross( up, m[2], m[0] );
   v3_normalize( m[0] );

   v3_cross( m[2], m[0], m[1] );
   v3_copy( pos, m[3] );
}

/*
 * -----------------------------------------------------------------------------
 * Section 4.d                  4x4 matrices
 * -----------------------------------------------------------------------------
 */

#define M4X4_IDENTITY   {{1.0f, 0.0f, 0.0f, 0.0f },\
                        { 0.0f, 1.0f, 0.0f, 0.0f },\
                        { 0.0f, 0.0f, 1.0f, 0.0f },\
                        { 0.0f, 0.0f, 0.0f, 1.0f }}
#define M4X4_ZERO       {{0.0f, 0.0f, 0.0f, 0.0f },\
                        { 0.0f, 0.0f, 0.0f, 0.0f },\
                        { 0.0f, 0.0f, 0.0f, 0.0f },\
                        { 0.0f, 0.0f, 0.0f, 0.0f }}

static void m4x4_projection( m4x4f m, f32 angle,
                             f32 ratio, f32 fnear, f32 ffar )
{
   f32 scale = tanf( angle * 0.5f * VG_PIf / 180.0f ) * fnear,
         r = ratio * scale,
         l = -r,
         t = scale,
         b = -t;

   m[0][0] =  2.0f * fnear / (r - l);
   m[0][1] =  0.0f;
   m[0][2] =  0.0f;
   m[0][3] =  0.0f;

   m[1][0] =  0.0f;
   m[1][1] =  2.0f * fnear / (t - b);
   m[1][2] =  0.0f;
   m[1][3] =  0.0f;

   m[2][0] =  (r + l) / (r - l);
   m[2][1] =  (t + b) / (t - b);
   m[2][2] = -(ffar + fnear) / (ffar - fnear);
   m[2][3] = -1.0f;

   m[3][0] =  0.0f;
   m[3][1] =  0.0f;
   m[3][2] = -2.0f * ffar * fnear / (ffar - fnear);
   m[3][3] =  0.0f;
} 

static void m4x4_translate( m4x4f m, v3f v )
{
   v4_muladds( m[3], m[0], v[0], m[3] );
   v4_muladds( m[3], m[1], v[1], m[3] );
   v4_muladds( m[3], m[2], v[2], m[3] );
}

static inline void m4x4_copy( m4x4f a, m4x4f b )
{
   v4_copy( a[0], b[0] );
   v4_copy( a[1], b[1] );
   v4_copy( a[2], b[2] );
   v4_copy( a[3], b[3] );
}

static inline void m4x4_identity( m4x4f a )
{
   m4x4f id = M4X4_IDENTITY;
   m4x4_copy( id, a );
}

static inline void m4x4_zero( m4x4f a )
{
   m4x4f zero = M4X4_ZERO;
   m4x4_copy( zero, a );
}

static inline void m4x4_mul( m4x4f a, m4x4f b, m4x4f d )
{
   f32 a00 = a[0][0], a01 = a[0][1], a02 = a[0][2], a03 = a[0][3],
         a10 = a[1][0], a11 = a[1][1], a12 = a[1][2], a13 = a[1][3],
         a20 = a[2][0], a21 = a[2][1], a22 = a[2][2], a23 = a[2][3],
         a30 = a[3][0], a31 = a[3][1], a32 = a[3][2], a33 = a[3][3],

         b00 = b[0][0], b01 = b[0][1], b02 = b[0][2], b03 = b[0][3],
         b10 = b[1][0], b11 = b[1][1], b12 = b[1][2], b13 = b[1][3],
         b20 = b[2][0], b21 = b[2][1], b22 = b[2][2], b23 = b[2][3],
         b30 = b[3][0], b31 = b[3][1], b32 = b[3][2], b33 = b[3][3];

  d[0][0] = a00*b00 + a10*b01 + a20*b02 + a30*b03;
  d[0][1] = a01*b00 + a11*b01 + a21*b02 + a31*b03;
  d[0][2] = a02*b00 + a12*b01 + a22*b02 + a32*b03;
  d[0][3] = a03*b00 + a13*b01 + a23*b02 + a33*b03;
  d[1][0] = a00*b10 + a10*b11 + a20*b12 + a30*b13;
  d[1][1] = a01*b10 + a11*b11 + a21*b12 + a31*b13;
  d[1][2] = a02*b10 + a12*b11 + a22*b12 + a32*b13;
  d[1][3] = a03*b10 + a13*b11 + a23*b12 + a33*b13;
  d[2][0] = a00*b20 + a10*b21 + a20*b22 + a30*b23;
  d[2][1] = a01*b20 + a11*b21 + a21*b22 + a31*b23;
  d[2][2] = a02*b20 + a12*b21 + a22*b22 + a32*b23;
  d[2][3] = a03*b20 + a13*b21 + a23*b22 + a33*b23;
  d[3][0] = a00*b30 + a10*b31 + a20*b32 + a30*b33;
  d[3][1] = a01*b30 + a11*b31 + a21*b32 + a31*b33;
  d[3][2] = a02*b30 + a12*b31 + a22*b32 + a32*b33;
  d[3][3] = a03*b30 + a13*b31 + a23*b32 + a33*b33;
}

static inline void m4x4_mulv( m4x4f m, v4f v, v4f d ) 
{
   v4f res;
  
   res[0] = m[0][0]*v[0] + m[1][0]*v[1] + m[2][0]*v[2] + m[3][0]*v[3];
   res[1] = m[0][1]*v[0] + m[1][1]*v[1] + m[2][1]*v[2] + m[3][1]*v[3];
   res[2] = m[0][2]*v[0] + m[1][2]*v[1] + m[2][2]*v[2] + m[3][2]*v[3];
   res[3] = m[0][3]*v[0] + m[1][3]*v[1] + m[2][3]*v[2] + m[3][3]*v[3];

