# Parametric surface

**Curvature of parametric surfacesparameterizeparametricparametricallyparametrizationparametrized surfacesurface parameterization**

A parametric surface is a surface in the Euclidean space \Bbb R^3 which is defined by a parametric equation with two parameters Parametric representation is a very general way to specify a surface, as well as implicit representation.wikipedia

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### Surface (mathematics)

**surfacesurfaces2-dimensional shape**

A parametric surface is a surface in the Euclidean space \Bbb R^3 which is defined by a parametric equation with two parameters Parametric representation is a very general way to specify a surface, as well as implicit representation.

In this case, one says that one has a parametric surface, which is parametrized by these two variables, called parameters.

### Surface area

**SurfaceAreafootprint**

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.

Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces.

### Second fundamental form

**extrinsic curvaturesecondshape tensor**

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 second fundamental form of a parametric surface

### First fundamental form

**firstfirst quadratic forms**

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.

be a parametric surface.

### Curve

**closed curvespace curvesmooth curve**

### Vector-valued function

**vector functionvector-valued functionsvector**

where \vec{r} is a vector-valued function of the parameters (u, v) and the parameters vary within a certain domain D in the parametric uv-plane.

### Euclidean space

**EuclideanspaceEuclidean vector space**

A parametric surface is a surface in the Euclidean space \Bbb R^3 which is defined by a parametric equation with two parameters Parametric representation is a very general way to specify a surface, as well as implicit representation.

### Parametric equation

**parametric curveparametricparametric equations**

A parametric surface is a surface in the Euclidean space \Bbb R^3 which is defined by a parametric equation with two parameters Parametric representation is a very general way to specify a surface, as well as implicit representation.

### Implicit surface

**implicit equationimplicit representationimplicit**

### Vector calculus

**vector analysisvectorvector algebra**

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.

### Stokes' theorem

**Stokes theoremStokes's theoremKelvin–Stokes theorem**

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.

### Divergence theorem

**Gauss's theoremGauss theoremdivergent-free**

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.

### Arc length

**rectifiable curvearclengthlength**

### Gaussian curvature

**Gauss curvaturecurvatureLiebmann's theorem**

### Mean curvature

**average curvaturemeanmean radius of curvature**

### Principal curvature

**principal curvaturesprincipal directionsprincipal radii of curvature**

### Cylinder

**cylindricalcylindersrod**

* The straight circular cylinder of radius R about x-axis has the following parametric representation: This is true for a circular cylinder, sphere, cone, torus, and a few other surfaces of revolution.

### Spherical coordinate system

**spherical coordinatessphericalspherical polar coordinates**

* Using the spherical coordinates, the unit sphere can be parameterized by

### Sphere

**sphericalhemisphereglobose**

* Using the spherical coordinates, the unit sphere can be parameterized by This is true for a circular cylinder, sphere, cone, torus, and a few other surfaces of revolution.

### Invertible matrix

**invertibleinversenonsingular**

for any constants a, b, c, d such that ad − bc ≠ 0, i.e. the matrix is invertible.

### Rational surface

**Castelnuovo's theoremCastelnuovo theoremMinimal rational surface**

### Rational function

**rational functionsrationalrational fraction**

### Algebraic surface

**algebraic surfacessurfacessurface**

### Surface of revolution

**surfaces of revolutionrevolutionof revolution**

This is true for a circular cylinder, sphere, cone, torus, and a few other surfaces of revolution.

### Taylor series

**Taylor expansionMaclaurin seriesTaylor polynomial**

The local shape of a parametric surface can be analyzed by considering the Taylor expansion of the function that parametrizes it.