We may ask for which values of u and v the Gauss mapping is singular. This will happen whenever Nu(u,v) and Nv(u,v) are linearly dependent, which implies that
Note that since
The area of the spherical image of a given domain on the surface is given by the integral of (Nu( u,v)·N v(u,v)) over the domain. Note that Nu(u,v) ·Nv( u,v) is a scalar multiple of N(u,v) and of Xu(u,v) ·Xv( u,v) because of the orthogonalities just noted. We define the Gaussian curvature K(u,v) by the following condition :
Note that the Gaussian curvature K(u,v) is positive if the frame given by Nu(u,v) followed by Nv(u,v) gives the same orientation to the tangent plane as the frame Xu(u,v) followed by Xv(u,v) , and K(u,v) is negative if these two orientations differ. The Gaussian curvature K(u,v) is zero precisely when the Gauss mapping is singular and the vectors Nu(u,v) and Nv(u,v) are linearly dependent. At this point, we point out the analogy between the way we just defined Gaussian Curvature on a surface and the way we defined total curvature of a space curve in lab 4.
Function Graph of the Gaussian CurvatureAs we select different parts of the domain, we can see where the Gaussian curvature is positive and where it is negative. The original domain in the uv-plane is shown in white in The Gaussian Curvature Graph . As in the previous demo on the Normal map of a surface, you may use the middle mouse button to choose a point in the domain around which the region is chosen. The area of the region is determined by the Width along U and Width along V sliders.
The main purpose of this demonstration is to visualize K(u,v) as a function of u and v , defined over the domain of the original surface.