   v4_copy( res, d );
}

static inline void m4x4_inv( m4x4f a, m4x4f d )
{
   f32 a00 = a[0][0], a01 = a[0][1], a02 = a[0][2], a03 = a[0][3],
         a10 = a[1][0], a11 = a[1][1], a12 = a[1][2], a13 = a[1][3],
         a20 = a[2][0], a21 = a[2][1], a22 = a[2][2], a23 = a[2][3],
         a30 = a[3][0], a31 = a[3][1], a32 = a[3][2], a33 = a[3][3],
         det,
         t[6];

   t[0] = a22*a33 - a32*a23;
   t[1] = a21*a33 - a31*a23;
   t[2] = a21*a32 - a31*a22;
   t[3] = a20*a33 - a30*a23;
   t[4] = a20*a32 - a30*a22;
   t[5] = a20*a31 - a30*a21;

   d[0][0] =  a11*t[0] - a12*t[1] + a13*t[2];
   d[1][0] =-(a10*t[0] - a12*t[3] + a13*t[4]);
   d[2][0] =  a10*t[1] - a11*t[3] + a13*t[5];
   d[3][0] =-(a10*t[2] - a11*t[4] + a12*t[5]);

   d[0][1] =-(a01*t[0] - a02*t[1] + a03*t[2]);
   d[1][1] =  a00*t[0] - a02*t[3] + a03*t[4];
   d[2][1] =-(a00*t[1] - a01*t[3] + a03*t[5]);
   d[3][1] =  a00*t[2] - a01*t[4] + a02*t[5];

   t[0] = a12*a33 - a32*a13;
   t[1] = a11*a33 - a31*a13;
   t[2] = a11*a32 - a31*a12;
   t[3] = a10*a33 - a30*a13;
   t[4] = a10*a32 - a30*a12;
   t[5] = a10*a31 - a30*a11;

   d[0][2] =  a01*t[0] - a02*t[1] + a03*t[2];
   d[1][2] =-(a00*t[0] - a02*t[3] + a03*t[4]);
   d[2][2] =  a00*t[1] - a01*t[3] + a03*t[5];
   d[3][2] =-(a00*t[2] - a01*t[4] + a02*t[5]);

   t[0] = a12*a23 - a22*a13;
   t[1] = a11*a23 - a21*a13;
   t[2] = a11*a22 - a21*a12;
   t[3] = a10*a23 - a20*a13;
   t[4] = a10*a22 - a20*a12;
   t[5] = a10*a21 - a20*a11;

   d[0][3] =-(a01*t[0] - a02*t[1] + a03*t[2]);
   d[1][3] =  a00*t[0] - a02*t[3] + a03*t[4];
   d[2][3] =-(a00*t[1] - a01*t[3] + a03*t[5]);
   d[3][3] =  a00*t[2] - a01*t[4] + a02*t[5];
   
   det = 1.0f / (a00*d[0][0] + a01*d[1][0] + a02*d[2][0] + a03*d[3][0]);
   v4_muls( d[0], det, d[0] );
   v4_muls( d[1], det, d[1] );
   v4_muls( d[2], det, d[2] );
   v4_muls( d[3], det, d[3] );
}

/* 
 * http://www.terathon.com/lengyel/Lengyel-Oblique.pdf 
 */
static void m4x4_clip_projection( m4x4f mat, v4f plane ){
   v4f c = 
   {
      (vg_signf(plane[0]) + mat[2][0]) / mat[0][0],
      (vg_signf(plane[1]) + mat[2][1]) / mat[1][1],
      -1.0f,
      (1.0f + mat[2][2]) / mat[3][2]
   };

   v4_muls( plane, 2.0f / v4_dot(plane,c), c );

   mat[0][2] = c[0];
   mat[1][2] = c[1];
   mat[2][2] = c[2] + 1.0f;
   mat[3][2] = c[3];
}

/*
 * Undoes the above operation
 */
static void m4x4_reset_clipping( m4x4f mat, float ffar, float fnear ){
   mat[0][2] = 0.0f; 
   mat[1][2] = 0.0f; 
   mat[2][2] = -(ffar + fnear) / (ffar - fnear); 
   mat[3][2] = -2.0f * ffar * fnear / (ffar - fnear); 
}

/*
 * -----------------------------------------------------------------------------
 * Section 5.a                       Boxes
 * -----------------------------------------------------------------------------
 */

static inline void box_addpt( boxf a, v3f pt )
{
   v3_minv( a[0], pt, a[0] );
   v3_maxv( a[1], pt, a[1] );
}

static inline void box_concat( boxf a, boxf b )
{
   v3_minv( a[0], b[0], a[0] );
   v3_maxv( a[1], b[1], a[1] );
}

static inline void box_copy( boxf a, boxf b )
{
   v3_copy( a[0], b[0] );
   v3_copy( a[1], b[1] );
}

static inline int box_overlap( boxf a, boxf b )
{
   return
   ( a[0][0] <= b[1][0] && a[1][0] >= b[0][0] ) &&
   ( a[0][1] <= b[1][1] && a[1][1] >= b[0][1] ) &&
   ( a[0][2] <= b[1][2] && a[1][2] >= b[0][2] )
   ;
}

static int box_within( boxf greater, boxf lesser )
{
   v3f a, b;
   v3_sub( lesser[0], greater[0], a );
   v3_sub( lesser[1], greater[1], b );

   if( (a[0] >= 0.0f) && (a[1] >= 0.0f) && (a[2] >= 0.0f) &&
       (b[0] <= 0.0f) && (b[1] <= 0.0f) && (b[2] <= 0.0f) )
   {
      return 1;
   }

   return 0;
}

static inline void box_init_inf( boxf box ){
   v3_fill( box[0],  INFINITY );
   v3_fill( box[1], -INFINITY );
}

