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 \mathbf {r} :\mathbb {R} ^{2}\to \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.
Examples
The simplest type of parametric surfaces is a bivariate surface, given by the graphs of functions of two variables (bivariate functions): z = f ( x , y ) , r ( x , y ) = ( x , y , f ( x , y ) ) . {\displaystyle z=f(x,y),\quad \mathbf {r} (x,y)=(x,y,f(x,y)).}
A rational surface is a surface that admits parameterizations by a rational function. A rational surface is an algebraic surface. Given an algebraic surface, it is commonly easier to decide if it is rational than to compute its rational parameterization, if it exists. Surfaces of revolution give another important class of surfaces that can be easily parametrized. If the graph z = f(x), a ≤ x ≤ b is rotated about the z-axis then the resulting surface has a parametrization r ( u , ϕ ) = ( u cos ϕ , u sin ϕ , f ( u ) ) , a ≤ u ≤ b , 0 ≤ ϕ < 2 π . {\displaystyle \mathbf {r} (u,\phi )=(u\cos \phi ,u\sin \phi ,f(u)),\quad a\leq u\leq b,0\leq \phi <2\pi .} It may also be parameterized r ( u , v ) = ( u 1 − v 2 1 + v 2 , u 2 v 1 + v 2 , f ( u ) ) , a ≤ u ≤ b , {\displaystyle \mathbf {r} (u,v)=\left(u{\frac {1-v^{2}}{1+v^{2}}},u{\frac {2v}{1+v^{2}}},f(u)\right),\quad a\leq u\leq b,} showing that, if the function f is rational, then the surface is rational. The straight circular cylinder of radius R about x-axis has the following parametric representation: r ( x , ϕ ) = ( x , R cos ϕ , R sin ϕ ) . {\displaystyle \mathbf {r} (x,\phi )=(x,R\cos \phi ,R\sin \phi ).}
Using the spherical coordinates, the unit sphere can be parameterized by r ( θ , ϕ ) = ( cos θ sin ϕ , sin θ sin ϕ , cos ϕ ) , 0 ≤ θ < 2 π , 0 ≤ ϕ ≤ π . {\displaystyle \mathbf {r} (\theta ,\phi )=(\cos \theta \sin \phi ,\sin \theta \sin \phi ,\cos \phi ),\quad 0\leq \theta <2\pi ,0\leq \phi \leq \pi .} This parametrization breaks down at the north and south poles where the azimuth angle θ is not determined uniquely. The sphere is a rational surface.
The same surface admits many different parametrizations. For example, the coordinate z-plane can be parametrized as
r ( u , v ) = ( a u + b v , c u + d v , 0 ) {\displaystyle \mathbf {r} (u,v)=(au+bv,cu+dv,0)}
for any constants a, b, c, d such that ad − bc ≠ 0, i.e. the matrix [ a b c d ] {\displaystyle {\begin{bmatrix}a&b\\c&d\end{bmatrix}}} is invertible.
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