Parametric surface

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.

Curvature Implicit surface Curve Parametric surface Graph of a function

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.

Area Sphere Parametric surface Minkowski content Cube

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

Quadratic form Parametric surface Principal curvature First fundamental form Gauss–Codazzi equations

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.

Parametric surface Canonical form Metric tensor Curvature Second fundamental form

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.

Space-filling curve Parametric equation Algebraic curve Surface (mathematics)Transcendental 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.

Parametric equation Function (mathematics)Dependent and independent variables Vector field Euclidean vector

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.

Three-dimensional space Dot product Reflection (mathematics)Non-Euclidean geometry Platonic solid

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.

Curve Parameter Vector-valued function Plane curve Function (mathematics)

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.

Surface (mathematics)Euclidean space Implicit function theorem Tangent space Normal (geometry)

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.

Vector field Partial derivative Multivariable calculus Oliver Heaviside Vector Analysis

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.

Differential form Exterior derivative Sir George Stokes, 1st Baronet Smith's Prize Kelvin–Stokes theorem

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.

Fluid dynamics Vector calculus Vector field Flux Electrostatics

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.

Elliptic integral Cycloid Semicubical parabola Ellipse Radian

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.

Differential geometry Hyperboloid Theorema Egregium Developable surface Hyperbolic geometry

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.

Minimal surface Young–Laplace equation Sophie Germain Curvature Differential geometry of surfaces

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.

Gaussian curvature Second fundamental form Curvature Darboux frame Minimal surface

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.

Solid geometry Steel and tin cans Surface (mathematics)Ellipse Cylindrical coordinate system

Spherical coordinate system

spherical coordinatessphericalspherical polar coordinates

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

Azimuth Fundamental plane (spherical coordinates)Polar coordinate system Celestial coordinate system Phi

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.

Ball (mathematics)Radius Diameter Solid geometry Surface (topology)

Invertible matrix

invertibleinversenonsingular

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

Determinant Linear algebra General linear group Matrix multiplication Identity matrix

Torus Surface (mathematics)Pappus's centroid theorem Sphere Minimal surface 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.

Brook Taylor Colin Maclaurin Taylor's theorem Analytic function Power series

Parametric surface

Surface (mathematics)

Surface area

Second fundamental form

First fundamental form

Curve

Vector-valued function

Euclidean space

Parametric equation

Implicit surface

Vector calculus

Stokes' theorem

Divergence theorem

Arc length

Gaussian curvature

Mean curvature

Principal curvature

Cylinder

Spherical coordinate system

Sphere

Invertible matrix

Rational surface

Rational function

Algebraic surface

Surface of revolution

Taylor series

Parametric surface

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Surface (mathematics)

Surface area

Second fundamental form

First fundamental form

Curve

Vector-valued function

Euclidean space

Parametric equation

Implicit surface

Vector calculus

Stokes' theorem

Divergence theorem

Arc length

Gaussian curvature

Mean curvature

Principal curvature

Cylinder

Spherical coordinate system

Sphere

Invertible matrix

Rational surface

Rational function

Algebraic surface

Surface of revolution

Taylor series