Wrong derivation for SH trigonometric terms

[wumo]

As wrote in 9.6.1 Basis functions:

It’s also fairly easy to compute the trigonometric terms recursively:

$$ \begin{align*} C_m &\equiv cos(m \phi) \\ S_m &\equiv sin(m \phi) \\ { x, y, z } &= { cos \phi sin \theta, sin \phi sin \theta, cos \theta } \end{align*} $$

Using the angle sum trigonometric identities:

$$ \begin{align*} cos(m \phi + \phi) &= cos(m \phi) cos(\phi) - sin(m \phi) sin(\phi) \Leftrightarrow C_{m + 1} = \frac{(x C_m - y S_m)}{sin(\theta)^{|m + 1|}} \\ sin(m \phi + \phi) &= sin(m \phi) sin(\phi) + cos(m \phi) sin(\phi) \Leftrightarrow S_{m + 1} = \frac{(x S_m - y C_m)}{sin(\theta)^{|m + 1|}} \end{align*} $$

If we define $C_m \equiv cos(m \phi)$ and $S_m \equiv sin(m \phi)$, then the identity formula(which is also typed wrong in the doc):

$$ sin(m \phi + \phi) = sin(m \phi) cos(\phi) + cos(m \phi) sin(\phi) $$

should be $S_{m+1} = S_{m} cos(\phi) + C_{m} sin(\phi)$, not $S_{m + 1} = \frac{(x S_m - y C_m)}{sin(\theta)^{|m + 1|}}$.

The correct definition for this should be (according to Appendix A1 in Stupid Spherical Harmonics (SH) Tricks):

$$ \begin{align*} C_m &\equiv cos(m \phi) (sin\theta)^m\\ S_m &\equiv sin(m \phi) (sin\theta)^m\\ { x, y, z } &= { cos \phi sin \theta, sin \phi sin \theta, cos \theta } \end{align*} $$

Then the angle sum trigonometric identities will be:

$$ \begin{align*} &sin(m \phi + \phi) = sin(m \phi) cos(\phi) + cos(m \phi) sin(\phi) \\ &\Leftrightarrow \frac{S_{m + 1}}{(sin\theta)^{m+1}} = \frac{S_m}{(sin\theta)^m} cos\phi + \frac{C_m}{(sin\theta)^m} sin\phi \\ &\Leftrightarrow S_{m + 1} = S_m sin\theta cos\phi + C_m sin\theta sin\phi \\ &\Leftrightarrow S_{m + 1} = xS_m + yC_m \end{align*} $$

The same derivation can be applied to $cos(m\phi)$:

$$ \begin{align*} cos(m \phi + \phi) = cos(m \phi) cos(\phi) - sin(m \phi) sin(\phi) \Leftrightarrow C_{m + 1} = xC_m - yS_m \end{align*} $$

The result is:

$$ \begin{align*} &cos(m \phi )= \frac{C_m}{(sin\theta)^m} \\ &sin(m \phi ) = \frac{S_m}{(sin\theta)^m} \end{align*} $$

[romainguy]

@pixelflinger

[pixelflinger]

Thanks, I will review this.

[pixelflinger]

Good catch for the typo in the angle sum trigonometric identities.