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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]]

Latest revision as of 05:40, 4 January 2015

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Yet, overall, less than 1% of websites presently have mobile versions of their websites. As a website owner, you can easily manage CMS-based website in a pretty easy and convenient style. Offshore Wordpress development services from a legitimate source caters dedicated and professional services assistance with very simplified yet technically effective development and designing techniques from experienced professional Wordpress developer India. This is because of the customization that works as a keystone for a SEO friendly blogging portal website. As with a terminology, there are many methods to understand how to use the terminology.

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