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
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.

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
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.

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
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.

Gaussian curvature

Gauss curvaturecurvatureLiebmann's theorem
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.

Mean curvature

average curvaturemeanmean radius of curvature
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.

Principal curvature

principal curvaturesprincipal directionsprincipal radii of curvature
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.

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.

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.