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[[File:Smoothstep and Smootherstep.svg|thumb|A plot of the smoothstep(x) and smootherstep(x) functions.]]
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'''Smoothstep''' is a scalar [[interpolation]] function commonly used in [[computer graphics]]<ref>[http://msdn.microsoft.com/en-us/library/bb509658(VS.85).aspx Smoothstep at Microsoft Developer Network]</ref><ref>[http://www.opengl.org/registry/doc/GLSLangSpec.Full.1.40.05.pdf GLSL Language Specification, Version 1.40]</ref> and [[Game engine|video game engines]].<ref>[http://unity3d.com/support/documentation/ScriptReference/Mathf.SmoothStep.html Unity game engine SmoothStep documentation]</ref> The function interpolates [[Smooth function|smoothly]] between two input values based on a third one that should be between the first two. The returned value is clamped between 0 and 1.
 
The slope of the smoothstep function tends toward zero at both edges.  This makes it easy to create a sequence of transitions using smoothstep to interpolate each segment rather than using a more sophisticated or expensive interpolation technique.
 
As pointed out in [[Microsoft Developer Network|MSDN]] and [[OpenGL]] documentation, smoothstep implements cubic [[Hermite interpolation]] after doing a clamp:
 
:<math> \operatorname{smoothstep}(t) = 3t^2 - 2t^3 </math>
 
An example implementation provided by AMD<ref>[http://ati.amd.com/developer/SIGGRAPH03/ATI_HardwareShading_SIGGRAPH2003.pps ATI R3x0 Pixel Shaders]</ref> follows.
 
<source lang="c">
float smoothstep(float edge0, float edge1, float x)
{
    // Scale, bias and saturate x to 0..1 range
    x = saturate((x - edge0)/(edge1 - edge0));
    // Evaluate polynomial
    return x*x*(3 - 2*x);
}
</source>
 
== Variations ==
[[Ken Perlin]] suggests<ref>[http://www.amazon.com/Texturing-Modeling-Third-Procedural-Approach/dp/1558608486 Texturing and Modeling, Third Edition: A Procedural Approach]</ref> an improved version of the smoothstep function which has zero 1st and 2nd order derivatives at t=0 and t=1:
 
:<math> \operatorname{smootherstep}(t) = 6t^5 - 15t^4 + 10t^3 </math>
 
Reference implementation:
<source lang="c">
float smootherstep(float edge0, float edge1, float x)
{
    // Scale, and clamp x to 0..1 range
    x = clamp((x - edge0)/(edge1 - edge0), 0.0, 1.0);
    // Evaluate polynomial
    return x*x*x*(x*(x*6 - 15) + 10);
}
</source>
 
== Origin ==
 
=== 3rd order equation ===
 
We start with a generic third order [[polynomial]] function and its first [[derivative]]:
:<math>\begin{alignat}{9}
f(t)  &&\; = \;&&  a_3 t^3 &&\; + \;&& a_2 t^2 &&\; + \;&& a_1 t &&\; + \;&& a_0 & \\
f'(t) &&\; = \;&& 3 a_3 t^2 &&\; + \;&& 2 a_2 t &&\; + \;&& a_1 &
\end{alignat}</math>
 
Applying the desired values for the function at both endpoints we get:
:<math>\begin{alignat}{13}
f(0) &&\; = \;&& 0 \;\;\;\;\;&& \Rightarrow &&\;\;\;\;\;  0 \;&& + &&\;  0 \;&& + &&\;  0 \;&& + &&\; a_0 &&\; = \;&& 0 & \\
f(1) &&\; = \;&& 1 \;\;\;\;\;&& \Rightarrow &&\;\;\;\;\; a_3 \;&& + &&\; a_2 \;&& + &&\; a_1 \;&& + &&\; a_0 &&\; = \;&& 1 &
\end{alignat}</math>
 
Applying the desired values for the first derivative of the function at both endpoints we get:
:<math>\begin{alignat}{11}
f'(0) &&\; = \;&& 0 \;\;\;\;&& \Rightarrow &&\;\;\;\;    0 \;&& + &&\;    0 \;&& + &&\; a_1 \;&& = \;&& 0 & \\
f'(1) &&\; = \;&& 0 \;\;\;\;&& \Rightarrow &&\;\;\;\; 3 a_3 \;&& + &&\; 2 a_2 \;&& + &&\; a_1 \;&& = \;&& 0 &
\end{alignat}</math>
 
Solving the system of 4 unknowns formed by the last 4 equations we obtain the values of the polynomial coefficients:
:<math>a_0 = 0 , \;\;\;\;\;\; a_1 = 0 , \;\;\;\;\;\; a_2 = 3 , \;\;\;\;\;\; a_3 = -2</math>
 
