Conformal mapping Conformal mappings locally correspond to similarities Figure 5.8.A conformal parameterization transforms a small circle into a small circle,i.e.,it is locally a similarity transform.(Image taken from Hormann et al.07].C2007 ACM,Inc.Included here by permission.)
Conformal mapping • Conformal mappings locally correspond to similarities
Mapping Build a local coordinate system on input triangle t. (xi,yi) (ui,Vi) The mapping is piecewise linear. Jtis2×2. (xjy) (Xkyk) uk,Vk) fi(x)=Jix+bt
Mapping • Build a local coordinate system on input triangle 𝑡. • The mapping is piecewise linear. • 𝐽𝑡 is 2 × 2. 𝑓𝑡(𝒙) = 𝐽𝑡𝒙 + 𝒃𝑡 𝑡 𝑇 (𝑥𝑖 , 𝑦𝑖) (𝑥𝑗 , 𝑦𝑗) (𝑥𝑘, 𝑦𝑘) (𝑢𝑖 , 𝑣𝑖) (𝑢𝑗 , 𝑣𝑗) (𝑢𝑘, 𝑣𝑘) 𝑢𝑗 − 𝑢𝑖 𝑢𝑘 − 𝑢𝑖 𝑣𝑗 − 𝑣𝑖 𝑣𝑘 − 𝑣𝑖 𝑥𝑗 − 𝑥𝑖 𝑥𝑘 − 𝑥𝑖 𝑦𝑗 − 𝑦𝑖 𝑦𝑘 − 𝑦𝑖 −1