/*
 * -----------------------------------------------------------------------------
 * Section 5.b                       Planes
 * -----------------------------------------------------------------------------
 */

static inline void tri_to_plane( f64 a[3], f64 b[3], 
      f64 c[3], f64 p[4] )
{
   f64 edge0[3];
   f64 edge1[3];
   f64 l;
   
   edge0[0] = b[0] - a[0];
   edge0[1] = b[1] - a[1];
   edge0[2] = b[2] - a[2];
   
   edge1[0] = c[0] - a[0];
   edge1[1] = c[1] - a[1];
   edge1[2] = c[2] - a[2];
   
   p[0] = edge0[1] * edge1[2] - edge0[2] * edge1[1];
   p[1] = edge0[2] * edge1[0] - edge0[0] * edge1[2];
   p[2] = edge0[0] * edge1[1] - edge0[1] * edge1[0];
   
   l = sqrt(p[0] * p[0] + p[1] * p[1] + p[2] * p[2]);
   p[3] = (p[0] * a[0] + p[1] * a[1] + p[2] * a[2]) / l;
   
   p[0] = p[0] / l;
   p[1] = p[1] / l;
   p[2] = p[2] / l;
}

static int plane_intersect3( v4f a, v4f b, v4f c, v3f p )
{
   f32 const epsilon = 1e-6f;
   
   v3f x;
   v3_cross( a, b, x );
   f32 d = v3_dot( x, c );
   
   if( (d < epsilon) && (d > -epsilon) ) return 0;

   v3f v0, v1, v2;
   v3_cross( b, c, v0 );
   v3_cross( c, a, v1 );
   v3_cross( a, b, v2 );

   v3_muls(       v0, a[3], p );
   v3_muladds( p, v1, b[3], p );
   v3_muladds( p, v2, c[3], p );
   v3_divs( p, d, p );
   
   return 1;
}

static int plane_intersect2( v4f a, v4f b, v3f p, v3f n )
{
   f32 const epsilon = 1e-6f;

   v4f c;
   v3_cross( a, b, c );
   f32 d = v3_length2( c );

   if( (d < epsilon) && (d > -epsilon) ) 
      return 0;

   v3f v0, v1, vx;
   v3_cross( c, b, v0 );
   v3_cross( a, c, v1 );

   v3_muls( v0, a[3], vx );
   v3_muladds( vx, v1, b[3], vx );
   v3_divs( vx, d, p );
   v3_copy( c, n );

   return 1;
}

static int plane_segment( v4f plane, v3f a, v3f b, v3f co )
{
   f32 d0 = v3_dot( a, plane ) - plane[3],
       d1 = v3_dot( b, plane ) - plane[3];

   if( d0*d1 < 0.0f )
   {
      f32 tot = 1.0f/( fabsf(d0)+fabsf(d1) );

      v3_muls( a, fabsf(d1) * tot, co );
      v3_muladds( co, b, fabsf(d0) * tot, co );
      return 1;
   }

   return 0;
}

static inline f64 plane_polarity( f64 p[4], f64 a[3] )
{
   return 
   (a[0] * p[0] + a[1] * p[1] + a[2] * p[2])
   -(p[0]*p[3] * p[0] + p[1]*p[3] * p[1] + p[2]*p[3] * p[2])
   ;
}

static f32 ray_plane( v4f plane, v3f co, v3f dir ){
   f32 d = v3_dot( plane, dir );
   if( fabsf(d) > 1e-6f ){
      v3f v0;
      v3_muls( plane, plane[3], v0 );
      v3_sub( v0, co, v0 );
      return v3_dot( v0, plane ) / d;
   }
   else return INFINITY;
}

/*
 * -----------------------------------------------------------------------------
 * Section 5.c            Closest point functions
 * -----------------------------------------------------------------------------
 */

/* 
 * These closest point tests were learned from Real-Time Collision Detection by 
 * Christer Ericson 
 */
static f32 closest_segment_segment( v3f p1, v3f q1, v3f p2, v3f q2, 
   f32 *s, f32 *t, v3f c1, v3f c2)
{
   v3f d1,d2,r;
   v3_sub( q1, p1, d1 );
   v3_sub( q2, p2, d2 );
   v3_sub( p1, p2, r );

   f32 a = v3_length2( d1 ),
         e = v3_length2( d2 ),
         f = v3_dot( d2, r );

   const f32 kEpsilon = 0.0001f;

   if( a <= kEpsilon && e <= kEpsilon )
   {
      *s = 0.0f;
      *t = 0.0f;
      v3_copy( p1, c1 );
      v3_copy( p2, c2 );

      v3f v0;
      v3_sub( c1, c2, v0 );

      return v3_length2( v0 );
   }
   
   if( a<= kEpsilon )
   {
      *s = 0.0f;
      *t = vg_clampf( f / e, 0.0f, 1.0f );
   }
   else
   {
      f32 c = v3_dot( d1, r );
      if( e <= kEpsilon )
      {
         *t = 0.0f;
         *s = vg_clampf( -c / a, 0.0f, 1.0f );
      }
      else
      {
         f32 b = v3_dot(d1,d2),
               d = a*e-b*b;

         if( d != 0.0f )
         {
            *s = vg_clampf((b*f - c*e)/d, 0.0f, 1.0f);
         }
         else
         {
            *s = 0.0f;
         }

