![]() ![]() To that end, the curvature is linearly interpolated, from its values at the vertices, at the centre of each triangle. In order to construct the full affine invariant metric we need to scale the equi-affine metric by the equi-affine Gaussian curvature. ( 57) are defined by the metric, according to Eq. The corresponding angles and areas in Eq. Finally, for each vertex, the Gaussian curvature with respect to the equi-affine metric is approximated using Eq. The equi-affine metric coefficients for each triangle are then evaluated from our local quadratic approximation of S( u, v) and its corresponding derivatives (see Raviv and Kimmel (2014) for more details). Three vertices belong to the triangle for which the metric is evaluated, and three to its nearest neighbouring triangles. The local coefficients matrix B is evaluated from six surface points (vertices). Where the matrix B 3×6 contains 18 parameters. ![]() ![]() (63) S ( u, v ) = x ( u, v ) y ( u, v ) z ( u, v ) ≈ B 1 u v u v u 2 v 2. Surfaces that are related by a curvature-preserving transformation (like the plane and the cylinder) are called isometric. On the other hand there is no way to form a sphere ( K constant, but strictly positive) from either cylinder or plane without stretching, tearing or gluing. For example, the plane, with Gaussian curvature, K ≡ 0, is easily rolled into a cylinder for which also K ≡ 0. This means that if we can bend a simply connected surface x into another simply connected surface y without stretching or tearing, there exists a continuous transformation from x to y that preserves the Gaussian curvature at every point. The Gaussian curvature, K, is a bending invariant. the cylinder), the Gaussian curvature is zero at all points on the surface. If the Gauss map of a surface comprises only a single point ( e.g. In the limit the quotient of the area of the surface element and its spherical image is 1 /K. Imagine shrinking the region progressively to an infinitesimal area about a point. An alternative definition of the Gaussian curvature follows from this result. In fact, the surface area of the Gauss-mapped region on the unit sphere is equal to the integral curvature of the region, ∫ surface K da. The Gauss map is closely related to the Gaussian curvature of the surface. An extreme example is the plane, which is mapped onto a single point, whose location depends on the orientation of the plane. This is a necessary feature of saddle-shaped surfaces, with negative Gaussian curvature.Ĭlearly the spherical image under the Gauss map of a highly curved surface patch will be larger than that of less curved patches of the same area, since the divergence in direction spanned by the normal vectors is wider for the highly curved patch. Notice that for the example illustrated the bounding curve on the surface and on the unit sphere are traversed in opposite senses. The normal vectors in the triangular ABC region of the saddle-shaped surface define a region on the unit sphere, A'B'C, given by the intersection of the unit sphere with the collection of normal vectors (each placed at the centre of a unit sphere) within the ABC region. ![]()
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