Mean Value - Idea in 2D

Look for an interpolant.

• Given a 2D boundary curve \(b(\theta)\), with values \(f(\theta)\)
• Mean-value associated to point \(p\):
• • \( \hat f(p)=\frac{\displaystyle \int_{\theta=0}^{2\pi} \kappa(\theta)\,f(\theta)\,\mathrm{d}\theta }{\displaystyle \int_{\theta=0}^{2\pi} \kappa(\theta)\,\mathrm{d}\theta} \,\), \(\;\kappa(\theta)=1/\|b(\theta)-p\|\)
  • Averaged value projected around a unit circle, weighted by the inverse distance.

• For a piecewise linear boundary
• • Over the edge \(p_i \, p_{i+1}\)
  • \(\displaystyle \int_{\theta=\theta_1}^{\theta=\theta_2} \kappa(\theta)\,f(\theta) \mathrm{d}\theta= \dots =\left(\frac{f(p_i)}{\|p_i-p\|}+\frac{f(p_{i+1})}{\|p_{i+1}-p\|}\right)\,{tan}\left(\frac{\theta_2-\theta_1}{2}\right)\)

• • \(\Rightarrow \displaystyle \omega_i = \frac{\tan(\alpha_{i-1}/2)+\tan(\alpha_i/2)}{\|p_i-p\|} \)