Isosurface Analysis

Surface analysis in crater.rs is based on the properties of scalar fields and their derivatives. Given a scalar field $F^i:{\mathbb{R}}^{m\times n}\to {\mathbb{R}}^m$, we can compute various surface properties.

Gradient Computation

The gradient of a scalar field $F^i$ at origins $X^{ij}$ is the vector of partial derivatives:

$$\nabla F^i(X^{ij})={\partial }^j{F}^i(X^{ij})=\left[\begin{array}{c}{\partial }^j{F}^i(X^{1j})\\ {\partial }^j{F}^i(X^{2j})\\ ⋮\\ {\partial }^j{F}^i(X^{mj})\end{array}\right]$$

In crater.rs, gradients are computed using automatic differentiation through the Burn framework.

Gradient Properties

• The gradient points in the direction of steepest increase of the scalar field
• The magnitude of the gradient indicates the rate of change
• At points on an isosurface $F(x)=c$, the gradient is normal to the surface

Surface Normals

For origins $X^{ij}$ on an isosurface $F^i(X^{ij})=c$, the unit normal vector is:

$$\hat{n}=\frac{\nabla F^i(X^{ij})}{|\nabla F^i(X^{ij})|}$$

Surface Curvature

Surface curvature measures how much the surface deviates from being flat. For a surface defined by $F(x)=c$, the mean curvature is:

$$H=\frac{1}{2}\text{tr}({\nabla }^{2}F)$$

Where ${\nabla }^{2}F$ is the Hessian matrix of second derivatives.