         *t = (b*(*s)+f) / e;

         if( *t < 0.0f )
         {
            *t = 0.0f;
            *s = vg_clampf( -c / a, 0.0f, 1.0f );
         }
         else if( *t > 1.0f )
         {
            *t = 1.0f;
            *s = vg_clampf((b-c)/a,0.0f,1.0f);
         }
      }
   }

   v3_muladds( p1, d1, *s, c1 );
   v3_muladds( p2, d2, *t, c2 );

   v3f v0;
   v3_sub( c1, c2, v0 );
   return v3_length2( v0 );
}

static int point_inside_aabb( boxf box, v3f point )
{
   if((point[0]<=box[1][0]) && (point[1]<=box[1][1]) && (point[2]<=box[1][2]) &&
      (point[0]>=box[0][0]) && (point[1]>=box[0][1]) && (point[2]>=box[0][2]) )
      return 1;
   else
      return 0;
}

static void closest_point_aabb( v3f p, boxf box, v3f dest )
{
   v3_maxv( p, box[0], dest );
   v3_minv( dest, box[1], dest );
}

static void closest_point_obb( v3f p, boxf box, 
                                  m4x3f mtx, m4x3f inv_mtx, v3f dest )
{
   v3f local;
   m4x3_mulv( inv_mtx, p, local );
   closest_point_aabb( local, box, local );
   m4x3_mulv( mtx, local, dest );
}

static f32 closest_point_segment( v3f a, v3f b, v3f point, v3f dest )
{
   v3f v0, v1;
   v3_sub( b, a, v0 );
   v3_sub( point, a, v1 );

   f32 t = v3_dot( v1, v0 ) / v3_length2(v0);
   t = vg_clampf(t,0.0f,1.0f);
   v3_muladds( a, v0, t, dest );
   return t;
}

static void closest_on_triangle( v3f p, v3f tri[3], v3f dest )
{
   v3f ab, ac, ap;
   f32 d1, d2;

   /* Region outside A */
   v3_sub( tri[1], tri[0], ab );
   v3_sub( tri[2], tri[0], ac );
   v3_sub( p, tri[0], ap );
   
   d1 = v3_dot(ab,ap);
   d2 = v3_dot(ac,ap);
   if( d1 <= 0.0f && d2 <= 0.0f ) 
   {
      v3_copy( tri[0], dest );
      v3_copy( (v3f){INFINITY,INFINITY,INFINITY}, dest );
      return;
   }

   /* Region outside B */
   v3f bp;
   f32 d3, d4;

   v3_sub( p, tri[1], bp );
   d3 = v3_dot( ab, bp );
   d4 = v3_dot( ac, bp );

   if( d3 >= 0.0f && d4 <= d3 )
   {
      v3_copy( tri[1], dest );
      v3_copy( (v3f){INFINITY,INFINITY,INFINITY}, dest );
      return;
   }
   
   /* Edge region of AB */
   f32 vc = d1*d4 - d3*d2;
   if( vc <= 0.0f && d1 >= 0.0f && d3 <= 0.0f )
   {
      f32 v = d1 / (d1-d3);
      v3_muladds( tri[0], ab, v, dest );
      v3_copy( (v3f){INFINITY,INFINITY,INFINITY}, dest );
      return;
   }

   /* Region outside C */
   v3f cp;
   f32 d5, d6;
   v3_sub( p, tri[2], cp );
   d5 = v3_dot(ab, cp);
   d6 = v3_dot(ac, cp);
   
   if( d6 >= 0.0f && d5 <= d6 )
   {
      v3_copy( tri[2], dest );
      v3_copy( (v3f){INFINITY,INFINITY,INFINITY}, dest );
      return;
   }

   /* Region of AC */
   f32 vb = d5*d2 - d1*d6;
   if( vb <= 0.0f && d2 >= 0.0f && d6 <= 0.0f )
   {
      f32 w = d2 / (d2-d6);
      v3_muladds( tri[0], ac, w, dest );
      v3_copy( (v3f){INFINITY,INFINITY,INFINITY}, dest );
      return;
   }

   /* Region of BC */
   f32 va = d3*d6 - d5*d4;
   if( va <= 0.0f && (d4-d3) >= 0.0f && (d5-d6) >= 0.0f )
   {
      f32 w = (d4-d3) / ((d4-d3) + (d5-d6));
      v3f bc;
      v3_sub( tri[2], tri[1], bc );
      v3_muladds( tri[1], bc, w, dest );
      v3_copy( (v3f){INFINITY,INFINITY,INFINITY}, dest );
      return;
   }

   /* P inside region, Q via barycentric coordinates uvw */
   f32 d = 1.0f/(va+vb+vc),
         v = vb*d,
         w = vc*d;

   v3_muladds( tri[0], ab, v, dest );
   v3_muladds( dest, ac, w, dest );
}

enum contact_type
{
   k_contact_type_default,
   k_contact_type_disabled,
   k_contact_type_edge
};

static enum contact_type closest_on_triangle_1( v3f p, v3f tri[3], v3f dest )
{
   v3f ab, ac, ap;
   f32 d1, d2;

   /* Region outside A */
   v3_sub( tri[1], tri[0], ab );
   v3_sub( tri[2], tri[0], ac );
   v3_sub( p, tri[0], ap );
   
   d1 = v3_dot(ab,ap);
   d2 = v3_dot(ac,ap);
   if( d1 <= 0.0f && d2 <= 0.0f ) 
   {
      v3_copy( tri[0], dest );
      return k_contact_type_default;
   }

   /* Region outside B */
   v3f bp;
   f32 d3, d4;

   v3_sub( p, tri[1], bp );
   d3 = v3_dot( ab, bp );
   d4 = v3_dot( ac, bp );

   if( d3 >= 0.0f && d4 <= d3 )
   {
      v3_copy( tri[1], dest );
      return k_contact_type_edge;
   }
   
