Parametric surface

A parametric surface is a surface in the Euclidean space R 3 {displaystyle {mathbb {R} }^{3}} which is defined by a parametric equation with two parameters r → : R 2 → R 3 . {displaystyle {vec {r}}:{mathbb {R} }^{2} ightarrow {mathbb {R} }^{3}.} Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form. The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization. A parametric surface is a surface in the Euclidean space R 3 {displaystyle {mathbb {R} }^{3}} which is defined by a parametric equation with two parameters r → : R 2 → R 3 . {displaystyle {vec {r}}:{mathbb {R} }^{2} ightarrow {mathbb {R} }^{3}.} Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form. The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization. The same surface admits many different parametrizations. For example, the coordinate z-plane can be parametrized as for any constants a, b, c, d such that ad − bc ≠ 0, i.e. the matrix [ a b c d ] {displaystyle {egin{bmatrix}a&b\c&dend{bmatrix}}} is invertible. The local shape of a parametric surface can be analyzed by considering the Taylor expansion of the function that parametrizes it. The arc length of a curve on the surface and the surface area can be found using integration.