Surface Parametrization Examples

Differential Geometry · Axiom Academy

EXAMPLE Surface Parametrization Examples Step-by-step derivations of parametrizations for common surfaces in R³ Differential Geometry • Surfaces in R³ Example 1: Parametrizing a Sphere Check Your Understanding: Sphere What is the point on the unit sphere (a=1) when θ = 0 and φ = π/2? Example 2: Parametrizing a Cylinder Check Your Understanding: Cylinder For the cylinder r(θ,z) = (2cos θ, 2sin θ, z), what geometric object do we get when θ is fixed at θ₀? Example 3: Parametrizing a Torus Check Your Understanding: Torus In the torus parametrization, what happens when we fix v = 0 and vary u from 0 to 2π? Example 4: Surface of Revolution Check Your Understanding: Surface of Revolution What surface do we get by rotating the line segment x = 2, z = t for 0 ≤ t ≤ 5 around the z-axis? Example 5: Graph of a Function z = f(x,y) Check Your Understanding: Graph Parametrization For the surface r(x,y) = (x, y, x² + y²), what are the tangent vectors r_x and r_y at the point (1, 0, 1)? Excellent work! You've learned to parametrize five fundamental types of surfaces. Here's what to remember: Sphere: Use spherical coordinates r(θ,φ) = (a sin φ cos θ, a sin φ sin θ, a cos φ) for a sphere of radius a Cylinder: Use r(θ,z) = (R cos θ, R sin θ, z) for a cylinder of radius R along the z-axis Torus: Use r(u,v) = ((R + r cos u) cos v, (R + r cos u) sin v, r sin u) with major radius R and minor radius r Surface of Revolution: Rotate a curve (f(t), g(t)) using r(t,θ) = (f(t) cos θ, f(t) sin θ, g(t))

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