   /* Edge region of AB */
   f32 vc = d1*d4 - d3*d2;
   if( vc <= 0.0f && d1 >= 0.0f && d3 <= 0.0f )
   {
      f32 v = d1 / (d1-d3);
      v3_muladds( tri[0], ab, v, dest );
      return k_contact_type_edge;
   }

   /* Region outside C */
   v3f cp;
   f32 d5, d6;
   v3_sub( p, tri[2], cp );
   d5 = v3_dot(ab, cp);
   d6 = v3_dot(ac, cp);
   
   if( d6 >= 0.0f && d5 <= d6 )
   {
      v3_copy( tri[2], dest );
      return k_contact_type_edge;
   }

   /* Region of AC */
   f32 vb = d5*d2 - d1*d6;
   if( vb <= 0.0f && d2 >= 0.0f && d6 <= 0.0f )
   {
      f32 w = d2 / (d2-d6);
      v3_muladds( tri[0], ac, w, dest );
      return k_contact_type_edge;
   }

   /* Region of BC */
   f32 va = d3*d6 - d5*d4;
   if( va <= 0.0f && (d4-d3) >= 0.0f && (d5-d6) >= 0.0f )
   {
      f32 w = (d4-d3) / ((d4-d3) + (d5-d6));
      v3f bc;
      v3_sub( tri[2], tri[1], bc );
      v3_muladds( tri[1], bc, w, dest );
      return k_contact_type_edge;
   }

   /* P inside region, Q via barycentric coordinates uvw */
   f32 d = 1.0f/(va+vb+vc),
         v = vb*d,
         w = vc*d;

   v3_muladds( tri[0], ab, v, dest );
   v3_muladds( dest, ac, w, dest );

   return k_contact_type_default;
}

static void closest_point_elipse( v2f p, v2f e, v2f o )
{
   v2f pabs, ei, e2, ve, t;

   v2_abs( p, pabs );
   v2_div( (v2f){ 1.0f, 1.0f }, e, ei );
   v2_mul( e, e, e2 );
   v2_mul( ei, (v2f){ e2[0]-e2[1], e2[1]-e2[0] }, ve );

   v2_fill( t, 0.70710678118654752f );

   for( int i=0; i<3; i++ ){
      v2f v, u, ud, w;

      v2_mul( ve, t, v );  /* ve*t*t*t */
      v2_mul( v, t, v );
      v2_mul( v, t, v );

      v2_sub( pabs, v, u );
      v2_normalize( u );
      
      v2_mul( t, e, ud );
      v2_sub( ud, v, ud );

      v2_muls( u, v2_length( ud ), u );

      v2_add( v, u, w );
      v2_mul( w, ei, w );

      v2_maxv( (v2f){0.0f,0.0f}, w, t );
      v2_normalize( t );
   }

   v2_mul( t, e, o );
   v2_copysign( o, p );
}

/*
 * -----------------------------------------------------------------------------
 * Section 5.d               Raycasts & Spherecasts
 * -----------------------------------------------------------------------------
 */

static int ray_aabb1( boxf box, v3f co, v3f dir_inv, f32 dist )
{
   v3f v0, v1;
   f32 tmin, tmax;

   v3_sub( box[0], co, v0 );
   v3_sub( box[1], co, v1 );

   v3_mul( v0, dir_inv, v0 );
   v3_mul( v1, dir_inv, v1 );
   
   tmin = vg_minf( v0[0], v1[0] );
   tmax = vg_maxf( v0[0], v1[0] );
   tmin = vg_maxf( tmin, vg_minf( v0[1], v1[1] ));
   tmax = vg_minf( tmax, vg_maxf( v0[1], v1[1] ));
   tmin = vg_maxf( tmin, vg_minf( v0[2], v1[2] ));
   tmax = vg_minf( tmax, vg_maxf( v0[2], v1[2] ));

   return (tmax >= tmin) && (tmin <= dist) && (tmax >= 0.0f);
}

/* Time of intersection with ray vs triangle */
static int ray_tri( v3f tri[3], v3f co, 
                    v3f dir, f32 *dist, int backfaces )
{
   f32 const kEpsilon = 0.00001f;

   v3f v0, v1, h, s, q, n;
   f32 a,f,u,v,t;

   f32 *pa = tri[0],
         *pb = tri[1],
         *pc = tri[2];

   v3_sub( pb, pa, v0 );
   v3_sub( pc, pa, v1 );
   v3_cross( dir, v1, h );
   v3_cross( v0, v1, n );

   if( (v3_dot( n, dir ) > 0.0f) && !backfaces ) /* Backface culling */
      return 0;
   
   /* Parralel */
   a = v3_dot( v0, h );

   if( a > -kEpsilon && a < kEpsilon )
      return 0;

   f = 1.0f/a;
   v3_sub( co, pa, s );

   u = f * v3_dot(s, h);
   if( u < 0.0f || u > 1.0f )
      return 0;
   
   v3_cross( s, v0, q );
   v = f * v3_dot( dir, q );
   if( v < 0.0f || u+v > 1.0f )
      return 0;

   t = f * v3_dot(v1, q);
   if( t > kEpsilon )
   {
      *dist = t;
      return 1;
   }
   else return 0;
}

/* time of intersection with ray vs sphere */
static int ray_sphere( v3f c, f32 r, 
                       v3f co, v3f dir, f32 *t )
{
   v3f m;
   v3_sub( co, c, m );

   f32 b  = v3_dot( m, dir ),
         c1 = v3_dot( m, m ) - r*r;

   /* Exit if r’s origin outside s (c > 0) and r pointing away from s (b > 0) */
   if( c1 > 0.0f && b > 0.0f ) 
      return 0;
   
   f32 discr = b*b - c1;

   /* A negative discriminant corresponds to ray missing sphere */
   if( discr < 0.0f )
      return 0;
   
