Curvature of a triangle mesh, definition and computation.


Defining and giving the formula to compute the curvature over a triangle mesh at some vertices.

Normal curvature

By intersecting a surface with a plane we see that the section defines a curve:

The curvature of this curve is called 'normal curvature' and noted \( k_n\) because the plane is spanned through the normal \( \boldsymbol n\) and an arbitrary tangent vector \( \boldsymbol t \). Now, given a point \( \boldsymbol p \) on the surface you can find an infinity of curvatures given the direction you choose for your tangent vector \( \boldsymbol t \). 

Principal curvatures

But how can we characterize the curvature of a surface at a given point when we seem to have an infinity of curvatures? Don't worry! It can be shown that when rotating the normal plane around \( \boldsymbol n \) you can always find two 'principal curvatures' \( k_1 \) and \( k_2 \) respectively defining the maximal and minimal curvature at a given point \( \boldsymbol p \) of the surface:

In addition, the principal curvatures \( k_1 \) and \( k_2 \) are associated to the 'principal directions' \( \boldsymbol t_1 \) and \( \boldsymbol t_2 \) which are always found to be orthogonal to each other! What's more, you can define any 'normal curvature' \( k_n \) as a combination of the principal curvatures \( k_1 \) and \( k_2 \) using Euler's theorem below:

\[ k_n = k_1\cos^2\theta + k_2\sin^2\theta \]

Where \( \theta \) is the angle between \( \boldsymbol t_1 \) (maximal curvature direction) and \( \boldsymbol t_n \) the normal curvature.

Calculation

One can compute the principal curvatures \( k_1 \) and \( k_2 \) according to the Mean curvature \( H \) and the Gaussian curvature \( k_g \), don't worry we will define both \( H \) and \( k_g \) in the next sections:

$$ \begin{align*} k_1 & = H + \sqrt{ H^2 - k_g} \\ k_2 & = H - \sqrt{ H^2 - k_g} \end{align*} $$

Gaussian curvature

The Gaussian is the product of principal curvatures \( k_g = k_1 * k_2 \) on a triangular mesh it can be computed at vertex \( \boldsymbol p_i \) as follows:

\[ k_g = \frac{ 2 \pi - \sum \theta_j } { A_i } \]

Where the angle \( \theta_j \) is defined as:

\( A_i \) can be computed as:

Interpreting the Gaussian curvature's value

It can help to locally characterize the shape:

Mean curvature

The mean curvature is the average of the principal curvatures:

$$ H= \frac { k_1 + k_2 } { 2 } $$

In the smooth setting the mean curvature is the average of the normal curvatures in \([0 - 2\pi]\)

$$ \begin{align*} k_h & = \frac{1}{2\pi} \int_0^{2\pi} { k_n(\theta) \, d\theta } \\ & \text{ recall the relationship between principal and normal curvature:} \\ k_h & = \frac{1}{2\pi} \int_0^{2\pi} { k_1\cos^2(\theta) + k_2\sin^2(\theta) \, d\theta } \\ \end{align*} $$

If you're lazy like me you can solve this with Maxima , just insert in Maxima's command line:

    ratsimp(1/(2*%pi)*integrate(k1*cos(t)^2 + k2*sin(t)^2, t, 0, 2*%pi));

step by step development:

