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### Intro

Similar to the analytical derivatives of Value Noise, Gradient Noise (the name used for variations and generalizations of Perlin Noise) accept analytic computation of derivatives. Just like with Value Noise derivatives, this allows for much faster lighting computations or any other computation that requires gradients/normals based on the noise since we no longer need to approximate it though numerical methods that involve taking with multiple samples of the noise.

### The code

Assuming we have some standard Gradient Noise implementation like the code on the left, the computation of the derivatives involves only a few more computations, as shown in the right.

// returns 3D value noise float noise( in vec3 x ) { // grid vec3 p = floor(x); vec3 w = fract(x); // quintic interpolant vec3 u = w*w*w*(w*(w*6.0-15.0)+10.0); // gradients vec3 ga = hash( p+vec3(0.0,0.0,0.0) ); vec3 gb = hash( p+vec3(1.0,0.0,0.0) ); vec3 gc = hash( p+vec3(0.0,1.0,0.0) ); vec3 gd = hash( p+vec3(1.0,1.0,0.0) ); vec3 ge = hash( p+vec3(0.0,0.0,1.0) ); vec3 gf = hash( p+vec3(1.0,0.0,1.0) ); vec3 gg = hash( p+vec3(0.0,1.0,1.0) ); vec3 gh = hash( p+vec3(1.0,1.0,1.0) ); // projections float va = dot( ga, w-vec3(0.0,0.0,0.0) ); float vb = dot( gb, w-vec3(1.0,0.0,0.0) ); float vc = dot( gc, w-vec3(0.0,1.0,0.0) ); float vd = dot( gd, w-vec3(1.0,1.0,0.0) ); float ve = dot( ge, w-vec3(0.0,0.0,1.0) ); float vf = dot( gf, w-vec3(1.0,0.0,1.0) ); float vg = dot( gg, w-vec3(0.0,1.0,1.0) ); float vh = dot( gh, w-vec3(1.0,1.0,1.0) ); // interpolation return va + u.x*(vb-va) + u.y*(vc-va) + u.z*(ve-va) + u.x*u.y*(va-vb-vc+vd) + u.y*u.z*(va-vc-ve+vg) + u.z*u.x*(va-vb-ve+vf) + u.x*u.y*u.z*(-va+vb+vc-vd+ve-vf-vg+vh); }
// returns 3D value noise (in .x) and its derivatives (in .yzw) vec4 noised( in vec3 x ) { // grid vec3 p = floor(x); vec3 w = fract(x); // quintic interpolant vec3 u = w*w*w*(w*(w*6.0-15.0)+10.0); vec3 du = 30.0*w*w*(w*(w-2.0)+1.0); // gradients vec3 ga = hash( p+vec3(0.0,0.0,0.0) ); vec3 gb = hash( p+vec3(1.0,0.0,0.0) ); vec3 gc = hash( p+vec3(0.0,1.0,0.0) ); vec3 gd = hash( p+vec3(1.0,1.0,0.0) ); vec3 ge = hash( p+vec3(0.0,0.0,1.0) ); vec3 gf = hash( p+vec3(1.0,0.0,1.0) ); vec3 gg = hash( p+vec3(0.0,1.0,1.0) ); vec3 gh = hash( p+vec3(1.0,1.0,1.0) ); // projections float va = dot( ga, w-vec3(0.0,0.0,0.0) ); float vb = dot( gb, w-vec3(1.0,0.0,0.0) ); float vc = dot( gc, w-vec3(0.0,1.0,0.0) ); float vd = dot( gd, w-vec3(1.0,1.0,0.0) ); float ve = dot( ge, w-vec3(0.0,0.0,1.0) ); float vf = dot( gf, w-vec3(1.0,0.0,1.0) ); float vg = dot( gg, w-vec3(0.0,1.0,1.0) ); float vh = dot( gh, w-vec3(1.0,1.0,1.0) ); // interpolation float v = va + u.x*(vb-va) + u.y*(vc-va) + u.z*(ve-va) + u.x*u.y*(va-vb-vc+vd) + u.y*u.z*(va-vc-ve+vg) + u.z*u.x*(va-vb-ve+vf) + u.x*u.y*u.z*(-va+vb+vc-vd+ve-vf-vg+vh); vec3 d = ga + u.x*(gb-ga) + u.y*(gc-ga) + u.z*(ge-ga) + u.x*u.y*(ga-gb-gc+gd) + u.y*u.z*(ga-gc-ge+gg) + u.z*u.x*(ga-gb-ge+gf) + u.x*u.y*u.z*(-ga+gb+gc-gd+ge-gf-gg+gh) + du * (vec3(vb-va,vc-va,ve-va) + u.yzx*vec3(va-vb-vc+vd,va-vc-ve+vg,va-vb-ve+vf) + u.zxy*vec3(va-vb-ve+vf,va-vb-vc+vd,va-vc-ve+vg) + u.yzx*u.zxy*(-va+vb+vc-vd+ve-vf-vg+vh) )); return vec4( v, d ); }

You can find a reference implementation here: https://www.shadertoy.com/view/4dffRH

In the case of 2D, the code gets naturally smaller:

// returns 3D value noise (in .x) and its derivatives (in .yz) vec3 noised( in vec2 x ) { vec2 i = floor( p ); vec2 f = fract( p ); vec2 u = f*f*f*(f*(f*6.0-15.0)+10.0); vec2 du = 30.0*f*f*(f*(f-2.0)+1.0); vec2 ga = hash( i + vec2(0.0,0.0) ); vec2 gb = hash( i + vec2(1.0,0.0) ); vec2 gc = hash( i + vec2(0.0,1.0) ); vec2 gd = hash( i + vec2(1.0,1.0) ); float va = dot( ga, f - vec2(0.0,0.0) ); float vb = dot( gb, f - vec2(1.0,0.0) ); float vc = dot( gc, f - vec2(0.0,1.0) ); float vd = dot( gd, f - vec2(1.0,1.0) ); return vec3( va + u.x*(vb-va) + u.y*(vc-va) + u.x*u.y*(va-vb-vc+vd), // value ga + u.x*(gb-ga) + u.y*(gc-ga) + u.x*u.y*(ga-gb-gc+gd) + // derivatives du * (u.yx*(va-vb-vc+vd) + vec2(vb,vc) - va)); }

An implementation can be found here: https://www.shadertoy.com/view/XdXBRH