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

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
      • $P_{x} = 5 \cdot cos(2 \pi \cdot t)$
      • $P_{y} = 5 \cdot sin(2 \pi \cdot t)$
    • $P(t) = ?$
    • $P(t) = (t, t \cdot cos(t), t*sin(t))$
      • $(x, y, z)$
      • 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

  • $SmoothStartN(t) = t^{n}$
  • Larger exponents result in steeper curve
  • Will always start and end at the same time, regardless of exponent value
  • Technique
    • exponentiating

SmoothStop

  • $SmoothStopN(t) = 1 - (1 - t)^{n}$
  • Larger exponents results in longer braking period at the end
  • Techniques
    • exponentiating
    • flipping

$Mix(a, b, weightB, t)= a + weightB(b-a)$

  • $Mix(SmoothStart2, SmoothStop2, blend, t)$
  • $SmoothStart2.2 = Mix(SmoothStart2, SmoothStart3, 0.2);$
    • Way faster than using the pow() function

Crossfade

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

Scale

  • $Scale(Function, t) = t \cdot Function(t)$

ReverseScale

  • $ReverseScale(Function, t) = (1-t) \cdot Function(t)$

$Arch2(t) = Scale(Flip(t)) = t \cdot (1-t)$

$SmoothStartArch3(t) = Scale(Arch2, t) = t^{2}(1-t)$

$SmoothStopArch3(t) = ReverseScale(Arch2, t) = t(1-t)^{2}$

$SmoothStepArch3(t) = ReverseScale(Scale(Arch2, t), t)$

$BellCurve6(t) = SmoothStop3(t) \cdot SmoothStart3(t)$

Juice it or lose it - a talk by Martin Jonasson & Petri Purho

The Art of Screenshake - Jan Willem Nijman - Vlambeer

References: