Aparametric surface is asurface in theEuclidean space${\displaystyle {\mathbb{R}}^3}$ which is defined by aparametric equation with two parameters⁠${\displaystyle \mathbf{r}:{\mathbb{R}}^2\to {\mathbb{R}}^3}$⁠. Parametric representation is a very general way to specify a surface, as well asimplicit representation. Surfaces that occur in two of the main theorems ofvector calculus,Stokes' theorem, and thedivergence theorem, are frequently given in a parametric form. The curvature andarc length ofcurves on the surface,surface area, differential geometric invariants such as thefirst andsecond fundamental forms,Gaussian,mean, andprincipal curvatures can all be computed from a given parametrization.

• The simplest type of parametric surfaces is abivariate surface, given by thegraphs of functions of two variables (bivariate functions):$${\displaystyle z=f(x,y),\mathbf{r}(x,y)=(x,y,f(x,y)).}$$
• Arational surface is a surface that admits parameterizations by arational function. A rational surface is analgebraic 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 graphz =f(x),a ≤x ≤b is rotated about thez-axis then the resulting surface has a parametrization It may also be parameterized$${\displaystyle \mathbf{r}(u,v)=\left(u\frac{1-v^2}{1+v^2},u\frac{2v}{1+v^2},f(u)\right),a\le u\le b,}$$ showing that, if the functionf is rational, then the surface is rational.
• The straight circularcylinder of radiusR aboutx-axis has the following parametric representation:$${\displaystyle \mathbf{r}(x,\varphi )=(x,R\cos \varphi ,R\sin \varphi ).}$$
• Using thespherical coordinates, the unitsphere can be parameterized by This parametrization breaks down at the north and south poles where theazimuth angleθ is not determined uniquely. The sphere is a rational surface.

The same surface admits many different parametrizations. For example, the coordinatez-plane can be parametrized as$${\displaystyle \mathbf{r}(u,v)=(au+bv,cu+dv,0)}$$for any constantsa,b,c,d such thatad −bc ≠ 0, i.e. the matrix isinvertible.

The local shape of a parametric surface can be analyzed by considering theTaylor expansion of the function that parametrizes it. The arc length of a curve on the surface and the surface area can be found usingintegration.

Let the parametric surface be given by the equation$${\displaystyle \mathbf{r}=\mathbf{r}(u,v),}$$where${\displaystyle \mathbf{r}}$ is avector-valued function of the parameters (u,v) and the parameters vary within a certain domainD in the parametricuv-plane. The first partial derivatives with respect to the parameters are usually denoted${\mathbf{r}}_u:=\frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }u}$ and${\displaystyle {\mathbf{r}}_v,}$ and similarly for the higher derivatives,${\displaystyle {\mathbf{r}}_{uu},{\mathbf{r}}_{uv},{\mathbf{r}}_{vv}.}$

Invector calculus, the parameters are frequently denoted (s,t) and the partial derivatives are written out using the∂-notation:$${\displaystyle \frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }s},\frac{\mathrm{\partial }\mathbf{r}}{\mathrm{\partial }t},\frac{{\mathrm{\partial }}^2\mathbf{r}}{\mathrm{\partial }{s}^2},\frac{{\mathrm{\partial }}^2\mathbf{r}}{\mathrm{\partial }s\mathrm{\partial }t},\frac{{\mathrm{\partial }}^2\mathbf{r}}{\mathrm{\partial }{t}^2}.}$$

The parametrization isregular for the given values of the parameters if the vectors$${\displaystyle {\mathbf{r}}_u,{\mathbf{r}}_v}$$are linearly independent. Thetangent plane at a regular point is the affine plane inR³ spanned by these vectors and passing through the pointr(u,v) on the surface determined by the parameters. Anytangent vector can be uniquely decomposed into alinear combination of${\displaystyle {\mathbf{r}}_u}$ and${\displaystyle {\mathbf{r}}_v.}$ Thecross product of these vectors is anormal vector to thetangent plane. Dividing this vector by its length yields a unitnormal vector to the parametrized surface at a regular point:$${\displaystyle \hat{\mathbf{n}}=\frac{{\mathbf{r}}_u\times {\mathbf{r}}_v}{\left|{\mathbf{r}}_u\times {\mathbf{r}}_v\right|}.}$$

In general, there are two choices of the unitnormal vector to a surface at a given point, but for a regular parametrized surface, the preceding formula consistently picks one of them, and thus determines anorientation of the surface. Some of the differential-geometric invariants of a surface inR³ are defined by the surface itself and are independent of the orientation, while others change the sign if the orientation is reversed.

Thesurface area can be calculated by integrating the length of the normal vector${\displaystyle {\mathbf{r}}_u\times {\mathbf{r}}_v}$ to the surface over the appropriate regionD in the parametricuv plane:$${\displaystyle A(D)={\iint }_D\left|{\mathbf{r}}_u\times {\mathbf{r}}_v\right|dudv.}$$

Although this formula provides a closed expression for the surface area, for all but very special surfaces this results in a complicateddouble integral, which is typically evaluated using acomputer algebra system or approximated numerically. Fortunately, many common surfaces form exceptions, and their areas are explicitly known. This is true for acircular cylinder,sphere,cone,torus, and a few othersurfaces of revolution.

