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Spherical Harmonics: Difference between revisions
→Definition
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\end{cases}</math> | \end{cases}</math> | ||
===Visualizations | ==Properties== | ||
Copied from Green<ref name="stupidsh">Robin Green (2003). ''Spherical Harmonic Lighting'' URL: [http://silviojemma.com/public/papers/lighting/spherical-harmonic-lighting.pdf http://silviojemma.com/public/papers/lighting/spherical-harmonic-lighting.pdf]</ref> | |||
* Orthonormal | |||
* Rotationally Invariant | |||
* Integration of two spherical harmonic functions is a dot product of their coefficients | |||
{{hidden | Proof| | |||
This follows from being orthonormal:<br> | |||
<math> | |||
\begin{align} | |||
\int_{S}\tilde{L}(s)\tilde{t}(s)ds &= \int_{S}\left(\sum_i L_i y_i(s)\right)\left(\sum_j t_j y_j(s)\right)ds \\ | |||
&= \sum_i\sum_j L_i t_j \int_{S} y_i(s) y_j(s)ds \\ | |||
&= \sum_i^{n^2} L_i t_i \int_{S}( y_i(s) y_i(s))ds\\ | |||
&\qquad\text{ because orthogonal}\\ | |||
&= \sum_i^{n^2} L_i t_i\\ | |||
&\qquad\text{ because orthonormal} | |||
\end{align} | |||
</math> | |||
}} | |||
==Visualizations== | |||
Below are distorted sphere visualizations where the radius corresponds to the value at each point. | Below are distorted sphere visualizations where the radius corresponds to the value at each point. | ||
* [https://www.shadertoy.com/view/lsfXWH iq's 0-3] | * [https://www.shadertoy.com/view/lsfXWH iq's 0-3] | ||