nv_scalar ffast_cos(const nv_scalar x)
{
// assert: 0 <= fT <= PI/2
// maximum absolute error = 1.1880e-03
// speedup = 2.14
nv_scalar x_sqr = x*x;
nv_scalar res = nv_scalar(3.705e-02);
res *= x_sqr;
res -= nv_scalar(4.967e-01);
res *= x_sqr;
res += nv_one;
return res;
}
vec3& cube_map_normal(int i, int x, int y, int cubesize, vec3& v)
{
nv_scalar s, t, sc, tc;
s = (nv_scalar(x) + nv_zero_5) / nv_scalar(cubesize);
t = (nv_scalar(y) + nv_zero_5) / nv_scalar(cubesize);
sc = s * nv_two - nv_one;
tc = t * nv_two - nv_one;
switch (i)
{
case 0:
v.x = nv_one;
v.y = -tc;
v.z = -sc;
break;
case 1:
v.x = -nv_one;
v.y = -tc;
v.z = sc;
break;
case 2:
v.x = sc;
v.y = nv_one;
v.z = tc;
break;
case 3:
v.x = sc;
v.y = -nv_one;
v.z = -tc;
break;
case 4:
v.x = sc;
v.y = -tc;
v.z = nv_one;
break;
case 5:
v.x = -sc;
v.y = -tc;
v.z = -nv_one;
break;
}
normalize(v);
return v;
}
nv_scalar fast_cos(const nv_scalar x)
{
// assert: 0 <= fT <= PI/2
// maximum absolute error = 2.3082e-09
// speedup = 1.47
nv_scalar x_sqr = x*x;
nv_scalar res = nv_scalar(-2.605e-07);
res *= x_sqr;
res += nv_scalar(2.47609e-05);
res *= x_sqr;
res -= nv_scalar(1.3888397e-03);
res *= x_sqr;
res += nv_scalar(4.16666418e-02);
res *= x_sqr;
res -= nv_scalar(4.999999963e-01);
res *= x_sqr;
res += nv_one;
return res;
}
// v is normalized
// theta in radians
void mat3::set_rot(const nv_scalar& theta, const vec3& v)
{
nv_scalar ct = nv_scalar(_cos(theta));
nv_scalar st = nv_scalar(_sin(theta));
nv_scalar xx = v.x * v.x;
nv_scalar yy = v.y * v.y;
nv_scalar zz = v.z * v.z;
nv_scalar xy = v.x * v.y;
nv_scalar xz = v.x * v.z;
nv_scalar yz = v.y * v.z;
a00 = xx + ct*(1-xx);
a01 = xy + ct*(-xy) + st*-v.z;
a02 = xz + ct*(-xz) + st*v.y;
a10 = xy + ct*(-xy) + st*v.z;
a11 = yy + ct*(1-yy);
a12 = yz + ct*(-yz) + st*-v.x;
a20 = xz + ct*(-xz) + st*-v.y;
a21 = yz + ct*(-yz) + st*v.x;
a22 = zz + ct*(1-zz);
}
/*
* Ok, simulate a track-ball. Project the points onto the virtual
* trackball, then figure out the axis of rotation, which is the cross
* product of P1 P2 and O P1 (O is the center of the ball, 0,0,0)
* Note: This is a deformed trackball-- is a trackball in the center,
* but is deformed into a hyperbolic sheet of rotation away from the
* center. This particular function was chosen after trying out
* several variations.
*
* It is assumed that the arguments to this routine are in the range
* (-1.0 ... 1.0)
*/
quat & trackball(quat& q, vec2& pt1, vec2& pt2, nv_scalar trackballsize)
{
vec3 a; // Axis of rotation
nv_scalar phi; // how much to rotate about axis
vec3 d;
nv_scalar t;
if (pt1.x == pt2.x && pt1.y == pt2.y)
{
// Zero rotation
q = quat_id;
return q;
}
// First, figure out z-coordinates for projection of P1 and P2 to
// deformed sphere
vec3 p1(pt1.x,pt1.y,tb_project_to_sphere(trackballsize,pt1.x,pt1.y));
vec3 p2(pt2.x,pt2.y,tb_project_to_sphere(trackballsize,pt2.x,pt2.y));
// Now, we want the cross product of P1 and P2
cross(a,p1,p2);
// Figure out how much to rotate around that axis.
d.x = p1.x - p2.x;
d.y = p1.y - p2.y;
d.z = p1.z - p2.z;
t = _sqrt(d.x * d.x + d.y * d.y + d.z * d.z) / (trackballsize);
// Avoid problems with out-of-control values...
if (t > nv_one)
t = nv_one;
if (t < -nv_one)
t = -nv_one;
phi = nv_two * nv_scalar(asin(t));
axis_to_quat(q,a,phi);
return q;
}
vec3 & spect_decomp(vec3 & kv, mat3 & S, mat3 & U)
{
#define X 0
#define Y 1
#define Z 2
#define W 3
vec3 Diag,OffD; // OffD is off-diag (by omitted index)
nv_scalar g,h,fabsh,fabsOffDi,t,theta,c,s,tau,ta,OffDq,a,b;
static char nxt[] = {Y,Z,X};
U = mat3_id;
Diag[X] = S(X,X); Diag[Y] = S(Y,Y); Diag[Z] = S(Z,Z);
OffD[X] = S(Y,Z); OffD[Y] = S(Z,X); OffD[Z] = S(X,Y);
for (int sweep=20; sweep>0; --sweep)
{
nv_scalar sm = (fabs(OffD[X])+fabs(OffD[Y])+fabs(OffD[Z]));
if (sm == nv_zero)
break;
for (int i=Z; i>=X; --i)
{
int p = nxt[i];
int q = nxt[p];
fabsOffDi = fabs(OffD[i]);
g = nv_scalar(100.0)*fabsOffDi;
if (fabsOffDi > nv_zero)
{
h = Diag[q] - Diag[p];
fabsh = fabs(h);
if (fabsh+g==fabsh)
{
t = OffD[i]/h;
}
else
{
theta = nv_zero_5*h/OffD[i];
t = nv_one/(fabs(theta)+sqrt(theta*theta+nv_one));
if (theta < nv_zero)
t = -t;
}
c = nv_one/sqrt(t*t+nv_one);
s = t*c;
tau = s/(c+nv_one);
ta = t*OffD[i];
OffD[i] = nv_zero;
Diag[p] -= ta;
Diag[q] += ta;
OffDq = OffD[q];
OffD[q] -= s*(OffD[p] + tau*OffD[q]);
OffD[p] += s*(OffDq - tau*OffD[p]);
for (int j=Z; j>=X; --j)
{
a = U(j,p); b = U(j,q);
U(j,p) -= s*(b + tau*a);
U(j,q) += s*(a - tau*b);
}
}
}
}
kv = Diag;
return kv;
}