   /* 
    * Ray now found to intersect sphere, compute smallest t value of 
    * intersection 
    */
   *t = -b - sqrtf( discr );
   
   /* If t is negative, ray started inside sphere so clamp t to zero */
   if( *t < 0.0f ) 
      *t = 0.0f;

   return 1;
}

/* 
 * time of intersection of ray vs cylinder
 * The cylinder does not have caps but is finite
 *
 * Heavily adapted from regular segment vs cylinder from:
 *    Real-Time Collision Detection
 */
static int ray_uncapped_finite_cylinder( v3f q, v3f p, f32 r, 
                                         v3f co, v3f dir, f32 *t )
{
   v3f d, m, n, sb;
   v3_muladds( co, dir, 1.0f, sb );

   v3_sub( q, p, d );
   v3_sub( co, p, m );
   v3_sub( sb, co, n );
   
   f32 md = v3_dot( m, d ),
         nd = v3_dot( n, d ),
         dd = v3_dot( d, d ),
         nn = v3_dot( n, n ),
         mn = v3_dot( m, n ),
         a  = dd*nn - nd*nd,
         k  = v3_dot( m, m ) - r*r,
         c  = dd*k - md*md;

   if( fabsf(a) < 0.00001f ) 
   {
      /* Segment runs parallel to cylinder axis */
      return 0;
   }

   f32 b     = dd*mn - nd*md,
         discr = b*b - a*c;

   if( discr < 0.0f ) 
      return 0; /* No real roots; no intersection */

   *t = (-b - sqrtf(discr)) / a;
   if( *t < 0.0f )
      return 0; /* Intersection behind ray */
   
   /* Check within cylinder segment */
   if( md + (*t)*nd < 0.0f ) 
      return 0;

   if( md + (*t)*nd > dd ) 
      return 0;

   /* Segment intersects cylinder between the endcaps; t is correct */
   return 1;
}

/*
 * Time of intersection of sphere and triangle. Origin must be outside the 
 * colliding area. This is a fairly long procedure.
 */
static int spherecast_triangle( v3f tri[3],
                                v3f co, v3f dir, f32 r, f32 *t, v3f n )
{
   v3f sum[3];
   v3f v0, v1;

   v3_sub( tri[1], tri[0], v0 );
   v3_sub( tri[2], tri[0], v1 );
   v3_cross( v0, v1, n );
   v3_normalize( n );
   v3_muladds( tri[0], n, r, sum[0] );
   v3_muladds( tri[1], n, r, sum[1] );
   v3_muladds( tri[2], n, r, sum[2] );

   int hit = 0;
   f32 t_min = INFINITY,
         t1;

   if( ray_tri( sum, co, dir, &t1, 0 ) ){
      t_min = vg_minf( t_min, t1 );
      hit = 1;
   }

   /* 
    * Currently disabled; ray_sphere requires |d| = 1. it is not very important.
    */
#if 0
   for( int i=0; i<3; i++ ){
      if( ray_sphere( tri[i], r, co, dir, &t1 ) ){
         t_min = vg_minf( t_min, t1 );
         hit = 1;
      }
   }
#endif

   for( int i=0; i<3; i++ ){
      int i0 =  i,
          i1 = (i+1)%3;

      if( ray_uncapped_finite_cylinder( tri[i0], tri[i1], r, co, dir, &t1 ) ){
         if( t1 < t_min ){
            t_min = t1;
            
            v3f co1, ct, cx;
            v3_add( dir, co, co1 );
            v3_lerp( co, co1, t_min, ct );

            closest_point_segment( tri[i0], tri[i1], ct, cx );
            v3_sub( ct, cx, n );
            v3_normalize( n );
         }

         hit = 1;
      }
   }

   *t = t_min;
   return hit;
}

/*
 * -----------------------------------------------------------------------------
 * Section 5.e                       Curves
 * -----------------------------------------------------------------------------
 */

static void eval_bezier_time( v3f p0, v3f p1, v3f h0, v3f h1, f32 t, v3f p )
{
   f32 tt = t*t,
         ttt = tt*t;

   v3_muls( p1, ttt, p );
   v3_muladds( p, h1, 3.0f*tt  -3.0f*ttt, p );
   v3_muladds( p, h0, 3.0f*ttt -6.0f*tt  +3.0f*t, p );
   v3_muladds( p, p0, 3.0f*tt  -ttt -3.0f*t +1.0f, p );
}

static void eval_bezier3( v3f p0, v3f p1, v3f p2, f32 t, v3f p )
{
   f32 u = 1.0f-t;

   v3_muls( p0, u*u, p );
   v3_muladds( p, p1, 2.0f*u*t, p );
   v3_muladds( p, p2, t*t, p );
}

/*
 * -----------------------------------------------------------------------------
 * Section 5.f                      Volumes
 * -----------------------------------------------------------------------------
 */

static f32 vg_sphere_volume( f32 r ){
   return (4.0f/3.0f) * VG_PIf * r*r*r;
}

static f32 vg_box_volume( boxf box ){
   v3f e;
   v3_sub( box[1], box[0], e );
   return e[0]*e[1]*e[2];
}

static f32 vg_cylinder_volume( f32 r, f32 h ){
   return VG_PIf * r*r * h;
}

static f32 vg_capsule_volume( f32 r, f32 h ){
   return vg_sphere_volume( r ) + vg_cylinder_volume( r, h-r*2.0f );
}

static void vg_sphere_bound( f32 r, boxf out_box ){
   v3_fill( out_box[0], -r );
   v3_fill( out_box[1],  r );
}

static void vg_capsule_bound( f32 r, f32 h, boxf out_box ){
   v3_copy( (v3f){-r,-h*0.5f,r}, out_box[0] );
   v3_copy( (v3f){-r, h*0.5f,r}, out_box[1] );
}


