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A **parametric surface** is a surface in the Euclidean space 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. 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. 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 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 - The straight circular cylinder of radius
*R*about*x*-axis has the following parametric representation: - Using the spherical coordinates, the unit sphere can be parameterized by
*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

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.

Let the parametric surface be given by the equation

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

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 **R**^{3} are defined by the surface itself and are independent of the orientation, while others change the sign if the orientation is reversed.

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:

The **first fundamental form** is a quadratic form

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 definite symmetric 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

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.

The second fundamental form

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 first and second fundamental forms of a surface determine its important differential-geometric invariants: the Gaussian curvature, the mean curvature, and the principal curvatures.

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

The **Gaussian curvature** *K* = *κ*_{1}*κ*_{2} and the **mean curvature** *H* = (*κ*_{1} + *κ*_{2})/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* − *F*^{2} is positive everywhere. Therefore, the sign of *K* coincides with the sign of *LN* − *M*^{2}, 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 *κ*_{1} and *κ*_{2} are the eigenvalues of *A*.^{[1]}

Now, if **v**_{1} = (*v*_{11}, *v*_{12}) is the eigenvector of *A* corresponding to principal curvature *κ*_{1}, the unit vector in the direction of is called the principal vector corresponding to the principal curvature *κ*_{1}.

Accordingly, if **v**_{2} = (*v*_{21},*v*_{22}) is the eigenvector of *A* corresponding to principal curvature *κ*_{2}, the unit vector in the direction of is called the principal vector corresponding to the principal curvature *κ*_{2}.

**^**Surface curvatures*Handouts, Principal Curvatures*

- Java applets demonstrate the parametrization of a helix surface
- m-ART(3d) - iPad/iPhone application to generate and visualize parametric surfaces.