Parametric Representation of Surfaces
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Tangent plane
The tangent plane to a surface S contains all the tangent vectors to S at a given point P. It is denoted [Equation goes here - download the original pdf to see it.] A curve on S is a continuous function [Equation goes here - download the original pdf to see it.] such that [Equation goes here - download the original pdf to see it.] lie on S. If [Equation goes here - download the original pdf to see it.] are differentiable, then a tangent vector to the curve is iven by Equation goes here - download the original pdf to see it.] It can be shown that the partial derivatives [Equation goes here - download the original pdf to see it.] are linearly independent and span the tangent plane [Equation goes here - download the original pdf to see it.]. Their vector product is normal to the tangent plane [Equation goes here - download the original pdf to see it.] A unit vector normal to the tangent plane, and hence normal to the surface, is [Equation goes here - download the original pdf to see it.] If S is the level surface [Equation goes here - download the original pdf to see it.] then the normal vector is also given by [Equation goes here - download the original pdf to see it.]
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Contents of
Parametric Representation of Surfaces
1 Surfaces in xyz-space
2 The need for a parametric representation
3 Standard parametric surfaces
4 Parametric representation of the sphere
5 Parametric representation of the cone
6 Tangent plane
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