Tangent Planes

Differential Geometry · Axiom Academy

Understanding the tangent plane to a surface through partial derivatives and linear approximations 1. Tangent Vectors X u and X v Consider a parametrized surface X(u,v) where u and v are parameters. The partial derivatives with respect to these parameters give us two fundamental tangent vectors: These vectors represent the rate of change of the surface position as we move in the u -direction and v -direction respectively. At each point p = X(u₀, v₀) , these vectors are tangent to the surface. 2. Definition of the Tangent Plane The tangent plane at a point p = X(u₀, v₀) is the plane that best approximates the surface near that point. It contains all tangent vectors to curves on the surface passing through p . The tangent plane is a two-dimensional vector space embedded in three-dimensional space. It represents the first-order linear approximation to the surface at point p . 3. Tangent Plane as span X u , X v A fundamental result in differential geometry states that the tangent plane is spanned by the two partial derivative vectors X u and X v , provided they are linearly independent (i.e., the surface is regular). This means any tangent vector at point p can be written as a linear combination v = aX u + bX v for some scalars a and b . 4. Equation of the Tangent Plane To find the explicit equation of the tangent plane, we use the normal vector, which is perpendicular to both X u and X v . This normal vector is given by the cross product:

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