• waveform
• data types
• functions
  • shaping functions shaping signals in the normalized range [0, 1]
  • Easing functions specify the rate of change of a parameter over time
  • Tweening functions: The purpose of a tweening function is to deliver a position for a specific time, given the tween’s essential characteristics. These characteristics are beginning position, final position, and duration. It is used in Flash for describe time related motion
  • 1D gray color gradient
• Pattern Master
• Unit generators
• Interpolation curves or interpolations
• Parametric acceleration curves for animations

Names or nomenclature comparison

All functions work in the range [0..1].

• take input function f which maps floating point number (x or position) in a normalised range [0.0, 1.0] and gives monotone output ( A monotonically non-decreasing function )
• make function diagram and color gradient ( image of continous color gradient) using above function
• make 3 types of periodic waves for each function type using ModifyPosition function
• make color diagram and color gradient for each GradientJoiningType of input function

typedef enum {no, steps, tubes } GradientJoiningType; 
double ModifyPosition(const double position, const GradientJoiningType GradientJoining){
	
	// input position should be in [0,1] range 
	double p = position; // p = local copy of position
	// if position > 1 then we have repetition of colors = periodic function = wave 
	
	
	switch(GradientJoining){
	
		case no : {break;} // return input position witout modifications
		
		// periodic waves with different joinings
		case steps : {	p = p * segments; // periodic = change range
				p = frac(p); 
 				break;}
 				
		case tubes : {	p = p * segments; // periodic = change range
				int ip = (int)p;
 				p = p-ip; // fractional part 
 				if (ip % 2) {p = 1.0-p;} // reverse gradient
				break;}
		default:{}
	}
	
	return p; // output in [0,1] range
}

So important variables:

• coordinate ( integer to float)
• position ( float )
• color (RGB) : float to integer in [0 , 255] range

Linear function graph = made using continous_data function

Step function graph. Made using discrete_data function

Sawtooth wave graph. Made using sawtooth_data function which includes steps type of joining

Sine wave

Piecewise linear functions: Boxcar function or square waveform or Rectangular function, the simplest step_function. It is made using square_wave function