/*
 * -----------------------------------------------------------------------------
 * Section 5.g                  Inertia Tensors
 * -----------------------------------------------------------------------------
 */

/*
 * Translate existing inertia tensor 
 */
static void vg_translate_inertia( m3x3f inout_inertia, f32 mass, v3f d ){
   /* 
    * I = I_0 + m*[(d.d)E_3 - d(X)d]
    *
    * I:   updated tensor
    * I_0: original tensor
    * m:   scalar mass
    * d:   translation vector 
    * (X): outer product
    * E_3: identity matrix
    */
   m3x3f t, outer, scale;
   m3x3_diagonal( t, v3_dot(d,d) );
   m3x3_outer_product( outer, d, d );
   m3x3_sub( t, outer, t );
   m3x3_diagonal( scale, mass );
   m3x3_mul( scale, t, t );
   m3x3_add( inout_inertia, t, inout_inertia );
}

/*
 * Rotate existing inertia tensor 
 */
static void vg_rotate_inertia( m3x3f inout_inertia, m3x3f rotation ){
   /*
    *  I = R I_0 R^T
    *
    *  I:   updated tensor
    *  I_0: original tensor
    *  R:   rotation matrix
    *  R^T: tranposed rotation matrix
    */

   m3x3f Rt;
   m3x3_transpose( rotation, Rt );
   m3x3_mul( rotation, inout_inertia, inout_inertia );
   m3x3_mul( inout_inertia, Rt, inout_inertia );
}
/*
 * Create inertia tensor for box
 */
static void vg_box_inertia( boxf box, f32 mass, m3x3f out_inertia ){
   v3f e, com;
   v3_sub( box[1], box[0], e );
   v3_muladds( box[0], e, 0.5f, com );

   f32 ex2 = e[0]*e[0],
       ey2 = e[1]*e[1],
       ez2 = e[2]*e[2],
       ix  = (ey2+ez2) * mass * (1.0f/12.0f),
       iy  = (ex2+ez2) * mass * (1.0f/12.0f),
       iz  = (ex2+ey2) * mass * (1.0f/12.0f);

   m3x3_identity( out_inertia );
   m3x3_setdiagonalv3( out_inertia, (v3f){ ix, iy, iz } );
   vg_translate_inertia( out_inertia, mass, com );
}

/*
 * Create inertia tensor for sphere
 */
static void vg_sphere_inertia( f32 r, f32 mass, m3x3f out_inertia ){
   f32 ixyz = r*r * mass * (2.0f/5.0f);
   
   m3x3_identity( out_inertia );
   m3x3_setdiagonalv3( out_inertia, (v3f){ ixyz, ixyz, ixyz } );
}

/*
 * Create inertia tensor for capsule
 */
static void vg_capsule_inertia( f32 r, f32 h, f32 mass, m3x3f out_inertia ){
   f32 density = mass / vg_capsule_volume( r, h ),
       ch  = h-r*2.0f, /* cylinder height */
       cm  = VG_PIf * ch*r*r * density, /* cylinder mass */
       hm  = VG_TAUf * (1.0f/3.0f) * r*r*r * density, /* hemisphere mass */
       
       iy  = r*r*cm * 0.5f,
       ixz = iy * 0.5f + cm*ch*ch*(1.0f/12.0f),

       aux0= (hm*2.0f*r*r)/5.0f;

   iy += aux0 * 2.0f;

   f32 aux1= ch*0.5f,
       aux2= aux0 + hm*(aux1*aux1 + 3.0f*(1.0f/8.0f)*ch*r);

   ixz += aux2*2.0f;

   m3x3_identity( out_inertia );
   m3x3_setdiagonalv3( out_inertia, (v3f){ ixz, iy, ixz } );
}

/*
 * -----------------------------------------------------------------------------
 * Section 6.a            PSRNG and some distributions
 * -----------------------------------------------------------------------------
 */

/* An implementation of the MT19937 Algorithm for the Mersenne Twister
 * by Evan Sultanik.  Based upon the pseudocode in: M. Matsumoto and
 * T. Nishimura, "Mersenne Twister: A 623-dimensionally
 * equidistributed uniform pseudorandom number generator," ACM
 * Transactions on Modeling and Computer Simulation Vol. 8, No. 1,
 * January pp.3-30 1998.
 *
 * http://www.sultanik.com/Mersenne_twister
 * https://github.com/ESultanik/mtwister/blob/master/mtwister.c
 */

#define MT_UPPER_MASK         0x80000000
#define MT_LOWER_MASK         0x7fffffff
#define MT_TEMPERING_MASK_B   0x9d2c5680
#define MT_TEMPERING_MASK_C   0xefc60000

#define MT_STATE_VECTOR_LENGTH 624

/* changes to STATE_VECTOR_LENGTH also require changes to this */
#define MT_STATE_VECTOR_M      397 

typedef struct vg_rand vg_rand;
struct vg_rand {
  u32 mt[MT_STATE_VECTOR_LENGTH];
  i32 index;
};

static void vg_rand_seed( vg_rand *rand, unsigned long seed ) {
   /* set initial seeds to mt[STATE_VECTOR_LENGTH] using the generator
    * from Line 25 of Table 1 in: Donald Knuth, "The Art of Computer
    * Programming," Vol. 2 (2nd Ed.) pp.102.
    */
   rand->mt[0] = seed & 0xffffffff;
   for( rand->index=1; rand->index<MT_STATE_VECTOR_LENGTH; rand->index++){
      rand->mt[rand->index] = (6069 * rand->mt[rand->index-1]) & 0xffffffff;
   }
}

