Calculation and visualization of Gaussian curvature on triangular mesh surface

This paper is based on the calculation and visualization of Gaussian curvature of triangular grid surface in the same mountain.

Long time no code to write, recently computed triangular mesh surface Gaussian curvature practiced hand, and realized the Gaussian curvature visualization, review a little bit of differential geometry knowledge. Sometimes it is necessary to write the code yourself, debug run, combined with the results of the test, in order to have a deeper understanding of the corresponding knowledge.

The Gaussian curvature of a point on a surface, that is, the product of two main curvature of the point. The vertices on the surface are mapped to the sphere of the unit sphere, the endpoints of the normals are mapped to the sphere, and the points on the surface are formed with the points on the sphere, which is called the spherical representation of the surface, also called the Gaussian map. The geometrical meaning of Gaussian curvature, that is the limit of area/surface area on the sphere, can be seen that the Gaussian curvature does reflect the local curvature of the surface.

Using the positive and negative of Gaussian curvature, it is very convenient to study the structure of the surface near a point, the Gaussian curvature k>0 is the ellipse point, the k<0 is the hyperbolic point, the k=0 is a plane or a parabolic point. And the Gaussian curvature is the intrinsic amount of the surface, which is related only to the first basic type of the surface, and is independent of the selection and parameterization of the axes.

To get to the problem, the Gaussian curvature of the triangular mesh surface needs to be solved by using discrete differential geometry, the formula I use is:

The geometric meaning of this formula is relatively straightforward, 2*pi-the angle of the point neighborhood triangle, and dividing by the area of the corresponding area, the degree of curvature of the point surface is carved out.

In fact, the method of derivation of the above formula is very ingenious, careful study, it uses the Gaussian mapping in the geometric sense, the discrete Gaussian curvature of the surface of the integral
Considering P-Point neighborhood normals mapped to the area on the unit sphere, which is approximate to 2*pi-the angle of the neighboring triangle of the point and
Do not carefully write, we look at the following picture, feel the beauty of this formula:

The specific coding is relatively simple, the gausscurvature array is calculated, normalized to [0,1], set three colors C1 gray yellow, C2 green, C3 red, linear weighted pseudo-color display. The k>0 is shown in green, the k<0 is red, the k=0 is grayed out, the brighter the color, the greater the absolute value of the Gaussian curvature. Implementation effects such as

The display effect is not good, has done the image processing person to know, needs to do a histogram equalization

After the histogram equalization, the display effect is:

This effect is much better, the bridge of the nose red for the typical hyperbolic point (two main curvature of the different numbers, the main direction of the two-section of the line, a forward to normal bending, a backward bending to the normal, forming a saddle surface, the tip of the green for the typical parabolic point (two main curvature of the same number, the surface in all directions toward the same side bent), The flatter area of the forehead (with one main curvature approaching 0) The Gaussian curvature has a smaller absolute value and a lighter color.

Put more experimental pictures, calculate the Gaussian curvature and pseudo-color display, it is true that some properties of the surface can be intuitively seen, should be able to guide the mesh denoising, smoothing, simplification, segmentation and other subsequent research

The calculation formula of discrete Gaussian curvature on triangular meshes is simple, and the program framework for visualizing the vertex properties of the mesh surface is constructed with this simple practice. You can also achieve the average curvature, the main curvature, and the calculation of the main direction after a chance.

Calculation and visualization of Gaussian curvature on triangular mesh surface