Introducing these coefficients back into the first equation gives the third order smoothstep function:
:<math>f(t) = -2t^3 + 3t^2</math>
 
=== 5th order equation ===
 
We start with a generic fifth order [[polynomial]] function, its first derivative and its second derivative:
:<math>\begin{alignat}{13}
f(t)  &&\; = \;&&    a_5 t^5 &&\; + \;&&    a_4 t^4 &&\; + \;&&  a_3 t^3 &&\; + \;&&  a_2 t^2 &&\; + \;&& a_1 t &&\; + \;&& a_0 & \\
f'(t)  &&\; = \;&&  5 a_5 t^4 &&\; + \;&&  4 a_4 t^3 &&\; + \;&& 3 a_3 t^2 &&\; + \;&& 2 a_2 t  &&\; + \;&& a_1  & \\
f''(t) &&\; = \;&& 20 a_5 t^3 &&\; + \;&& 12 a_4 t^2 &&\; + \;&& 6 a_3 t  &&\; + \;&& 2 a_2    &
\end{alignat}</math>
 
Applying the desired values for the function at both endpoints we get:
:<math>\begin{alignat}{17}
f(0) &&\; = \;&& 0 \;\;\;\;\;&& \Rightarrow &&\;\;\;\;\;  0 \;&& + &&\;  0 \;&& + &&\;  0 \;&& + &&\;  0 \;&& + &&\;  0 \;&& + &&\; a_0 &&\; = \;&& 0 & \\
f(1) &&\; = \;&& 1 \;\;\;\;\;&& \Rightarrow &&\;\;\;\;\; a_5 \;&& + &&\; a_4 \;&& + &&\; a_3 \;&& + &&\; a_2 \;&& + &&\; a_1 \;&& + &&\; a_0 &&\; = \;&& 1 &
\end{alignat}</math>
 
Applying the desired values for the first derivative of the function at both endpoints we get:
:<math>\begin{alignat}{15}
f'(0) &&\; = \;&& 0 \;\;\;\;&& \Rightarrow &&\;\;\;\;    0 \;&& + &&\;    0 \;&& + &&\;    0 \;&& + &&\;    0 \;&& + &&\; a_1 \;&& = \;&& 0 & \\
f'(1) &&\; = \;&& 0 \;\;\;\;&& \Rightarrow &&\;\;\;\; 5 a_5 \;&& + &&\; 4 a_4 \;&& + &&\; 3 a_3 \;&& + &&\; 2 a_2 \;&& + &&\; a_1 \;&& = \;&& 0 &
\end{alignat}</math>
 
Applying the desired values for the second derivative of the function at both endpoints we get:
:<math>\begin{alignat}{15}
f''(0) &&\; = \;&& 0 \;\;\;\;&& \Rightarrow &&\;\;\;\;      0 \;&& + &&\;      0 \;&& + &&\;    0 \;&& + &&\; 2 a_2 \;&& = \;&& 0 & \\
f''(1) &&\; = \;&& 0 \;\;\;\;&& \Rightarrow &&\;\;\;\; 20 a_5 \;&& + &&\; 12 a_4 \;&& + &&\; 6 a_3 \;&& + &&\; 2 a_2 \;&& = \;&& 0 &
\end{alignat}</math>
 
Solving the system of 6 unknowns formed by the last 6 equations we obtain the values of the polynomial coefficients:
:<math>a_0 = 0 , \;\;\;\;\;\; a_1 = 0 , \;\;\;\;\;\; a_2 = 0 , \;\;\;\;\;\; a_3 = 10 , \;\;\;\;\;\; a_4 = -15 , \;\;\;\;\;\; a_5 = 6</math>
 
Introducing these coefficients back into the first equation gives the fifth order smoothstep function:
:<math>f(t) = 6t^5 - 15t^4 + 10t^3</math>
 
== References ==
{{reflist}}
 
== External links ==
* [http://www.fundza.com/rman_shaders/smoothstep/index.html Using smoothstep] (in the [[RenderMan Shading Language]]) by Prof. Malcolm Kesson.
* [http://sol.gfxile.net/interpolation/ Interpolation tricks] by Jari Komppa
 
[[Category:Computer graphics algorithms]]

Revision as of 11:40, 26 February 2014

Next - GEN Gallery is a full incorporated Image Gallery plugin for Word - Press which has a Flash slideshow option. This one is one of the most beneficial features of Word - Press as this feature allows users to define the user roles. PSD files are incompatible to browsers and are suppose to be converted into wordpress compatible files so that it opens up in browser. After confirming the account, login with your username and password at Ad - Mob. It is found that most of the visitors only look for the results that are displayed on the first page of the search engines and so if you get the service from professional service providers then they strive for the first page ranking of your site and improve the online visibility.

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