Calculation and visualization of Gauss curvature on triangular mesh surface

Turn from: http://www.cnblogs.com/haoyul/p/6910679.html

For a long time did not write code, recently took the calculation of triangular mesh surface Gauss curvature practiced practicing, and realized the Gaussian curvature visualization, reviewed a little differential geometry knowledge. Feel sometimes still want to write the code, debugging and running, combined with the test results, in order to have a deeper understanding of the corresponding knowledge.

The Gaussian curvature of a point on a surface, the product of two principal curvatures of that point. The vertices on the surface are mapped to the center of the unit ball, the endpoints of the normal are mapped to the sphere, and the points on the surface are established with the points on the sphere, called the spherical representation of the surface, also known as the Gaussian map. The geometrical meaning of Gaussian curvature, i.e. the limit of area/surface area on the sphere, can be seen that the Gaussian curvature does reflect the local curvature of the surface.

By using the positive and negative property of Gaussian curvature, it is very convenient to study the structure of the curved surface in one point, the Gaussian curvature k>0 as the elliptic point, the k<0 as the hyperbolic point, and the k=0 as plane or parabolic point. And the Gaussian curvature is the intrinsic quantity of the surface, which is related to the first basic type of the surface, and has nothing to do with the selection and parameterization of the axes.

In order to solve the Gauss curvature of the triangular mesh surface, we need to use the discrete differential geometry, the formula I used is:

The geometric meaning of this formula is more intuitive, 2*pi-the angle of the point neighborhood triangle and dividing the area of the corresponding area, the curvature of the point surface is carved.

The area of a (v) depends on the size of the triangular apex, and the definition of acute and obtuse angles is different.

In fact, the method of deriving the above formula is very ingenious, carefully studied, it utilizes the Gaussian mapping in the geometric sense of the discrete Gaussian curvature of the surface integral
Considering the area of the P-point neighborhood normal map to the unit sphere, that is approximate to the 2*pi-angle of the point neighborhood triangle and
Not carefully written, we look at the following picture, feel the beauty of this formula:

The specific encoding is relatively simple, the gausscurvature array is obtained, normalized to [0,1], set three kinds of colors C1 gray yellow, C2 green, C3 red, linear weighted pseudo color display. K>0 display is green, k<0 is shown as red, k=0 is grayed out, the color is brighter, the absolute value of Gaussian curvature is larger. The implementation of the results as shown below

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

The display effect of histogram equalization is:

This effect is much better, the nose of the red for the typical hyperbolic point (two main curvature of the difference, the main direction of the two section of the line, a forward bend to the normal, a backward bend to the normal, forming a saddle surface, and a typical parabolic point at the tip of the nose (two principal curvature same number, curved surface in all directions toward the same side), The flat area of the forehead (with a principal curvature of nearly 0) the Gaussian curvature has a smaller absolute value and a lighter color.

Put out more test pictures, calculate the Gaussian curvature and pseudo color display, it can intuitively see some of the surface properties, 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 this simple practice is to construct a framework for visualizing the vertex properties of grid surface. In the future, we can also realize the average curvature, the principal curvature, the main direction of the calculation.