This can also be expressed as asurface integral over thescalar field 1:$${\displaystyle {\int }_{S}1dS.}$$

Thefirst fundamental form is aquadratic form$${\displaystyle \mathrm{I}=Ed{u}^2+2Fdudv+Gd{v}^2}$$on thetangent plane to the surface which is used to calculate distances and angles. For a parametrized surface${\displaystyle \mathbf{r}=\mathbf{r}(u,v),}$ its coefficients can be computed as follows:$${\displaystyle E={\mathbf{r}}_u\cdot {\mathbf{r}}_u,F={\mathbf{r}}_u\cdot {\mathbf{r}}_v,G={\mathbf{r}}_v\cdot {\mathbf{r}}_v.}$$

Arc length of parametrized curves on the surfaceS, the angle between curves onS, and the surface area all admit expressions in terms of the first fundamental form.

If(u(t),v(t)),a ≤t ≤b represents a parametrized curve on this surface then its arc length can be calculated as the integral:$${\displaystyle {\int }_a^b\sqrt{E{u}^{\prime }(t{)}^2+2F{u}^{\prime }(t)v^{\prime }(t)+G{v}^{\prime }(t{)}^2}dt.}$$

The first fundamental form may be viewed as a family ofpositive definitesymmetric bilinear forms on the tangent plane at each point of the surface depending smoothly on the point. This perspective helps one calculate the angle between two curves onS intersecting at a given point. This angle is equal to the angle between the tangent vectors to the curves. The first fundamental form evaluated on this pair of vectors is theirdot product, and the angle can be found from the standard formula$${\displaystyle \cos \theta =\frac{\mathbf{a}\cdot \mathbf{b}}{\left|\mathbf{a}\right|\left|\mathbf{b}\right|}}$$expressing thecosine of the angle via the dot product.

Surface area can be expressed in terms of the first fundamental form as follows:$${\displaystyle A(D)={\iint }_D\sqrt{EG-F^2}dudv.}$$

ByLagrange's identity, the expression under thesquare root is precisely${\displaystyle {\left|{\mathbf{r}}_u\times {\mathbf{r}}_v\right|}^2}$, and so it is strictly positive at the regular points.

The second fundamental form$${\displaystyle \mathrm{I}\mathrm{I}=Ld{u}^2+2Mdudv+Nd{v}^2}$$is a quadratic form on the tangent plane to the surface that, together with the first fundamental form, determines the curvatures of curves on the surface. In the special case when(u,v) = (x,y) and the tangent plane to the surface at the given point is horizontal, the second fundamental form is essentially the quadratic part of theTaylor expansion ofz as a function ofx andy.

For a general parametric surface, the definition is more complicated, but the second fundamental form depends only on thepartial derivatives of order one and two. Its coefficients are defined to be the projections of the second partial derivatives of${\displaystyle \mathbf{r}}$ onto the unit normal vector${\displaystyle \hat{\mathbf{n}}}$ defined by the parametrization:$${\displaystyle L={\mathbf{r}}_{uu}\cdot \hat{\mathbf{n}},M={\mathbf{r}}_{uv}\cdot \hat{\mathbf{n}},N={\mathbf{r}}_{vv}\cdot \hat{\mathbf{n}}.}$$

Like the first fundamental form, the second fundamental form may be viewed as a family of symmetric bilinear forms on the tangent plane at each point of the surface depending smoothly on the point.

The first and second fundamental forms of a surface determine its important differential-geometricinvariants: theGaussian curvature, themean curvature, and theprincipal curvatures.

The principal curvatures are the invariants of the pair consisting of the second and first fundamental forms. They are the rootsκ₁,κ₂ of the quadratic equation

TheGaussian curvatureK =κ₁κ₂ and themean curvatureH = (κ₁ +κ₂)/2 can be computed as follows:$${\displaystyle K=\frac{LN-M^2}{EG-F^2},H=\frac{EN-2FM+GL}{2(EG-F^2)}.}$$

Up to a sign, these quantities are independent of the parametrization used, and hence form important tools for analysing the geometry of the surface. More precisely, the principal curvatures and the mean curvature change the sign if the orientation of the surface is reversed, and the Gaussian curvature is entirely independent of the parametrization.

The sign of the Gaussian curvature at a point determines the shape of the surface near that point: forK > 0 the surface is locallyconvex and the point is calledelliptic, while forK < 0 the surface is saddle shaped and the point is calledhyperbolic. The points at which the Gaussian curvature is zero are calledparabolic. In general, parabolic points form a curve on the surface called theparabolic line. The first fundamental form ispositive definite, hence its determinantEG −F² is positive everywhere. Therefore, the sign ofK coincides with the sign ofLN −M², the determinant of the second fundamental.

The coefficients of thefirst fundamental form presented above may be organized in asymmetric matrix:And the same for the coefficients of thesecond fundamental form, also presented above:

Defining now matrix${\displaystyle A=F_1^{-1}{F}_2}$, the principal curvaturesκ₁ andκ₂ are theeigenvalues ofA.^[1]

Now, ifv₁ = (v₁₁,v₁₂) is theeigenvector ofA corresponding to principal curvatureκ₁, theunit vector in the direction of${\displaystyle {\mathbf{t}}_1=v_{11}{\mathbf{r}}_u+v_{12}{\mathbf{r}}_v}$ is called the principal vector corresponding to the principal curvatureκ₁.

Accordingly, ifv₂ = (v₂₁,v₂₂) is theeigenvector ofA corresponding to principal curvatureκ₂, the unit vector in the direction of${\displaystyle {\mathbf{t}}_2=v_{21}{\mathbf{r}}_u+v_{22}{\mathbf{r}}_v}$ is called the principal vector corresponding to the principal curvatureκ₂.

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