/*
 * Generates a pseudo-randomly generated long.
 */
static u32 vg_randu32( vg_rand *rand ) {
   u32 y;
   /* mag[x] = x * 0x9908b0df for x = 0,1 */
   static u32 mag[2] = {0x0, 0x9908b0df}; 
   if( rand->index >= MT_STATE_VECTOR_LENGTH || rand->index < 0 ){
      /* generate STATE_VECTOR_LENGTH words at a time */
      int kk;
      if( rand->index >= MT_STATE_VECTOR_LENGTH+1 || rand->index < 0 ){
         vg_rand_seed( rand, 4357 );
      }
      for( kk=0; kk<MT_STATE_VECTOR_LENGTH-MT_STATE_VECTOR_M; kk++ ){
         y = (rand->mt[kk] & MT_UPPER_MASK) | 
             (rand->mt[kk+1] & MT_LOWER_MASK);
         rand->mt[kk] = rand->mt[kk+MT_STATE_VECTOR_M] ^ (y>>1) ^ mag[y & 0x1];
      }
      for( ; kk<MT_STATE_VECTOR_LENGTH-1; kk++ ){
         y = (rand->mt[kk] & MT_UPPER_MASK) | 
             (rand->mt[kk+1] & MT_LOWER_MASK);
         rand->mt[kk] = 
            rand->mt[ kk+(MT_STATE_VECTOR_M-MT_STATE_VECTOR_LENGTH)] ^ 
                        (y >> 1) ^ mag[y & 0x1];
      }
      y = (rand->mt[MT_STATE_VECTOR_LENGTH-1] & MT_UPPER_MASK) | 
          (rand->mt[0] & MT_LOWER_MASK);
      rand->mt[MT_STATE_VECTOR_LENGTH-1] = 
         rand->mt[MT_STATE_VECTOR_M-1] ^ (y >> 1) ^ mag[y & 0x1];
      rand->index = 0;
   }
   y = rand->mt[rand->index++];
   y ^= (y >> 11);
   y ^= (y << 7) & MT_TEMPERING_MASK_B;
   y ^= (y << 15) & MT_TEMPERING_MASK_C;
   y ^= (y >> 18);
   return y;
}

/*
 * Generates a pseudo-randomly generated f64 in the range [0..1].
 */
static inline f64 vg_randf64( vg_rand *rand ){
   return (f64)vg_randu32(rand)/(f64)0xffffffff;
}

static inline f64 vg_randf64_range( vg_rand *rand, f64 min, f64 max ){
   return vg_lerp( min, max, (f64)vg_randf64(rand) );
}

static inline void vg_rand_dir( vg_rand *rand, v3f dir ){
   dir[0] = vg_randf64(rand);
   dir[1] = vg_randf64(rand);
   dir[2] = vg_randf64(rand);

   /* warning: *could* be 0 length.
    * very unlikely.. 1 in (2^32)^3. but its mathematically wrong. */

   v3_muls( dir, 2.0f, dir );
   v3_sub( dir, (v3f){1.0f,1.0f,1.0f}, dir );

   v3_normalize( dir );
}

static inline void vg_rand_sphere( vg_rand *rand, v3f co ){
   vg_rand_dir(rand,co);
   v3_muls( co, cbrtf( vg_randf64(rand) ), co );
}

static void vg_rand_disc( vg_rand *rand, v2f co ){
   f32 a = vg_randf64(rand) * VG_TAUf;
   co[0] = sinf(a);
   co[1] = cosf(a); 
   v2_muls( co, sqrtf( vg_randf64(rand) ), co );
}

static void vg_rand_cone( vg_rand *rand, v3f out_dir, f32 angle ){
   f32 r = sqrtf(vg_randf64(rand)) * angle * 0.5f,
       a = vg_randf64(rand) * VG_TAUf;

   out_dir[0] = sinf(a) * sinf(r);
   out_dir[1] = cosf(a) * sinf(r);
   out_dir[2] = cosf(r);
}

static void vg_hsv_rgb( v3f hsv, v3f rgb ){
   i32 i = floorf( hsv[0]*6.0f );
   f32 v = hsv[2],
       f = hsv[0] * 6.0f - (f32)i,
       p = v * (1.0f-hsv[1]),
       q = v * (1.0f-f*hsv[1]),
       t = v * (1.0f-(1.0f-f)*hsv[1]);

   switch( i % 6 ){
      case 0: rgb[0] = v; rgb[1] = t; rgb[2] = p; break;
      case 1: rgb[0] = q; rgb[1] = v; rgb[2] = p; break;
      case 2: rgb[0] = p; rgb[1] = v; rgb[2] = t; break;
      case 3: rgb[0] = p; rgb[1] = q; rgb[2] = v; break;
      case 4: rgb[0] = t; rgb[1] = p; rgb[2] = v; break;
      case 5: rgb[0] = v; rgb[1] = p; rgb[2] = q; break;
   }
}

static void vg_rgb_hsv( v3f rgb, v3f hsv ){
   f32 min = v3_minf( rgb ),
       max = v3_maxf( rgb ),
       range = max-min,
       k_epsilon = 0.00001f;

   hsv[2] = max;
   if( range < k_epsilon ){
      hsv[0] = 0.0f;
      hsv[1] = 0.0f;
      return;
   }

   if( max > k_epsilon ){
      hsv[1] = range/max;
   }
   else {
      hsv[0] = 0.0f;
      hsv[1] = 0.0f;
      return;
   }

   if( rgb[0] >= max )
      hsv[0] = (rgb[1]-rgb[2])/range;
   else if( max == rgb[1] )
      hsv[0] = 2.0f+(rgb[2]-rgb[0])/range;
   else
      hsv[0] = 4.0f+(rgb[0]-rgb[1])/range;

   hsv[0] = vg_fractf( hsv[0] * (60.0f/360.0f) );
}