Notes on 1D Nonlinear Transformations for Games

My notes from Squirrel Eiserloh’s presentation on 1D nonlinear transformations for game development.

Author

Christian Mills

Published

December 29, 2021

Overview

Here are some notes I took while watching Squirrel Eiserloh’s presentation covering how 1D nonlinear transformations can be used by game programmers.

Motivations

Juice it or lose it talk: Makes the case for thinking about is something linear or non-linear, is it mechanical or organic
The art of screenshake:

Implicit versus Parametric Equations

Implicit equations are rules:
- Equation for a circle: $x^{2} + Y^{2} = 25$
  - A point is either on the circle or not
Parametric functions
- Yield an output for an input value
  - $P_{x} = 5 \cdot c o s (2 π \cdot t)$
  - $P_{y} = 5 \cdot s i n (2 π \cdot t)$
- $P (t) = ?$
- $P (t) = (t, t \cdot c o s (t), t * s i n (t))$
  - $(x, y, z)$
  - Generates a spiral that increases in radius along the x axis
- Anything you can express in terms of a single float as input
- A common float input is “time”

Parametric Manipulations

Do NOT mess with the interpolation itself (e.g. color, position, AI disposition, etc.)
Instead just mess the parameter

Parametric Opportunities

Anytime you have a single float to change
Anytime you can express something in terms of a single float
Pretty much whenever you use time

The Big Idea

You can make any parametric equation more interesting without modifying the function itself, without knowing anything about the function

The Two Most Important Number Ranges

$[0, 1]$
- Useful for fractions
  - % shadow
  - % luminance
  - % falloff
  - % complete
  - % damage
  - % experience
  - % cost
  - % penalty
  - % fog
  - % AI aggression
  - % chance to hit
  - % chance to drop loot
  - % time to complete
  - Fuzzy Logic
  - Most anything parametric
$[- 1, 1]$
- Useful for deviations
  - noise
  - perturbation
  - terrain and map generation
  - variation
  - distribution
  - sinusoidal
  - AI response curves

Normalized Non-Linear Functions

$[0, 1]$
Functions for which:
- $P (0) = 0$
- $P (1) = 1$
- $P (t)! = t$
Examples
- Position over time
- Scale over time
- Alpha over time
- Color over time
- Strength over time
- Aggression over time
Also called
- easing functions
- filter functions
- lerping functions
- tweening functions

Range Mapping

can be applied during middle of range-mapping

out RangeMap(in, inStart, inEnd, outStart, outEnd)
{
    // Puts in [0, inEnd - inStart]
    out = in - inStart;
    // Puts in [0,1]
    out /= (inEnd - inStart);
    // in [0,1]
    out = ApplySomeEasingFunction(out);
    // Puts in [0, outRange]
    out *= (outEnd - outStart);
    // Puts in [outStart, outEnd]
    return out + outStart
}

SmoothStart

$S m o o t h S t a r t N (t) = t^{n}$
Larger exponents result in steeper curve
Will always start and end at the same time, regardless of exponent value
Technique
- exponentiating

SmoothStop

$S m o o t h S t o p N (t) = 1 - (1 - t)^{n}$
Larger exponents results in longer braking period at the end
Techniques
- exponentiating
- flipping

$M i x (a, b, w e i g h t B, t) = a + w e i g h t B (b - a)$

$M i x (S m o o t h S t a r t 2, S m o o t h S t o p 2, b l e n d, t)$
$S m o o t h S t a r t 2.2 = M i x (S m o o t h S t a r t 2, S m o o t h S t a r t 3, 0.2);$
- Way faster than using the pow() function

Crossfade

Like Mix, but use t itself as the mix weight
Also called SmoothStep

Scale

$S c a l e (F u n c t i o n, t) = t \cdot F u n c t i o n (t)$

ReverseScale

$R e v e r s e S c a l e (F u n c t i o n, t) = (1 - t) \cdot F u n c t i o n (t)$

$A r c h 2 (t) = S c a l e (F l i p (t)) = t \cdot (1 - t)$

$S m o o t h S t a r t A r c h 3 (t) = S c a l e (A r c h 2, t) = t^{2} (1 - t)$

$S m o o t h S t o p A r c h 3 (t) = R e v e r s e S c a l e (A r c h 2, t) = t (1 - t)^{2}$

$S m o o t h S t e p A r c h 3 (t) = R e v e r s e S c a l e (S c a l e (A r c h 2, t), t)$

$B e l l C u r v e 6 (t) = S m o o t h S t o p 3 (t) \cdot S m o o t h S t a r t 3 (t)$