Normal Vectors

Differential Geometry · Axiom Academy

Understanding the geometry of normal vectors on surfaces For a parametric surface X (u,v), the normal vector is obtained by taking the cross product of the tangent vectors X u and X v . The unit normal vector N is then found by normalizing this cross product. This formula ensures that N has unit length and points perpendicular to both tangent directions on the surface. 2. Normal Vector and Tangent Plane The normal vector is perpendicular to the tangent plane at each point. This means that N is orthogonal to every tangent vector at that point. The tangent plane is spanned by X u and X v , and the normal vector provides the third dimension, completing the local coordinate system. 3. Two Choices for Orientation At any point on a surface, there are two possible unit normal vectors: one pointing "up" and one pointing "down." The choice depends on the order of the cross product. The orientation is determined by the parameterization. A consistent choice of normal orientation across the entire surface is called an orientation of the surface. 4. Gauss Map: Surface to Unit Sphere The Gauss map is a remarkable function that takes each point on a surface to a point on the unit sphere, mapping the normal vector N (u,v) to its tip on the sphere. This map captures how the normal direction changes across the surface and is intimately connected to Gaussian curvature. 5. Normal Variations Along Surface

This is the written version of the interactive lesson above. See the full Differential Geometry course.