Computational Assessment of Curvatures and Principal Directions of Implicit Surfaces from 3D Scalar Data

Abstract

An implicit method based on high-order differentiation to determine the mean, Gaussian and principal curvatures of implicit surfaces from a three-dimensional scalar field is presented and assessed. The method also determines normal vectors and principal directions. Compared to explicit methods, the implicit approach shows robustness and improved accuracy to measure curvatures of implicit surfaces. This is evaluated on simple cases where curvature is known in closed-form. The method is applied to compute the curvatures of wrinkled flames on large triangular unstructured meshes (namely a 3D isosurface of temperature).

A Derivation of Formulas for Principal Directions of Implicit Surfaces

The methodology of Lehmann et al. [18] is used to build the curvature tensor in the ($\varvec{u},\varvec{v},\varvec{n}$) frame. In this frame, the orthogonal projector T onto the tangent space and the Hessian matrix of $\phi$ are respectively:

$$\begin{aligned} T=Id_3-\varvec{n}\cdot \varvec{n}^t= \left( \begin{array}{ccc} 1 &{} 0 &{} 0\\ 0 &{} 1 &{} 0\\ 0 &{} 0 &{} 0 \end{array} \right) \quad and \quad \nabla ^2\phi = \left( \begin{array}{ccc} \phi _{uu} &{} \phi _{uv} &{} \phi _{uN}\\ \phi _{vu} &{} \phi _{vv} &{} \phi _{vN}\\ \phi _{Nu} &{} \phi _{Nv} &{} \phi _{NN} \end{array} \right) _{(\varvec{u},\varvec{v},\varvec{n})} \end{aligned}$$

The curvature tensor has then a simple expression in the normal frame:

$$\begin{aligned} E=\dfrac{T\cdot \nabla ^2\phi \cdot T}{\vert \phi _n\vert }= \left( \begin{array}{ccc} \tfrac{\phi _{uu}}{\vert \phi _n\vert } &{} \tfrac{\phi _{uv}}{\vert \phi _n\vert } &{} 0\\ \tfrac{\phi _{uv}}{\vert \phi _n\vert } &{} \tfrac{\phi _{vv}}{\vert \phi _n\vert } &{} 0\\ 0 &{} 0 &{} 0 \end{array} \right) _{(\varvec{u},\varvec{v},\varvec{n})} \end{aligned}$$

(12)

Its expression is far more complicated in an arbitrary basis $(\varvec{x},\varvec{y},\varvec{z})$ where we could not find eigenvectors. In the normal frame, main eigenvalues and corresponding eigenvectors of the curvature tensor are then found equal to:

$$\begin{aligned} \kappa _1= & {} K_h-\sqrt{\vert \kappa _h^2-\kappa _k\vert }\cdot \zeta \qquad \qquad \kappa _2 = K_h+\sqrt{\vert \kappa _h^2-\kappa _k\vert }\cdot \zeta \end{aligned}$$

(13)

$$\begin{aligned} \varvec{t_1}= & {} \dfrac{1}{D_1}\left[ \begin{array}{l} \phi _{uv}\\ \kappa _1\phi _n-\phi _{uu}\\ 0 \end{array} \right] _{\varvec{u},\varvec{v},\varvec{n}} \qquad \varvec{t_2} = \dfrac{1}{D_2}\left[ \begin{array}{l} \kappa _2\phi _n-\phi _{vv}\\ \phi _{uv}\\ 0 \end{array} \right] _{\varvec{u},\varvec{v},\varvec{n}} \end{aligned}$$

(14)

with $\kappa _k=\tfrac{\phi _{uu}\phi _{vv}-\phi _{uv}^2}{\phi _n^2}$, $\kappa _h=\tfrac{\phi _{uu}+\phi _{vv}}{2\vert \phi _n\vert }$, $D_1=D_2=\sqrt{\phi _{uv}^2+(\kappa _1 \phi _n-\phi _{uu})^2}$ and $\zeta =\pm 1$.

Since derivatives in the ($\varvec{u},\varvec{v},\varvec{n}$) basis are linked to the derivatives in the default basis ($\varvec{x},\varvec{y},\varvec{z}$) and coordinates of $\varvec{u}$, $\varvec{v}$, $\varvec{n}$:

$$\begin{aligned} \phi _n= & {} \varvec{\nabla }\phi \cdot \varvec{n}=\phi _xn_x+\phi _yn_y+\phi _zn_z= \Vert \varvec{\nabla }\phi \Vert =\vert \phi _n\vert \end{aligned}$$

(15a)

$$\begin{aligned} \phi _{uv}= & {} \varvec{u}^t\cdot \nabla ^2\phi \cdot \varvec{v}= \phi _{xx}u_xv_x+\phi _{yy}u_yv_y+\phi _{zz}u_zv_z \nonumber \\+ & {} \phi _{xy}(u_xv_y+u_yv_x)+ \phi _{xz}(u_xv_z+u_zv_x)+\phi _{yz}(u_yv_z+u_zv_y)\end{aligned}$$

(15b)

$$\begin{aligned} \phi _{uu}= & {} \varvec{u}^t\cdot \nabla ^2\phi \cdot \varvec{u}\end{aligned}$$

(15c)

$$\begin{aligned} \phi _{vv}= & {} \varvec{v}^t\cdot \nabla ^2\phi \cdot \varvec{v} \end{aligned}$$

(15d)

Replacing $\phi _n$, $\phi _{uv}$, $\phi _{uv}$ and $\phi _{uv}$ into curvature expressions (2b), (3b) and making some formal simplication, we recover the intrinsic expressions of curvatures (2a), (3a) that are independent on the choice of $\varvec{u},\varvec{v}$. To sum up, principal curvatures do not depend on the choice of ($\varvec{u},\varvec{v}$) whereas principal directions depend on their coordinates. A bad choice of ($\varvec{u},\varvec{v}$) may then conduct to null vectors and then bad estimations. To circumvent bad choices, we then propose a criteria on $\zeta$ to ensure that $\varvec{t_1}$ and $\varvec{t_2}$ are non-null vectors in Sect. 1.

To better understand the role of the parameter $\zeta$ in avoiding degeneracies, it is advised to compute curvatures and principal directions by hand with $\zeta =\pm 1$ at $u=v=0$ and $n=|R|$ for the following case studies:

• $\phi (u,v,n)=u^2+n^2-R^2$ $\Rightarrow$ cylinder of axis $\varvec{v}$ that requires $\zeta =+1$

• $\phi (u,v,n)=R^2-u^2-n^2$ $\Rightarrow$ cylinder of axis $\varvec{v}$ that requires $\zeta =-1$

• $\phi (u,v,n)=v^2+n^2-R^2$ $\Rightarrow$ cylinder of axis $\varvec{u}$ that requires $\zeta =-1$

• $\phi (u,v,n)=R^2-v^2-n^2$ $\Rightarrow$ cylinder of axis $\varvec{u}$ that requires $\zeta =+1$

• $\phi (u,v,n)=\pm u*v+n$ $\Rightarrow$ saddle shaped surface ($\zeta =\pm 1$)

This work was granted access to the HPC resources of the RZG of the Max Planck Society made available within the Distributed European Computing Initiative by the PRACE-2IP, receiving funding from the European Community’s Seventh Framework Programme (FP7/2007–2013) under grant agreement n $^{\circ }$ RI-283493. The research leading to these results has received funding from the European Research Council under the ERC grant agreement n $^{\circ }$ 247322, GREENEST. The authors thank Xiang He for his help in code development and Bruno Denet for his feedback in using the GTS library.