Triangle wave = made from linear function using tubes type of joining

Linear Linear_no

Linear_steps

Linear_tubes

Quadratic Quadratic_no

Quadratic_steps

Quadratic_tubes

Cubic Cubic_no

Cubic_steps

Cubic_tubes

CubicInv CubicInv_no

CubicInv_steps

CubicInv_tubes

Sqrt Sqrt_no

Sqrt_steps

Sqrt_tubes

Root Root_no

Root_steps

Root_tubes

Gamma Gamma_no

Gamma_steps

Gamma_tubes

LSin LSin_no

effect of a sine wave superimposed on a ramp function

It is based on the The Colour Map Test Image by Peter Kovesi

LSin_steps

LSin_tubes

SinExp SinExp_no

SinExp_steps

SinExp_tubes

Sin Sin_no

Sin_steps

Sin_tubes

Smooth Smooth_no

Smooth_steps

Smooth_tubes

Tanh Tanh_no

Tanh_steps

Tanh_tubes

smoothstep_Cubic_Polynomial smoothstep_Cubic_Polynomial_no

smoothstep_Cubic_Polynomial_steps

smoothstep_Cubic_Polynomial_tubes

inv_smoothstep_Cubic_Polynomial inv_smoothstep_Cubic_Polynomial_no

inv_smoothstep_Cubic_Polynomial_steps

inv_smoothstep_Cubic_Polynomial_tubes

smoothstep_Quartic_Polynomial smoothstep_Quartic_Polynomial_no

smoothstep_Quartic_Polynomial_steps

smoothstep_Quartic_Polynomial_tubes

inv_smoothstep_Quartic_Polynomial inv_smoothstep_Quartic_Polynomial_no

inv_smoothstep_Quartic_Polynomial_steps

inv_smoothstep_Quartic_Polynomial_tubes

smoothstep_Quintic_Polynomial smoothstep_Quintic_Polynomial_no

smoothstep_Quintic_Polynomial_steps

smoothstep_Quintic_Polynomial_tubes

smoothstep_Quadratic_Rational smoothstep_Quadratic_Rational_no

smoothstep_Quadratic_Rational_steps

smoothstep_Quadratic_Rational_tubes

inv_smoothstep_Quadratic_Rational inv_smoothstep_Quadratic_Rational_no

inv_smoothstep_Quadratic_Rational_steps

inv_smoothstep_Quadratic_Rational_tubes

smoothstep_Cubic_Rational smoothstep_Cubic_Rational_no

smoothstep_Cubic_Rational_steps

smoothstep_Cubic_Rational_tubes

inv_smoothstep_Cubic_Rational inv_smoothstep_Cubic_Rational_no

inv_smoothstep_Cubic_Rational_steps

inv_smoothstep_Cubic_Rational_tubes

smoothstep_Rational smoothstep_Rational_no

smoothstep_Rational_steps

smoothstep_Rational_tubes

inv_smoothstep_Rational inv_smoothstep_Rational_no

inv_smoothstep_Rational_steps

inv_smoothstep_Rational_tubes

smoothstep_Piecewise_Quadratic smoothstep_Piecewise_Quadratic_no

smoothstep_Piecewise_Quadratic_steps

smoothstep_Piecewise_Quadratic_tubes

inv_smoothstep_Piecewise_Quadratic inv_smoothstep_Piecewise_Quadratic_no

inv_smoothstep_Piecewise_Quadratic_steps

inv_smoothstep_Piecewise_Quadratic_tubes

smoothstep_Piecewise_Polynomial smoothstep_Piecewise_Polynomial_no

smoothstep_Piecewise_Polynomial_steps

smoothstep_Piecewise_Polynomial_tubes

inv_smoothstep_Piecewise_Polynomial inv_smoothstep_Piecewise_Polynomial_no

inv_smoothstep_Piecewise_Polynomial_steps

inv_smoothstep_Piecewise_Polynomial_tubes

smoothstep_Trigonometric smoothstep_Trigonometric_no

smoothstep_Trigonometric_steps

smoothstep_Trigonometric_tubes

inv_smoothstep_Trigonometric inv_smoothstep_Trigonometric_no

inv_smoothstep_Trigonometric_steps

inv_smoothstep_Trigonometric_tubes

almostIdentity almostIdentity_no

almostIdentity_steps

almostIdentity_tubes

almostIdentity2 almostIdentity2_no

almostIdentity2_steps

almostIdentity2_tubes

almostUnitIdentity almostUnitIdentity_no

almostUnitIdentity_steps

almostUnitIdentity_tubes

integralSmoothstep integralSmoothstep_no

integralSmoothstep_steps

integralSmoothstep_tubes

expImpulse expImpulse_no

expImpulse_steps

expImpulse_tubes

quaImpulse quaImpulse_no

quaImpulse_steps

quaImpulse_tubes

polyImpulse polyImpulse_no

polyImpulse_steps

polyImpulse_tubes

expSustainedImpulse expSustainedImpulse_no

expSustainedImpulse_steps

expSustainedImpulse_tubes

cubicPulse cubicPulse_no

cubicPulse_steps

cubicPulse_tubes

expStep expStep_no

expStep_steps

expStep_tubes

gain gain_no

gain_steps

gain_tubes

parabola parabola_no

parabola_steps

parabola_tubes

pcurve pcurve_no

pcurve_steps

pcurve_tubes

sinc sinc_no

sinc_steps

sinc_tubes

trunc_fallof trunc_fallof_no

trunc_fallof_steps

trunc_fallof_tubes

MRE = minimal reproducible ( working) example = short simple code

code in ./src/mwe/ directory

One file program that makes all images

• d.c
• results of the program are in the text files (./src/*.txt)

Compile and run the program

gcc d.c -Wall -Wextra -lm
a.out > c.txt
gnuplot
plot "c.txt" with lines 
# save image as a c.png

code in ./src/g/ directory

One file program that makes all images. To run it:

make

• add new functions named: Give_s
• add new enum named: s in ColorTransferFunctionType
• use s.c program to compute code for c functions (GiveRGB_Gray and GiveColor) and for gnuplot
• copy code

// 
double Give_s(const double position){
	
	
 double s = position;
 return s;
}

S directory contains c program for creating code for plot.gp and g.c

curves by kynd.inf

Screen shot from online WebGL demo

available animation transition types: (the default transition type, 'linear', was omitted) from the wiki of the Sparrow Framework — the Open Source Game Engine for iOS.

