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The tangent plane at a regular point is the affine plane in R 3 spanned by these vectors and passing through the point r(u, v) on the surface determined by the parameters. Any tangent vector can be uniquely decomposed into a linear combination of r u {\displaystyle \mathbf {r} _{u}} and r v . {\displaystyle \mathbf {r} _{v}.}
are two tangent vectors at point P. Let n(u, v) be the unit normal vector and let (E, F, G) and (L, M, N) be the coefficients of the first and second fundamental forms of this surface, respectively. The Weingarten equation gives the first derivative of the unit normal vector n at point P in terms of the tangent vectors r u and r v:
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. It is the best approximation of the surface by a plane at p , and can be obtained as the limiting position of the planes passing through 3 distinct points on the surface close to p as these points converge to p .
is the normal curvature of the surface at a regular point for the unit tangent direction . H F {\displaystyle H_{F}} is the Hessian matrix of F {\displaystyle F} (matrix of the second derivatives). The proof of this formula relies (as in the case of an implicit curve) on the implicit function theorem and the formula for the normal curvature of ...
The second fundamental form of a parametric surface S in R 3 was introduced and studied by Gauss.First suppose that the surface is the graph of a twice continuously differentiable function, z = f(x,y), and that the plane z = 0 is tangent to the surface at the origin.
As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point. Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the ...
It is an irregular point that remains irregular, whichever parametrization is chosen (otherwise, there would exist a unique tangent plane). Such an irregular point, where the tangent plane is undefined, is said singular. There is another kind of singular points. There are the self-crossing points, that is the points where the surface crosses ...
Indeed, consider a point in the plane, say (x 0,y 0). This point lies on a tangent line if and only if there exists a t such that (, (,)) = = . This is a cubic in t and as such has at least one real solution. It follows that at least one tangent line to γ must pass through any given point in the plane.