Tangent Plane

From Department of Mathematics at UTSA
Revision as of 10:45, 6 October 2021 by Lila (talk | contribs) (→‎Definition)
Jump to navigation Jump to search

Intuitively, it seems clear that, in a plane, only one line can be tangent to a curve at a point. However, in three-dimensional space, many lines can be tangent to a given point. If these lines lie in the same plane, they determine the tangent plane at that point. A tangent plane at a regular point contains all of the lines tangent to that point. A more intuitive way to think of a tangent plane is to assume the surface is smooth at that point (no corners). Then, a tangent line to the surface at that point in any direction does not have any abrupt changes in slope because the direction changes smoothly.

Definition

Let be a point on a surface , and let be any curve passing through and lying entirely in . If the tangent lines to all such curves at lie in the same plane, then this plane is called the tangent plane to at .

Resources

Videos