$$ \newcommand{annotation}[1]{ ~~~~~~\raise 0.8em {\color{grey}\scriptsize{ #1 }} } \begin{align*} k_h & = \frac{1}{2\pi} \left[ \int_0^{2\pi} { k_1 \cos^2(\theta) \, d\theta } \, {\color{red}+} \int_0^{2\pi} {k_2\sin^2(\theta) \, d\theta } \right] \annotation{ \int a + b = \int a + \int b } \\ k_h & = \frac{1}{2\pi} \left[ {\color{red}k_1} \int_0^{2\pi} { \cos^2(\theta) \, d\theta } + {\color{red}k_2} \int_0^{2\pi} {\sin^2(\theta) \, d\theta }\right] \annotation{ \int a.f(x) = a.\int f(x) } \\ k_h & = \frac{1}{2\pi} \left[ k_1\int_0^{2\pi} { {\color{red}\frac{1+ \cos(2\theta)}{2}} \, d\theta } + k_2 \int_0^{2\pi} {{\color{red}\frac{1 - \cos(2\theta)}{2}} \, d\theta }\right] \annotation{ \href{https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Power-reduction_formulae}{\text{Power reduction formulae}} } \\ k_h & = \frac{1}{{\color{red}4}\pi} \left[ k_1 \int_0^{2\pi} { {1+ \cos(2\theta)} \, d\theta } + k_2 \int_0^{2\pi} {{1 - \cos(2\theta)} \, d\theta } \right]\\ k_h & = \frac{1}{4\pi} \left[ k_1 \int_0^{2\pi} {1\, d\theta} \, {\color{red}+} k_1 \int_0^{2\pi}{ \cos(2\theta) \, d\theta } + k_2 \int_0^{2\pi} {1\, d\theta} \,{\color{red}-} k_2 \int_0^{2\pi}{ \cos(2\theta) \, d\theta }\right] \\ k_h & = \frac{1}{4\pi} \left[ k_1 {\color{red} \left [ x \right ]_{0}^{2\pi}} + k_1 \int_0^{2\pi}{ \cos(2\theta) \, d\theta } + \cdots \right] \annotation{ \frac{d(x + c)}{dx} = 1 } \\ k_h & = \frac{1}{4\pi} \left[ k_1 {\color{red} \left [ 2\pi - 0 \right ]} + k_1 {\color{red} \left [ \frac{\sin(2\theta)}{2} \right ]_{0}^{2\pi}} + \cdots \right] \annotation{ \text{chain rule:} f'(g(x)) = g'(x)f'(g(x)) \text{ so, } \frac{d}{dx} ( \frac{1}{2} \, \sin(2x) + c ) = \frac{2}{2} \, \cos(2x) } \\ k_h & = \frac{1}{4\pi} \left[ k_1 {\color{red} 2\pi} + k_1 {\color{red} \left [ \frac{\sin(4\pi)}{2} - \frac{\sin(0)}{2} \right]} + \cdots \right]\\ k_h & = \frac{1}{4\pi} \left[ k_1 2\pi + k_1 {\color{red} 0} + \cdots \right] ~~~~~~\raise 0.8em {\color{grey}\scriptsize{ sin(0) = sin(2\pi) = sin(4\pi) = 0 }} \\ k_h & = \frac{1}{4\pi} \left[ k_1 2\pi + k_2 {\color{red} 2\pi } - k_2 {\color{red} 0} \right]\\ k_h & = \frac{1}{2} \left[ k_1 + k_2 \right] \\ \end{align*} $$ šŸ„²

In addition, the mean-curvature can also be expressed using the Laplace operator. Below is the mean-curvature normal which norm corresponds to the amount of curvature \( H \):

$$ \frac{\Delta p_i}{2} = -Hn $$

We designate the norm as the absolute mean-curvature:

$$ |H| = \frac{ \| \Delta p_i \|}{ 2 } $$

Mean-curvature sign: the sign of the dot product between a mesh's normal nāƒ— (for instance averaging at vertices face's normals) and the mean-curvature normal \( -\Delta p_i . \vec n_i \) gives the sign/orientation of the mean-curvature H. Which is why we say the mean-curvature is an extrinsic property since its since depends on the orientation of the normal of the mesh.





