- Parametric surface
A parametric surface is a
surface in theEuclidean space R3 which is defined by aparametric equation with two parameters. Parametric representation is the most general way to specify a surface. Surfaces that occur in two of the main theorems ofvector calculus ,Stokes' theorem anddivergence theorem , are frequently given in a parametric form. The curvature andarc length ofcurve s 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.Examples
* The simplest type of parametric surfaces is given by the graphs of functions of two variables:
::z = f(x,y), quad vec r(x,y) = (x, y, f(x,y)).
* Surfaces of revolution give another important class of surfaces that can be easily parametrized. If the graph "y" = "f"("x"), "a" ≤ "x" ≤ "b" is rotated about the "z"-axis then the resulting surface has a parametrization
:: vec r(u,phi) = (ucosphi, usinphi, f(u)), quad aleq uleq b, 0leqphi < 2pi.
* The straight circular cylinder of radius "R" about "x"-axis has the following parametric representation:
::vec r(x, phi) = (x, Rcosphi, Rsinphi).
* Using the
spherical coordinates , the unitsphere can be parameterized by::vec r( heta,phi) = (sin heta cosphi, sin heta sin phi, cos heta), quad 0 leq heta leq pi, -pi < phi leq pi.
: This parametrization breaks down at the north and south poles where the polar angle "φ" is not determined uniquely.
The same surface admits many different parametrizations. For example, the coordinate "z"-plane can parametrised as
:vec r(u,v)=(au+bv,cv+dv, 0)
for any constants "a", "b", "c", "d" such that "ad" − "bc" ≠ 0, i.e. the matrix egin{bmatrix}a & b\ c & dend{bmatrix} is invertible.
Local differential geometry
The local shape of a parametric surface can be analyzed by considering the
Taylor expansion of the function that parametrizes it. The arc length of a curve on the surface and the surface area can be found using integration.Notation
Let the parametric surface be given by the equation
: vec{r}=vec{r}(u,v),
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. The first partial derivatives with respect to the parameters are usually denoted vec{r}_u and vec{r}_v, and similarly for the higher derivatives, vec{r}_{uu}, vec{r}_{uv}, vec{r}_{vv}.In
vector calculus , the parameters are frequently denoted ("s","t") and the partial derivatives are written out using the ∂-notation:: frac{partialvec{r{partial s}, frac{partialvec{r{partial t}, frac{partial^2vec{r{partial s^2}, frac{partial^2vec{r{partial spartial t}, frac{partial^2vec{r{partial t^2}.
Tangent plane and normal vector
The parametrization is regular for the given values of the parameters if the vectors
: vec{r}_u, vec{r}_v
are linearly independent. The
tangent plane at a regular point is the affine plane in R3 spanned by these vectors and passing through the point "r"("u", "v") on the surface determined by the parameters. Any tangent vector can be uniquely decomposed into alinear combination of vec{r}_u and vec{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 parametrised surface at a regular point:: vec{n}=frac{vec{r}_u imesvec{r}_v}{left|vec{r}_u imesvec{r}_v ight.
In general, there are two choices of the unit
normal vector to a surface at a given point, but for a regular parametrised surface, the preceding formula consistently picks one of them, and thus determines an orientation of the surface. Some of the differential-geometric invariants of a surface in R3 are defined by the surface itself and are independent of the orientation, while others change the sign if the orientation is reversed.urface area
The
surface area can be calculated by integrating the length of the normal vector vec{r}_u imesvec{r}_v to the surface over the appropriate region "D" in the parametric "uv" plane:: A(D) = iint_Dleft |vec{r}_u imesvec{r}_v ight |du dv.
Although this formula provides a closed expression for the surface area, for all but very special surfaces this results in a complicated
double 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 true for a circular cylinder,sphere , cone,torus , and a few other surfaces of revolution.First fundamental form
The first fundamental form is a
quadratic form : I = E du^2 + 2F du dv + G dv^2
on the
tangent plane to the surface which is used to calculate distances and angles. For a parametrized surface vec r=vec r(u,v), its coefficients can be computed as follows:: E=vec r_ucdotvec r_u, quadF=vec r_ucdotvec r_v, quadG=vec r_vcdot vec r_v.
Arc length of parametrised curves on the surface "S", the angle between curves on "S", and the surface area all admit expressions in terms of the first fundamental form.If ("u"("t"), "v"("t")), "a" ≤ "t" ≤ "b" represents a parametrised curve on this surface then its arc length can be calculated as the integral:
: int_a^b sqrt{E,u'(t)^2 + 2F,u'(t)v'(t) + G,v'(t)^2}, dt.
The first fundamental form may be viewed as a family of positive definite
symmetric bilinear form s 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 on "S" 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: cos heta = frac{vec{a}cdotvec{b{left|vec{a} ight| |vec{b}
expressing the
cosine of the angle via the dot product.Surface area can be expressed in terms of the first fundamental form as follows:
: A(D) = iint_D sqrt{EG-F^2}, du dv.
The expression under the square root is precisely vec{r}_u imesvec{r}_v, and so it is strictly positive at the regular points.
econd fundamental form
The second fundamental form
: mathrm{II} = L du^2 + 2M du dv + N dv^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 the
Taylor expansion of "z" as a function of "x" and "y".For a general parametric surface, the definition is more complicated, but the second fundamental form depends only on the
partial derivative s of order one and two. Its coefficients are defined to be the projections of the second partial derivatives of vec{r} onto the unit normal vector vec{n} defined by the parametrization:: L = vec r_{uu}cdot vec n, quadM = vec r_{uv}cdot vec n, quadN = vec r_{vv}cdot vec n. quad
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.
Curvature
The first and second fundamental forms of a surface determine its important differential-geometric invariants: the
Gaussian curvature , themean curvature , and theprincipal curvature s.The principal curvatures are the invariants of the pair consisting of the second and first fundamental forms. They are the roots "κ"1, "κ"2 of the quadratic equation
: det(mathrm{II}-kappamathrm{I})=0, quaddetleft|egin{matrix}L-kappa E & M-kappa F \ M-kappa F & N-kappa G end{matrix} ight| = 0.
The Gaussian curvature "K" = "κ"1"κ"2 and the mean curvature "H" = 1/2("κ"1 + "κ"2) can be computed as follows:
:K={LN-M^2over EG-F^2}, quad H={EN-2FM+GLover 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: for "K" > 0 the surface is locally convex and the point is called "elliptic", while for "K" < 0 the surface is saddle shaped and the point is called "hyperbolic". The points at which the Gaussian curvature is zero are called "parabolic". In general, parabolic points form a curve on the surface called the "parabolic line". The first fundamental form is positive definite, hence its determinant "EG" − "F"2 is positive everywhere. Therefore, the sign of "K" coincides with the sign of "LN" − "M"2, the determinant of the second fundamental form.
ee also
*
Spline (mathematics)
*Surface normal External links
* [http://www.math.umn.edu/~nykamp/m2374/readings/planeparamex/ Java applets demonstrate the parametrization of a helix surface]
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