Parametric surface

A parametric apparent is a apparent in the Euclidean amplitude R3 which is authentic by a parametric blueprint with two parameters. Parametric representation is the a lot of accepted way to specify a surface. Surfaces that action in two of the capital theorems of agent calculus, Stokes' assumption and the alteration theorem, are frequently accustomed in a parametric form. The curvature and arc breadth of curves on the surface, apparent area, cogwheel geometric invariants such as the aboriginal and additional axiological forms, Gaussian, mean, and arch curvatures can all be computed from a accustomed parametrization.

Examples

The simplest blazon of parametric surfaces is accustomed by the graphs of functions of two variables:

Surfaces of anarchy accord addition important chic of surfaces that can be calmly parametrized. If the blueprint z = f(x), a ≤ x ≤ b is rotated about the z-axis again the consistent apparent has a parametrization

The beeline annular butt of ambit R about x-axis has the afterward parametric representation:

Using the all-around coordinates, the assemblage apple can be parameterized by

This parametrization break down at the arctic and south poles area the arctic bend θ is not bent uniquely.

The aforementioned apparent admits abounding altered parametrizations. For example, the alike z-plane can be parametrized as

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

Local differential geometry

The bounded appearance of a parametric apparent can be analyzed by because the Taylor amplification of the action that parametrizes it. The arc breadth of a ambit on the apparent and the apparent breadth can be begin application integration.

editNotation

Let the parametric apparent be accustomed by the equation

where is a vector-valued action of the ambit (u, v) and the ambit alter aural a assertive breadth D in the parametric uv-plane. The aboriginal fractional derivatives with account to the ambit are usually denoted and and analogously for the college derivatives,

In agent calculus, the ambit are frequently denoted (s,t) and the fractional derivatives are accounting out application the ∂-notation:

editTangent even and accustomed vector

The parametrization is approved for the accustomed ethics of the ambit if the vectors

are linearly independent. The departure even at a approved point is the affine even in R3 spanned by these vectors and casual through the point r(u, v) on the apparent bent by the parameters. Any departure agent can be abnormally addle into a beeline aggregate of and The cantankerous artefact of these vectors is a accustomed agent to the departure plane. Dividing this agent by its breadth yields a assemblage accustomed agent to the parametrised apparent at a approved point:

In general, there are two choices of the assemblage accustomed agent to a apparent at a accustomed point, but for a approved parametrised surface, the above-mentioned blueprint consistently picks one of them, and appropriately determines an acclimatization of the surface. Some of the differential-geometric invariants of a apparent in R3 are authentic by the apparent itself and are absolute of the orientation, while others change the assurance if the acclimatization is reversed.

editSurface area

The apparent breadth can be affected by amalgam the breadth of the accustomed agent to the apparent over the adapted arena D in the parametric uv plane:

Although this blueprint provides a bankrupt announcement for the apparent area, for all but actual appropriate surfaces this after-effects in a complicated bifold integral, which is about evaluated application a computer algebra arrangement or approximated numerically. Fortunately, abounding accepted surfaces anatomy exceptions, and their areas are absolutely known. This is accurate for a annular cylinder, sphere, cone, torus, and a few added surfaces of revolution.

This can aswell be bidding as a apparent basic over the scalar acreage 1:

First fundamental form

The aboriginal axiological anatomy is a boxlike form

on the departure even to the apparent which is acclimated to account distances and angles. For a parametrized apparent its coefficients can be computed as follows:

Arc breadth of parametrised curves on the apparent S, the bend amid curves on S, and the apparent breadth all accept expressions in agreement of the aboriginal axiological form.

If (u(t), v(t)), a ≤ t ≤ b represents a parametrised ambit on this apparent again its arc breadth can be affected as the integral:

The aboriginal axiological anatomy may be beheld as a ancestors of absolute audible symmetric bilinear forms on the departure even at anniversary point of the apparent depending calmly on the point. This bend helps one account the bend amid two curves on S intersecting at a accustomed point. This bend is according to the bend amid the departure vectors to the curves. The aboriginal axiological anatomy evaluated on this brace of vectors is their dot product, and the bend can be begin from the accepted formula

expressing the cosine of the bend via the dot product.

Surface breadth can be bidding in agreement of the aboriginal axiological anatomy as follows:

The announcement beneath the aboveboard basis is absolutely , and so it is carefully absolute at the approved points.

Second fundamental form

The additional axiological form

is a boxlike anatomy on the departure even to the apparent that, calm with the aboriginal axiological form, determines the curvatures of curves on the surface. In the appropriate case if (u, v) = (x, y) and the departure even to the apparent at the accustomed point is horizontal, the additional axiological anatomy is about the boxlike allotment of the Taylor amplification of z as a action of x and y.

For a accepted parametric surface, the analogue is added complicated, but the additional axiological anatomy depends alone on the fractional derivatives of adjustment one and two. Its coefficients are authentic to be the projections of the additional fractional derivatives of assimilate the assemblage accustomed agent authentic by the parametrization:

Like the aboriginal axiological form, the additional axiological anatomy may be beheld as a ancestors of symmetric bilinear forms on the departure even at anniversary point of the apparent depending calmly on the point.

Curvature

The aboriginal and additional axiological forms of a apparent actuate its important differential-geometric invariants: the Gaussian curvature, the beggarly curvature, and the arch curvatures.

The arch curvatures are the invariants of the brace consisting of the additional and aboriginal axiological forms. They are the roots κ1, κ2 of the boxlike equation

The Gaussian curvature K = κ1κ2 and the beggarly curvature H = 1/2(κ1 + κ2) can be computed as follows:

Positive definite

Up to a sign, these quantities are absolute of the parametrization used, and appropriately anatomy important accoutrement for analysing the geometry of the surface. More precisely, the arch curvatures and the beggarly curvature change the assurance if the acclimatization of the apparent is reversed, and the Gaussian curvature is absolutely absolute of the parametrization.

The assurance of the Gaussian curvature at a point determines the appearance of the apparent abreast that point: for K > 0 the apparent is locally arched and the point is alleged elliptic, while for K < 0 the apparent is saddle shaped and the point is alleged hyperbolic. The credibility at which the Gaussian curvature is aught are alleged parabolic. In general, emblematic credibility anatomy a ambit on the apparent alleged the emblematic line. The aboriginal axiological anatomy is absolute definite, appropriately its account EG − F2 is absolute everywhere. Therefore, the assurance of K coincides with the assurance of LN − M2, the account of the additional fundamental