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diff --git a/trochoid.cginc b/trochoid.cginc
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+#ifndef __TROCHOID_MATH
+#define __TROCHOID_MATH
+
+#if defined(_TROCHOID)
+
+#include "globals.cginc"
+
+#define PI 3.14159265
+#define TAU PI * 2.0
+
+float3 _trochoid_map(float theta, float r0, float vert_z)
+{
+  r0 = pow(r0, _Trochoid_r_Power);
+  r0 *= 100;
+
+  float R = _Trochoid_R;
+  float r = _Trochoid_r;
+  float d = _Trochoid_d;
+
+  theta *= max(R, r) * _Trochoid_theta_k;
+  float theta_t = theta + _Time[2] * _Trochoid_t_k;
+
+  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;
+  float3 trochoid_scale = float3(_Trochoid_X_Scale, _Trochoid_Y_Scale, _Trochoid_Z_Scale);
+  result *= trochoid_scale;
+  return result;
+}
+
+float3 trochoid_map(float3 pos)
+{
+  float theta = atan2(pos.y, pos.x);
+  float r0 = length(pos.xy) * theta;
+  float z = pos.z;
+  float3 result = _trochoid_map(theta, r0, z);
+  return result;
+}
+
+float3 trochoid_normal(float3 pos)
+{
+    // Parameters of the trochoid
+    float R = _Trochoid_R;
+    float r = _Trochoid_r;
+    float d = _Trochoid_d;
+    float theta_k = max(R, r) * _Trochoid_theta_k;
+    float t_k = _Trochoid_t_k;
+    float time = _Time[2];
+    float r_power = _Trochoid_r_Power;
+
+    // Intermediate variables based on input position
+    float r_orig = max(length(pos.xy), 1e-5); // Avoid division by zero
+    float theta = atan2(pos.y, pos.x);
+    
+    float r0 = r_orig * theta;
+    float theta_scaled = theta * theta_k;
+    float theta_t = theta_scaled + time * t_k;
+
+    // Components of the trochoid function before final scaling
+    float C1 = cos(theta_t);
+    float S1 = sin(theta_t);
+    float common_factor = (R - r) / r;
+    float C2 = cos(common_factor * theta_t);
+    float S2 = sin(common_factor * theta_t);
+
+    float p_x = (R - r) * C1 + d * C2;
+    float p_y = (R - r) * S1 - d * S2;
+    float p_z = pos.z + cos(theta_t * 5.0) * 0.1 + theta * 0.0002;
+
+    // Partial derivatives of the components with respect to theta_t
+    float dp_x_dt = -(R - r) * S1 - d * common_factor * S2;
+    float dp_y_dt =  (R - r) * C1 - d * common_factor * C2;
+    float dp_z_dt = -sin(theta_t * 5.0) * 5.0 * 0.1;
+
+    // Partial derivatives of intermediate variables w.r.t. pos.x and pos.y
+    float d_theta_dx = -pos.y / (r_orig * r_orig);
+    float d_theta_dy =  pos.x / (r_orig * r_orig);
+    float d_r_orig_dx = pos.x / r_orig;
+    float d_r_orig_dy = pos.y / r_orig;
+
+    // The r0-dependent scaling factor and its derivative
+    float F = 100.0 * pow(r0, r_power);
+    float dF_dr0 = 100.0 * r_power * pow(r0, r_power - 1.0);
+
+    // Chain rule for tangents
+    // Tangent with respect to pos.x
+    float d_r0_dx = d_r_orig_dx * theta + r_orig * d_theta_dx;
+    float d_theta_t_dx = d_theta_dx * theta_k;
+    float dF_dx = dF_dr0 * d_r0_dx;
+
+    float dp_x_dx = dp_x_dt * d_theta_t_dx;
+    float dp_y_dx = dp_y_dt * d_theta_t_dx;
+    float dp_z_dx = dp_z_dt * d_theta_t_dx + 0.0002 * d_theta_dx;
+    
+    float3 trochoid_scale = float3(_Trochoid_X_Scale, _Trochoid_Y_Scale, _Trochoid_Z_Scale);
+    float3 T_x = trochoid_scale * (dF_dx * float3(p_x, p_y, p_z) + F * float3(dp_x_dx, dp_y_dx, dp_z_dx));
+
+    // Tangent with respect to pos.y
+    float d_r0_dy = d_r_orig_dy * theta + r_orig * d_theta_dy;
+    float d_theta_t_dy = d_theta_dy * theta_k;
+    float dF_dy = dF_dr0 * d_r0_dy;
+    
+    float dp_x_dy = dp_x_dt * d_theta_t_dy;
+    float dp_y_dy = dp_y_dt * d_theta_t_dy;
+    float dp_z_dy = dp_z_dt * d_theta_t_dy + 0.0002 * d_theta_dy;
+
+    float3 T_y = trochoid_scale * (dF_dy * float3(p_x, p_y, p_z) + F * float3(dp_x_dy, dp_y_dy, dp_z_dy));
+    
+    return normalize(cross(T_x, T_y));
+}
+
+#endif  // _TROCHOID
+
+#endif  // __TROCHOID_MATH
+