A parametric surface is a surface in the Euclidean space R³ which is defined by a parametric equation with two parameters. Parametric representation is the most general way to specify a surface. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and 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. An open surface with X-, Y-, and Z-contours shown. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ... Graph of a butterfly curve, a parametric equation discovered by Temple H. Fay In mathematics, a parametric equation explicitly relates two or more variables in terms of one or more independent parameters. ... Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... Stokes theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ... In vector calculus, the divergence theorem, also known as Gauss theorem, Ostrogradskys theorem, or Gauss-Ostrogradsky theorem is a result that relates the flow (that is, flux) of a vector field through a surface to the behaviour of the vector field inside the surface. ... Determining the length of an irregular arc segment—also called rectification of a curve—was historically difficult. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... Area is the measure of how much exposed area any two dimensional object has. ... In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of R3. ... In differential geometry, the second fundamental form is a quadratic form on the tangent space of a hypersurface, usually denoted by II. It is an equivalent way to describe the shape operator (denoted by S) of a hypersurface, where denoted covariant derivative and n a field of normal vectors on... Curvature is the amount by which a geometric object deviates from being flat. ... In mathematics, mean curvature of a surface is a notion from differential geometry. ... Principal curvature is the inverse of the radius of the osculating circle. ...

Examples

• The simplest type of parametric surfaces is given by the graphs of functions of two variables:

• 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

• 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

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 The parabola y=x2 rotated about the z-axis A surface of revolution is a surface created by rotating a curve lying on some plane (the generatrix) around a straight line (the axis of rotation) that lies on the same plane. ... A right circular cylinder An elliptic cylinder In mathematics, a cylinder is a quadric surface, with the following equation in Cartesian coordinates: This equation is for an elliptic cylinder, a generalization of the ordinary, circular cylinder (a = b). ... This article describes some of the common coordinate systems that appear in elementary mathematics. ... For other uses, see Sphere (disambiguation). ...

for any constants a, b, c, d such that ad − bc ≠ 0, i.e. the matrix is invertible. In linear algebra, an n-by-n (square) matrix A is called invertible or non-singular if there exists an n-by-n matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...

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. As the degree of the taylor series rises, it approaches the correct function. ... This article is about the concept of integrals in calculus. ...

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, A graph of the vector-valued function <2Cos(t),4Sin(t),t> A vector-valued function is a mathematical function that maps real numbers onto vectors. ...

In vector calculus, the parameters are frequently denoted (s,t) and the partial derivatives are written out using the ∂-notation: Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ...

Tangent plane and normal vector

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 parametrised surface at a regular point: In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... For the cross product in algebraic topology, see Künneth theorem. ...

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. The torus is an orientable surface. ...

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: Area is the measure of how much exposed area any two dimensional object has. ...

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 true for a circular cylinder, sphere, cone, torus, and a few other surfaces of revolution. In mathematical analysis, there is a serious distinction between a double integral and an iterated integral. ... A computer algebra system (CAS) is a software program that facilitates symbolic mathematics. ... A right circular cylinder An elliptic cylinder In mathematics, a cylinder is a quadric surface, with the following equation in Cartesian coordinates: This equation is for an elliptic cylinder, a generalization of the ordinary, circular cylinder (a = b). ... For other uses, see Sphere (disambiguation). ... This article is about the geometric object, for other uses see Cone. ... In geometry, a torus (pl. ... The parabola y=x2 rotated about the z-axis A surface of revolution is a surface created by rotating a curve lying on some plane (the generatrix) around a straight line (the axis of rotation) that lies on the same plane. ...

First fundamental form

Main article: First fundamental form

The first fundamental form is a quadratic form In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of R3. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ...

I = Edu² + 2Fdudv + Gdv²

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: In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. ...

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. Determining the length of an irregular arc segment—also called rectification of a curve—was historically difficult. ...

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:

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 In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ... In mathematics, a bilinear form on a vector space V over a field F is a mapping V × V → F which is linear in both arguments. ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ...

expressing the cosine of the angle via the dot product. In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

Surface area can be expressed in terms of the first fundamental form as follows:

The expression under the square root is precisely , and so it is strictly positive at the regular points.

Second fundamental form

Main article: Second fundamental form

The second fundamental form In differential geometry, the second fundamental form is a quadratic form on the tangent space of a hypersurface, usually denoted by II. It is an equivalent way to describe the shape operator (denoted by S) of a hypersurface, where denoted covariant derivative and n a field of normal vectors on...

II = Ldu² + 2Mdudv + Ndv²

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 horisontal, the second fundamental form is essentially the quadratic part of the Taylor expansion of z as a function of x and y. As the degree of the taylor series rises, it approaches the correct function. ...

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: In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). ...

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

Main article: Curvature

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. In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. ... In mathematics, an invariant is something that does not change under a set of transformations. ... Curvature is the amount by which a geometric object deviates from being flat. ... In mathematics, mean curvature of a surface is a notion from differential geometry. ... Principal curvature is the inverse of the radius of the osculating circle. ...

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 curvature K = κ₁κ₂ and the mean curvature H = 1/2(κ₁ + κ₂) 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 form. Look up Convex set in Wiktionary, the free dictionary. ... In linear algebra, a positive-definite matrix is a Hermitian matrix which in many ways is analogous to a positive real number. ...

See also

• Spline (mathematics)
• Surface normal

One type of spline, a bézier curve In the mathematical subfield of numerical analysis, a spline is a special function defined piecewise by polynomials. ... A surface normal, or just normal to a flat surface is a three-dimensional vector which is perpendicular to that surface. ...

External links

• Java applets demonstrate the parametrization of a helix surface