• jQuery easing; illustrated by James Padolsey
• online WebGL demo
• larsenwork: easing-gradients/ Linear gradients often have hard edges where they start and/or end. We can avoid those by controlling the color mix with easing functions.
• FlexMonkey: Interpolation-Playground-
• Easing Functions Cheat Sheet

• 1D-RGB-color-gradient
• Golan Levin: Pattern_Master for Processing
• p5.js-func: Function Generators for p5.js
• Shaping functions in GLSL by Patricio Gonzalez Vivo & Jen Lowe
• curves by kynd.inf

Easing functions

• each easing function has 3 variants: ease-in, ease-out, and ease-in-out
• Easing Equations by @nicmulvaney
• EASING EQUATIONS by Robert Penner - github
• EASING EQUATIONS by Robert Penner - www
• Tweening chapter of Robert Penner's book: Programming Macromedia Flash MX
• AHEasing: A supplemental library of easing functions for C, C++, and Objective-C by Warren Moore
• Parametric acceleration curves in Core Animation, September 9, 2008 by Matt Gallagher
• stackoverflow question: how-to-create-custom-easing-function-with-core-animation
• Robert Penner's easing equations converted to coffeescript. by Jim Jeffers
• Understanding Easing (Explaining Penner’s equations) – JavaScript and ActionScript
• easing-functions in CSS
• easy.c

by Iñigo Quilez

• GraphToy = a tool to visualize GLSL functions in WebGL, only function of 2 variables: f(x,t)
• useful little functions
• smoothsteps/
• smoothstep integral
• mega polynomial curve

Smoothstep

• wikipedia
• Smoothstep - Shader Graph Basics - Episode 15 by Ben Cloward
• smoothstep-looking-for-a-continuous-family-of-interpolation-functions
• RSL: Using smoothstep
• interpolations tricks or How I Learned to Stop Worrying and Love the 0..1 Range by Jari Komppa

Shaping functions

• shape.c

Polynomial Shaping Functions: This page presents a collection of polynomial functions for shaping, tweening, and easing signals in the range [0...1]. Functions include:

• Blinn-Wyvill Approximation to the Raised Inverted Cosine
• Double-Cubic Seat
• Double-Cubic Seat with Linear Blend
• Double-Odd-Polynomial Seat
• Symmetric Double-Polynomial Sigmoids
• Quadratic Through a Given Point

Exponential Shaping Functions

• Exponential Ease-In and Ease-Out
• Double-Exponential Seat
• Double-Exponential Sigmoid
• The Logistic Sigmoid

Circular & Elliptical Shaping Functions

• Circular Interpolation: Ease-In and Ease-Out
• Double-Circle Seat
• Double-Circle Sigmoid
• Double-Elliptic Seat
• Double-Elliptic Sigmoid
• Double-Linear with Circular Fillet
• Circular Arc Through a Given Points

Bezier and Other Parametric Shaping Functions

• Quadratic Bezier
• Cubic Bezier
• Cubic Bezier (Nearly) Through Two Given Points

List of interpolations

• Linear Interpolation
• Smooth Step
• Smoother Step
• Smoothest Step
• Squared
• Inverse Squared
• Cubed
• Sin
• Catmull-Rom
• Elastic In
• Elastic Out
• Wobble
• Gaussian

This work is a spiritual descendent (not to say derivative work) of works done by the following individuals:

• Robert Penner: Created the first flexible, reusable Easing functions ...which are the cornerstone of all JS animation techniques today
• Golan Levin
• Warren Moore
• George McGinley Smith
• James Padolsey
• Authors of jQuery
• Matt Gallagher
• Jesse Crossen
• Iñigo Quilez
• Andrey Sitnik
• Jim Jeffers

• lerp = linear interpolation
• Easing functions specify the rate of change of a parameter over time.
• Exponential decay
• Exponential growth: Exponential growth is the inverse of logarithmic growth
• Logarithmic growth: Logarithmic growth is the inverse of exponential growth and is very slow
• linear growth ( linear function): y = f(x) ; proportional or directly proportional if their corresponding elements have a constant ratio
• Quadratic_growth
• Hyperbolic_growth: If the output of a function is inversely proportional to its input, or inversely proportional to the difference from a given value x0, the function will exhibit hyperbolic growth, with a singularity at x0
• Big O notation, see Orders of common functions

git add README.md
git commit -m "first commit"
git branch -M main
git remote add origin git@github.com:adammaj1/Waveform.git
git push -u origin main

mkdir images
git add *.png
git mv *.png ./png
git commit -m "move"
git push -u origin main

then link the images:

![](./png/n.png "description of n.png image") 

to overwrite

git mv -f 

~/Dokumenty/c/Waveform

File LICENSE must be in main repo directory to be read by github

Github

• GitHub Flavored Markdown Spec
• md cheat sheet
• CommonMark Spec
• Markdown parser

Math equation inside README.md

• $\sqrt{3x-1}+(1+x)^2$