website articles
useful little functions

### Intro

When writing shader or during any procedural creation process (texturing, modeling, shading, animating...) you often find yourself modifying signals in different ways so they behave the way you want. It is common to use smoothstep() to threshold some values, or to pow() shape a signal, ir clamp() to clip, fmod() to repeat, a mix() to blend, exp() to attenuate, etc etc. These are convenient functions because they are available to you by default in most systems, as hardware instructions or as function calls in the langauge. However there are some operations that are often used that don't exist in any language that you still use a lot. Have you ever found yourself subtracting to smoothstep()'s to isolate some range or create a ring? Or performing some smooth clipping to avoid dividing by huge numbers? Of course you have. I have. In fact, these are some of such little useful functions that I have collected over the years:

### Almost Identity (I)

Imagine you don't want to change a value unless it's zero or very close to it, in which case you want to replace the value with a small constant. Then, rather than doing a conditional branch which introduces a discontinuity, you can smoothly blend your value with your threshold. Let m be the threshold (anything above m stays unchanged), and n the value things will take when your input is zero. Then, the following function does the soft clipping (in a cubic fashion):

float almostIdentity( float x, float m, float n ) { if( x>m ) return x; const float a = 2.0*n - m; const float b = 2.0*m - 3.0*n; const float t = x/m; return (a*t + b)*t*t + n; } ### Almost Unit Identity

This is another near-identiy function, but this one maps the unit interval to itself. But it is special in that not only remaps 0 to 0 and 1 to 1, but has a 0 derivative at the origin and a derivative of 1 at 1, making it ideal for transitioning things from being stationary to being in motion as if they had been in motion the whole time. It's equivalent to the Almost Identiy above with n=0 and m=1, basically. And since it's a cubic just like smoothstep() and therefore very fast to evaluate:

float almostIdentity( float x ) { return x*x*(2.0-x); } ### Almost Identity (II)

A different way to achieve a near identity that can also be used as smooth-abs() is through the square root of a biased square, instead of a cubic polynomail. I saw this technique first in a shader by user "omeometo" in Shadertoy. This approach can be a bit slower than the cubic above, depending on the hardware. And while it has zero derivative, it has a non-zero second derivative, which could cause problems in some situations:

float almostIdentity( float x, float n ) { return sqrt(x*x+n); } ### Exponential Impulse

Great for triggering behaviours or making envelopes for music or animation, and for anything that grows fast and then slowly decays. Use k to control the stretching of the function. Btw, its maximum, which is 1, happens at exactly x = 1/k.

float expImpulse( float x, float k ) { const float h = k*x; return h*exp(1.0-h); } ### Sustained Impulse

Similar to the previous, but it allows for control on the width of attack (through the parameter "k") and the release (parameter "f") independently. Also, it ensures the impulse releases at a value of 1.0 instead of 0.

float expSustainedImpulse( float x, float f, float k ) { float s = max(x-f,0.0) return min( x*x/(f*f), 1+(2.0/f)*s*exp(-k*s)); } ### Polynomial Impulse

Another impulse function that doesn't use exponentials can be designed by using polynomicals. Use k to control falloff of the function. For example, a quadratic can be used, which peaks at x = sqrt(1/k).

float quaImpulse( float k, float x ) { return 2.0*sqrt(k)*x/(1.0+k*x*x); }

You can easily generalize it to other powers to get different falloff shapes, where n is the degree of the polynomial:
float polyImpulse( float k, float n, float x ) { return (n/(n-1.0))*pow((n-1.0)*k,1.0/n) * x/(1.0+k*pow(x,n)); }

These generalized impulses peak at x = [k(n-1)]-1/n. ### Cubic Pulse

Of course you found yourself doing smoothstep(c-w,c,x)-smoothstep(c,c+w,x) very often, probably because you were trying to isolate some features in a signal. Then, this cubicPulse() will be your new best friend. You can also use it as a cheap replacement for a gaussian.

float cubicPulse( float c, float w, float x ) { x = fabs(x - c); if( x>w ) return 0.0; x /= w; return 1.0 - x*x*(3.0-2.0*x); } ### Exponential Step

A natural attenuation is an exponential of a linearly decaying quantity: yellow curve, exp(-x). A gaussian, is an exponential of a quadratically decaying quantity: light green curve, exp(-x2). You can generalize and keep increasing powers, and get a sharper and sharper s-shaped curves, until you get a step() in the limit.

float expStep( float x, float k, float n ) { return exp( -k*pow(x,n) ); } ### Gain

Remapping the unit interval into the unit interval by expanding the sides and compressing the center, and keeping 1/2 mapped to 1/2, that can be done with the gain() function. This was a common function in RSL tutorials (the Renderman Shading Language). k=1 is the identity curve, k<1 produces the classic gain() shape, and k>1 produces "s" shaped curces. The curves are symmetric (and inverse) for k=a and k=1/a.

float gain(float x, float k) { const float a = 0.5*pow(2.0*((x<0.5)?x:1.0-x), k); return (x<0.5)?a:1.0-a; } k<1 on the left, k>1 on the right

### Parabola

A nice choice to remap the 0..1 interval into 0..1, such that the corners are mapped to 0 and the center to 1. In other words, parabola(0) = parabola(1) = 0, and parabola(1/2) = 1.

float parabola( float x, float k ) { return pow( 4.0*x*(1.0-x), k ); } ### Power curve

This is a generalziation of the Parabola() above. It also maps the 0..1 interval into 0..1 by keeping the corners mapped to 0. But in this generalziation you can control the shape one either side of the curve, which comes handy when creating leaves, eyes, and many other interesting shapes.

float pcurve( float x, float a, float b ) { const float k = pow(a+b,a+b) / (pow(a,a)*pow(b,b)); return k * pow( x, a ) * pow( 1.0-x, b ); }

Note that k is chosen such that pcurve() reaches exactly 1 at its maximum for illustration purposes, but in many applications the curve needs to be scaled anyways so the slow computation of k can be simply avoided. ### Sinc curve

A phase shifted sinc curve can be useful if it starts at zero and ends at zero, for some bouncing behaviors (suggested by Hubert-Jan). Give k different integer values to tweak the amount of bounces. It peaks at 1.0, but that take negative values, which can make it unusable in some applications.

float sinc( float x, float k ) { const float a = PI*((k*x-1.0); return sin(a)/a; } 