a x φ ( x ) v$_{1}$ and v$_{2}$ Dφ ( φ ( x ))( v$_{1}$ ) space normal to v$_{2}$ v$_{1}$ b θ a v$_{1}$ b D φ ( φ (x))(v$_{1}$) v$_{2}$ θ x y φ (x) h ^ h Figure 2.4: A 2-dimensional illustration of the dilation calculation. Figure 2.3: Illustration of the dilation and nonorthogonal projection involved in retrac- tion for a 3-simplex. Then { v$_{1}$ , w$_{1}$ , ..., w$_{ℓ}$$_{−}$$_{2}$ , v$_{2}$ } is an orthogonal basis for R $^{ℓ}$, and φ$_{π}$ is given by φ$_{π}$ ( v$_{1}$ ) = α w $^{′}$, φ$_{π}$ ( w$_{i}$ ) = w$_{i}$ , i ∈ { 1 , ..., ℓ − 2 } , and φ$_{π}$ ( v$_{2}$ ) = 0 , (2.15) 33