In other words, to find the mean-curvature normal of a triangle mesh, we need to apply the discrete Laplace operator at some vertex \( \boldsymbol p_i \) as follows:

$$ \Delta \vec p_i = \frac{1} {2A_i} \sum\limits_{{j \in \mathcal N(i)}} { (\cot \alpha_{ij} + \cot \beta_{ij}) (p_j - p_i) } $$

Note: Alternate discretization of the Laplacian operator could be used. For instance, a rather crude option is to compute the barycenter of our mesh cell. Since we set uniformly to 1 all weights instead of the cotangent weight scheme, we loose a great deal of accuracy:

$$ \Delta \vec p_i = \frac{1} { |\mathcal N(i)| } \sum\limits_{{j \in \mathcal N(i)}} { 1 . (p_j - p_i) } \\ = \left ( \frac{1} { |\mathcal N(i)| } \sum\limits_{{j \in \mathcal N(i)}} { p_j } \right ) - p_i $$

\(|N(i)|\) : number of adjacent vertices to \(i\).

Interpreting the Mean curvature's value

It does not characterize the shape but gives the amount of curvature in one way or another:

Smoothing

Interestingly the mean-curvature normal is related to mesh smoothing algorithm, since smoothing (sometimes designated as curvature flow) can be seen as minimizing the mean-curvature.

Gaussian VS Mean curvature

Notice how \( H < 0 \) or \( H > 0 \) virtually describes the same shape, the difference between a peak and a pit is a matter of perspective. Ultimately it is set according to the convention you use to define the mean-curvature's sign.

Code

Gaussian curvature in libigl
Mean-curvature in libigl

Code sample explained:

// a list of 3d vectors that represents the mean-curvature normals (H x n)
MatrixXd HN; // size: nb_verts x 3
SparseMatrix<double> L; // Laplacian matrix (nb_verts x nb_verts)
SparseMatrix<double> M; // Mass matrix (diagonal matrix where each diagonal element a_i == the cell area at the ith vertex)
SparseMatrix<double> Minv; // inverse of the mass matrix (so 1.f/a_i)
igl::cotmatrix(V,F,L); // computes L, the variables V and F are just the list of Vertex and triangle Face index
igl::massmatrix(V,F,igl::MASSMATRIX_TYPE_VORONOI,M); // Compute M
igl::invert_diag(M,Minv); // Compute the inverse of M
HN = -Minv*(L*V); // Minv*L => is the Laplace operator, multiplying Minv*L against the list of vertices V is equivalent to applying the operator (\Delta p)
H = HN.rowwise().norm(); //up to sign <- absolute mean-curvature

More accurate approaches to compute mean curvature also includes:

- Approach based on dihedral angles
- Quadric fitting

JavaScript

JavaScript library to compute Gauss/Mean and principal curvatures. The corresponding online demo is here. Find more information at the main project page.

Other hacks

When accuracy is not an issue, for instance you just roughly want to detect pits and darken them or colorize them for artistic purpose, there are other more hacky/simpler ways to compute curvature:
- https://blender.stackexchange.com/a/147371/126011
- https://computergraphics.stackexchange.com/a/1719/16151

References

Discrete Differential-Geometry Operators for Triangulated 2-Manifolds
Mark Meye & Al
Polygon mesh processing by Mario Bosch & Al
Discrete Differential Operators for Computer Graphics (Mark Meyer's thesis)
Alec Jacobson geometry processing course: "discrete-mean-curvature-normal-via-discrete-laplace"

Related

By Keenan Crane: Lecture 15: Curvature of Surfaces (Discrete Differential Geometry)
By Justin Solomon: Shape Analysis (Lecture 7): Approximating Gaussian/mean/principal curvatures on triangle meshes

One comment

The ‘barycentric cell area’ is not “sum of the triangles’ area adjacent to i times 3”, but “sum of the triangles’ area adjacent to i divided by 3”

From page 10 in [http://www.geometry.caltech.edu/pubs/DMSB_III.pdf]:
“It is interesting to notice that using the barycentric area as an averaging
region results in an operator very similar to the definition of the mean cur-
vature normal by Desbrun et al. [DMSB99], since A_Barycenter is a third of the
whole 1-ring area A_1-ring used in their derivation.”
——————————
Rodolphe: Nice catch, thank you!

Felix - 10-10-’22 21:51
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