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#include "globals.cginc"
#ifndef __TROCHOID_MATH
#define __TROCHOID_MATH
#if defined(_TROCHOID)
#define PI 3.14159265
#define TAU PI * 2.0
float3 trochoid_map(float theta, float r0, float3 vert_z)
{
r0 *= r0;
r0 *= 100;
float R = _Trochoid_R;
float r = _Trochoid_r;
float d = _Trochoid_d;
theta *= max(R, r);
float theta_t = theta + _Time[2];
float x = (R - r) * cos(theta_t) + d * cos((R - r) * theta_t / r);
float y = (R - r) * sin(theta_t) - d * sin((R - r) * theta_t / r);
float z = vert_z + cos(theta_t * 5) * .1 + theta * .0002;
float3 result = float3(x, y, z) * r0;
result.xy *= 0.1;
return result;
}
float trochoid_normal(float3 objPos, float2 uv)
{
float theta = uv.x * TAU;
float r0 = length(objPos.xyz);
float x = objPos.x;
float y = objPos.y;
float z = objPos.z;
float e = 5E-2;
float small_step = 1E-2 * e;
float du_dt = (trochoid_map(theta + small_step, r0, z) - trochoid_map(theta - small_step, r0, z)) / small_step;
small_step = 1E-4 * e;
float du_dr = (trochoid_map(theta, r0 + small_step, z) - trochoid_map(theta, r0 - small_step, z)) / small_step;
small_step = 1E-5 * e;
float du_dz = (trochoid_map(theta, r0, z + small_step) - trochoid_map(theta, r0, z - small_step)) / small_step;
// U(T(x, y, z), R(x, y, z), Z(x, y, z))
// T(x, y, z) = atan2(y, x)
// R(x, y, z) = length(float3(x, y, z))
// Z(x, y, z) = z
// U(a, b, c) = trochoid_map(a, b, c)
// dU/dx = dU/dT dT/dx + dU/dR dR/dx + dU/dZ dZ/dx
// dU/dy = dU/dT dT/dy + dU/dR dR/dy + dU/dZ dZ/dy
// dU/dz = dU/dT dT/dz + dU/dR dR/dz + dU/dZ dZ/dz
// dT/dx = d/dx atan2(y, x) = -y / (x**2 + y**2)
// dT/dy = d/dx atan2(y, x) = x / (x**2 + y**2)
// dT/dz = d/dz atan2(y, x) = 0
// dR/dx = d/dx sqrt(x**2 + y**2 + z**2) = x / sqrt(x**2 + y**2 + z**2)
// dR/dy = d/dy sqrt(x**2 + y**2 + z**2) = y / sqrt(x**2 + y**2 + z**2)
// dR/dz = d/dy sqrt(x**2 + y**2 + z**2) = z / sqrt(x**2 + y**2 + z**2)
// dZ/dx = 0
// dZ/dy = 0
// dZ/dz = 1
float xy_norm = sqrt(x * x + y * y);
float dt_dx = -y / xy_norm;
float dt_dy = x / xy_norm;
float dt_dz = 0;
float xyz_norm = sqrt(x * x + y * y + z * z);
float dr_dx = x / xyz_norm;
float dr_dy = y / xyz_norm;
float dr_dz = z / xyz_norm;
float dz_dx = 0;
float dz_dy = 0;
float dz_dz = 1;
float3 normal =
normalize(
float3(
du_dt * dt_dx + du_dr * dr_dx + du_dz * dz_dx,
du_dt * dt_dy + du_dr * dr_dy + du_dz * dz_dy,
du_dt * dt_dz + du_dr * dr_dz + du_dz * dz_dz));
return UnityObjectToWorldNormal(normal);
}
#endif // _TROCHOID
#endif // __TROCHOID_MATH
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