Касательные плоскости к параметрической поверхности

Consider a paraboloid surface where r represents the position vector of any point on the surface. Using x and y as parameters, the surface can be described parametrically.

A tangent plane is a flat surface that touches this curved surface at a single unique point. Consequently, the tangent plane contains the tangent vectors to any curved paths along the surface that pass through this point.

The partial derivative r_x gives the tangent vector in the x direction, and the partial derivative r_y gives the tangent vector in the y direction.

Evaluating these at point A gives the tangent vectors in the x and y directions contained in the tangent plane.

Their cross product gives a normal vector n perpendicular to the tangent plane.

For any point P in the tangent plane, the vector v from A to P lies in the plane and is therefore perpendicular to the plane’s normal vector n.

As a result, the dot product of the vector n and v is zero. Expanding this dot product and simplifying gives the final equation of the tangent plane at point A to the given surface.