The sectional curvature is said to be constant if it has the same value at every point and for every two-dimensional tangent plane at that point.
32.
Near to any point, " S " can be approximated by its tangent plane at that point, which is an affine subspace of Euclidean space.
33.
The first fundamental form may be viewed as a family of positive definite symmetric bilinear forms on the tangent plane at each point of the surface depending smoothly on the point.
34.
Think of the problem the other way : if you project a straight line through the origin of the tangent plane at a point, you get a curve on the surface.
35.
In other words, any affine transformation maps the tangent plane to the surface at a point to the tangent plane to the image of the surface at the image of the point.
36.
In other words, any affine transformation maps the tangent plane to the surface at a point to the tangent plane to the image of the surface at the image of the point.
37.
The " tangent plane " to a surface at a given point " p " is defined in an analogous way to the tangent line in the case of curves.
38.
A tangent plane can be " rolled " along " S ", and as it does so the point of contact traces out a curve on " S ".
39.
The "'tangent plane "'at a regular point is the unique plane passing through, and having a direction parallel to the two row vectors of the Jacobian matrix.
40.
In differential geometry, the "'Bj�rling problem "'is the problem of finding a minimal surface passing through a given curve with prescribed normal ( or tangent planes ).
