Function Repository Resource:

BiasedRescale

Source Notebook

Bias a value in the unit interval either lower or higher

Contributed by: Jan Mangaldan

ResourceFunction["BiasedRescale"][a,x]

biases x toward lower values if a<1/2 and toward higher values if a>1/2 as x runs from 0 to 1.

ResourceFunction["BiasedRescale"][a,x,{min,max}]

biases x as x runs from min to max.

ResourceFunction["BiasedRescale"][a,x,{min,max},{y_min,y_max}]

biases x as x runs from min to max, with the result rescaled to run from y_min to y_max.

Details and Options

a can be any real number between 0 and 1.

The result of ResourceFunction["BiasedRescale"][a,x,{min,max}] is always between 0 and 1.

ResourceFunction["BiasedRescale"] can take the Method option for the bias function to use.

Possible settings for the Method option are "Schlick" (default) and "Perlin".

With the default option setting Method→"Schlick", ResourceFunction["BiasedRescale"] uses Schlick's bias function x/((1/a-2)(1-x)+1).

With Method→"Perlin", ResourceFunction["BiasedRescale"] uses Perlin's bias function x^-log₂(a).

ResourceFunction["BiasedRescale"] is commonly used in computer graphics.

Examples

Basic Examples (3)

Evaluate numerically:

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Plot over a subset of the reals:

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Plot different bias functions:

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Scope (4)

Evaluate for symbolic x:

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BiasedRescale threads over lists in its first and second arguments:

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Plot different bias functions with a rescaled domain:

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Plot different bias functions with a rescaled domain and range:

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Options (1)

Method (1)

Compare the Schlick and Perlin bias functions:

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Applications (2)

Adjust the middle color of a built-in color gradient:

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Demonstrate the effect of BiasedRescale on easing in/easing out:

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Manipulate[
Graphics[Text[
"Zoom!", {ResourceFunction["BiasedRescale"][a, x, {-1, 2}], 0}], PlotRange -> {{-0.1, 1.1}, {-0.1, 0.1}}], {{a, 1/2}, 0, 1}, {x, -1, 2, ControlType -> Animator, AnimationRunning -> False}]

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Properties and Relations (1)

BiasedRescale[1/2,x,{min,max}] is equivalent to Rescale[x,{min,max}]:

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Possible Issues (2)

BiasedRescale is left unevaluated if the first argument is non-numeric:

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BiasedRescale is left unevaluated if the first argument is not a real number between 0 and 1:

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Neat Examples (6)

Use BiasedRescale with the resource function GaussianQuadratureWeights to evaluate a Cauchy principal value integral.

The function to be integrated and the integration limits:

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f[x_] := E^x/x;
xmin = -1.; xmax = 3.;

Even-order Gaussian quadrature abscissas and weights:

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The transformation function to be applied to the abscissas and weights:

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$tr[x_] = ResourceFunction["BiasedRescale"][Rescale[0, {xmin, xmax}], x, {xmin, xmax}, {xmin, xmax}];$

Transform the abscissas and weights:

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Compute the integral:

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Compare with the result of NIntegrate:

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Version History

1.0.1 – 16 February 2021
1.0.0 – 28 December 2020

Source Metadata

Citation:
- Perlin K., Hoffert E.M., Hypertexture. SIGGRAPH ‘89: Proceedings of the 16th annual conference on Computer graphics and interactive techniques, 253-262, 1989 DOI: 10.1145/74333.74359
- Schlick C., Fast alternatives to Perlin’s bias and gain functions. Graphics Gems IV, 401-403, 1994 DOI: 10.1016/B978-0-12-336156-1.50050-1

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License Information

This work is licensed under a Creative Commons Attribution 4.0 International License