The simplest type of parametric surfaces is given by the graphs of functions of two variables:
A
rational surface is a surface that admits parameterizations by a
rational function. A rational surface is an
algebraic surface. Given an algebraic surface, it is commonly easier to decide if it is rational than to compute its rational parameterization, if it exists.
Surfaces of revolution give another important class of surfaces that can be easily parametrized. If the graph z = f(x), a ≤ x ≤ b is rotated about the z-axis then the resulting surface has a parametrization
It may also be parameterized
showing that, if the function f is rational, then the surface is rational.
The straight circular
cylinder of radius R about x-axis has the following parametric representation:
This parametrization breaks down at the north and south poles where the azimuth angle θ is not determined uniquely. The sphere is a rational surface.
The same surface admits many different parametrizations. For example, the coordinate z-plane can be parametrized as
for any constants a, b, c, d such that ad − bc ≠ 0, i.e. the matrix 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
where 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 and and similarly for the higher derivatives,
In
vector calculus, the parameters are frequently denoted (s,t) and the partial derivatives are written out using the ∂-notation:
The parametrization is regular for the given values of the parameters if the vectors
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 a
linear combination of and The
cross product of these vectors is a
normal vector to the
tangent plane. Dividing this vector by its length yields a unit
normal vector to the parametrized surface at a regular point:
In general, there are two choices of the unit
normal vector to a surface at a given point, but for a regular parametrized 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.
Surface area
The
surface area can be calculated by integrating the length of the normal vector to the surface over the appropriate region D in the parametric uv plane:
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 a
computer algebra system or approximated numerically. Fortunately, many common surfaces form exceptions, and their areas are explicitly known. This is true for a
circular cylinder,
sphere,
cone,
torus, and a few other
surfaces of revolution.
This can also be expressed as a
surface integral over the scalar field 1:
on the
tangent plane to the surface which is used to calculate distances and angles. For a parametrized surface its coefficients can be computed as follows:
Arc length of parametrized 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 parametrized curve on this surface then its arc length can be calculated as the integral:
The first fundamental form may be viewed as a family of
positive definitesymmetric bilinear forms 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 their
dot product, and the angle can be found from the standard formula
expressing the
cosine of the angle via the dot product.
Surface area can be expressed in terms of the first fundamental form as follows:
By
Lagrange's identity, the expression under the square root is precisely , and so it is strictly positive at the regular points.
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 derivatives of order one and two.
Its coefficients are defined to be the projections of the second partial derivatives of onto the unit normal vector defined by the parametrization:
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.
The principal curvatures are the invariants of the pair consisting of the second and first fundamental forms. They are the roots κ1, κ2 of the quadratic equation
The Gaussian curvatureK = κ1κ2 and the mean curvatureH = (κ1 + κ2)/2 can be computed as follows:
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 − F2 is positive everywhere. Therefore, the sign of K coincides with the sign of LN − M2, the determinant of the second fundamental.
The coefficients of the
first fundamental form presented above may be organized in a symmetric matrix:
Defining now matrix , the principal curvatures κ1 and κ2 are the
eigenvalues of A.[1]
Now, if v1 = (v11, v12) is the
eigenvector of A corresponding to principal curvature κ1, the unit vector in the direction of is called the principal vector corresponding to the principal curvature κ1.
Accordingly, if v2 = (v21,v22) is the
eigenvector of A corresponding to principal curvature κ2, the unit vector in the direction of is called the principal vector corresponding to the principal curvature κ2.