Parametric Surfaces

Differential Geometry · Axiom Academy

Understanding surfaces through parameterization in differential geometry 1. Parametric Surface Definition A parametric surface is a mapping from a domain in the plane to three-dimensional space: Here, D is a subset of ℝ² (the parameter domain), and X(u,v) produces points in ℝ³. We can write this in component form: Domain D: The region in the (u,v)-plane where the parameterization is defined Parameters (u,v): Independent variables that "coordinate" the surface Image X(D): The actual surface in ℝ³ The parameters (u,v) create two families of curves on the surface called coordinate curves : u-curves: Hold v constant, vary u. These curves show how the surface changes as we move in the u-direction v-curves: Hold u constant, vary v. These curves show how the surface changes as we move in the v-direction Together, they form a coordinate grid on the surface, like longitude and latitude lines on Earth The coordinate curves intersect orthogonally on many important surfaces (orthogonal parameterizations). The unit sphere centered at the origin can be parameterized using spherical coordinates (θ, φ): θ ∈ [0, 2π]: Azimuthal angle (longitude), sweeps around the z-axis φ ∈ [0, π]: Polar angle (colatitude), measures down from the north pole θ-curves (φ = constant): Circles of latitude (parallels) φ-curves (θ = constant): Meridians (lines of longitude) Domain: θ ∈ [0, 2π], z ∈ [0, h] Domain: u, v ∈ ℝ (or bounded regions) 5. Implicit vs Parametric Representations

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