Exploring Surface Curvature and Differentials
Exploring Surface Curvature and Differentials
This Demonstration shows the relationship between the principal curvatures of a surface and the differential of the unit normal of the surface. The differential at a point may be viewed as a linear transformation of the tangent plane at the point, since the derivative of the unit normal is orthogonal to the normal. The eigenvalues of are the negatives of the principal curvatures of the surface at the point, and correspondingly the eigenvectors are the principal directions. At a point one may rotate an orthogonal basis for the tangent plane to diagonalize the matrix for by aligning the basis with the principal directions. One may see how a small tangent vector is mapped by and compare it to the actual difference in the normals at the points and . The lines that are drawn on the surface are lines of curvature, which are curves whose tangents at every point match one of the principal directions.
dN
dN
dN
P
dN
dP
dN
P
P+dP