Jacobians and Hessians of Mean Value Coordinates for Closed Triangular Meshes

As written in [2], a 3D point η can be expressed as a linear sum of the 3D positions p i of the vertices of a triangular mesh M by: η = i wi•pi i wi = i λ i • p i . For a point x onto the surface (a two-dimensional parameter), we note as usual φ i [x] the linear function on M that takes value 1 on vertex i and 0 on other vertices, and p[x] its 3D position. The definition of the weights λ i should guarantee linear precision (i.e. η = i λ i (η)p i ).

Since Bη(M ) p[x]-η |p[x]-η| dS η (x) = 0 (the integral of the unit outward normal onto the unit sphere is 0), we have

B η (M ) being the projection of the manifold M onto the unit sphere centered in η.

Writing that ∀x p

The weights λ i are given by

And the weights w i such that λ i = wi j wj are given by

This definition guarantees linear precision; it gives a linear interpolation of the function onto the triangles of the cage; and it extends it in a regular way to the entire 3D space.

Computing the weights w i : The support of the function φ i [x] is only composed of the adjacent triangles to the vertex i. Then, we can rewrite Eq. 4 as w i = T ∈N 1(i) w T i , with

Given a triangle T with vertices t 1 , t 2 , t 3 , we see that

This last integral is simply the integral of the unit outward normal on the spherical triangle T .

By noting 1), it can be easily expressed as

This comes from the fact that the integral of the unit outward normal on a closed surface is always 0.

Finally, we obtain

This point was discussed in [2]. As the authors pointed out, by noting A T the 3 by 3 matrix {p t1 -η, p t2 -η, p t3 -η}, we can derive the weights w T tj by We now present the derivatives of the Mean Value Coordinates. Deforming the cage mesh with f (p i ) = p i induces a deformation of the 3D space by f = i λ i • p i . In the rest of the document, for any function h : E → F , we note ∂ x h, ∂ y h, ∂ z h its derivative by x, y, and z, -→ h its gradient, Jh its jacobian, and Hh its hessian.

The deformation function f as defined acts now on R 3 entirely. The derivatives of f can be expressed as a linear sum of positions p i = {x i , y i , z i }:

Consequently, it allows to specify implicit equations on the cage in a linear system by giving specified rotations and scales on 3D locations, or to minimize the norm of the hessian to force rigidity, as done in the case of Green Coordinates in [1].

Since

We also have ∀c = x, y, z

or

From these expressions, we see that, in order to get -→ λ i (η) and Hλ i (η), we first need to obtain -→ w i (η) and Hw i (η) for each vertex i of the cage.

Mean Value Coordinates define an interpolation process. The function represented onto the vertices of the cage (in our case, a space transformation) is extended to the interior of the triangles with linear interpolation on each triangle. Then it is extended to the space by means of a surfacic integration of the function (see Eq. 1).

Since we represent the cage as triangle mesh in the 3D case, the deformation function cannot be anything more than continuous onto the edges of the cage in 3D. Therefore Jacobians and Hessians of the deformation cannot be evaluated everywhere on the surface of the cage, and we do not provide any formula for Jacobians and Hessians of the deformation onto the surface of the cage.

In the general case where det(A

with

and eq 1 (x) = cos(x) sin(x)-x sin(x) 3 and eq 2 (x) = x sin(x) two well defined functions on ]0, π[ that admit well controlled Taylor expansion around 0,

Special case: η ∈ Support(T ), / ∈ T :

with eq 2 (x) =

x sin(x) , eq 1 (x) = cos(x) sin(x)-x sin(x) 3

, and eq 3 (x) = cos(x)-1 sin(x) 2 being functions well defined on ]0, π[ and that admit controlable Taylor expansion around 0.

We note

with

and eq 6 (x) = d(eq1) dx (x)cos(x)/sin(x), eq 7 (x) = d(eq1) dx (x)sin(x), eq 8 (x) = d(eq2) dx (x)cos(x)/sin(x), and eq 9 (x) = d(eq2) dx (x)sin(x) being functions well defined on ]0, π[ and that admit controllable Taylor expansion around 0.

Special case: η ∈ Support(T ), / ∈ T :

and eq 1 (x) = cos(x) sin(x)-x sin(x)3

, eq 4 (x) =

cos(x) sin(x)

+3(sin(x) cos(x)-x) sin(x) 5