wave

modulus, a.k.a. mod, %, modulo

the modulus operator A % B gives the remainder after every whole B has been removed from A.

10 % 3 = 1
   (10 - 3 = 7)
   ( 7 - 3 = 4)
   ( 4 - 3 = 1)
   ( 1 < 3, so stop)

mod is very handy for range control. it allows you to chop up one big range into a repeating series of little ranges. when you use the % operator, say x % y, you are mapping values from the range [0, x] to the range [0, y-1]. for example, if y = 5:

x     : 0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17..
x % y : 0  1  2  3  4  0  1  2  3  4  0  1  2  3  4  0  1  2..

periodic functions

ranges of values that repeat are called periodic. functions that generate repeating value ranges are called periodic functions. periodic functions are usually designed to return values in a convenient range, like [-1, 1], or [0, 1].

terminology

a graph of the values generated by a periodic function is called a waveform.

waveforms have several properties described by the following terms:

period: the interval between repeats
frequency: a count of how many repeats occur during some interval. hertz is the count of repeats per second.
amplitude: the distance to crest or valley of the wave; the largest absolute value generated by the function
phase: the position of a feature of the waveform relative to the same feature on another waveform of the same amplitude and frequency. when waveforms are in phase, their features line up. phase shift is a difference in phase, a translation of the graph of the waveform.
- in phase [swf] [fla]
- out of phase [swf] [fla]

structure

for a given periodic function f(t), the following general structure outlines how to control aspects of the waveform:

result = amplitude * f( frequency * t + phaseShift );

example waveforms

square

sawtooth

triangle

pulse

sinusoid

noise

engines

the input to a periodic function is a continually increasing value, like time. so we can think of periodic functions as convenient engines—with time values as fuel, they can drive properties like position, opacity, orientation, etc.

examples:

square wave—blink [swf] [fla]
sawtooth wave—wrap [swf] [fla]
pulse wave—bounce [swf] [fla]
sine+cosine wave—orbit [swf] [fla]
noise wave—flicker [swf] [fla]