Parametric surface

A parametric surface is a surface defined by a parametric equation, involving two parameters. Typically they will be surfaces in three dimensions.

The simplest example of a parametric surface is the x-y plane. Here the surface is defined by the equation

$S:\mathbf {R} ^{2}\to \mathbf {R} ^{3}$, $S:(s,t)\to (s,t,0).\,$

The mapping S is a parameterization of the surface and the variables, s, t are said to be the parameters of the mapping. Any pair of value of s and t will give a point on the surface.

Many different parametrizations can give the same surface, for example the parametrisation

$S(s,t)=(s+t,s-t,0)\,$

also gives the x-y plane.

Surfaces can be defined in other ways, the plane can be defined as an algebraic surface which is the set of zeros of a polynomial equation. The x-y plane can be defined as the zeros of the function

$f:\mathbf {R} ^{3}\to \mathbf {R}$, $f(x,y,z)=z\,$

giving the surface

$f^{-1}(0),z=0.\,$

This can be generalised to the zeros of any implicit function. Other methods for defining surfaces include minimal surfaces defined through a process of minimising energy, soap bubbles are an example of this.

The unit sphere can be parameterized by

$s(\theta ,\phi )=(\sin \theta \;\cos \phi ,\sin \theta \;\sin \phi ,\cos \theta )\,$

where $0\leq \theta \leq \pi \,$ and $-\pi <\phi \leq \pi \,$ are the two parameters. This parametrisation breaks down at the north and south poles where the more than one set of parameters give the same point.

Local differential geometry

The local shape of a surface can be characterised by considering the partial derivatives of the parametrisation.

Notation: here lower case letters will be used for points and curves in the parameter space, which will be taken to be the plane, upper case will be used for points and curves on the surface. Likewise lower case vectors ${\vec {u}}$ will be tangent vectors in the plane and upper case vectors ${\vec {U}}$ will be the corresponding tangent vectors to the surface in R³.

For any point on a parameterized surface S(s,t), two tangent vectors are defined by taking the partial derivatives ${\vec {U}}={\partial S \over \partial s}$ and ${\vec {V}}={\partial S \over \partial t}$. Provided neither ${\vec {U}},{\vec {V}}$ are zero and they are not parallel then they define a tangent plane. The tangent plane will have a normal vector ${\vec {N}}={\vec {U}}\times {\vec {V}}$ which will be at right angles to any tangent vectors, this can be made into a unit normal vector by dividing by its length. The tangent plane does not depend on the particular parametrisation chosen, and the unit length normal vector will only change up to sign (that is point in the opposite direction).

Directional derivatives

The partial derivatives can be expanded to give a directional derivative, a map from the set of tangent vectors at a point, p, in the plane to the set of tangent vectors to the surface at S(p). If ${\vec {w}}=(\alpha ,\beta )$ is a tangent vector in the plane then its directional derivative will be

$dS\langle {\vec {w}}\rangle =\alpha {\vec {U}}+\beta {\vec {V}}=\alpha {\partial S \over \partial s}+\beta {\partial S \over \partial t}.$

A parametrised curve in the plane which has tangent vector ${\vec {w}}$ will be mapped to a curve on the surface with tangent vector $dS\left\langle {\vec {w}}\right\rangle$.

The second directional derivative is constructed by differentiating the first directional derivative. This will give a bi-linear map on pairs of tangent vectors. If ${\vec {u}}=(\alpha ,\beta ),\ {\vec {v}}=(\gamma ,\delta )$ then

$d^{2}S\langle {\vec {u}},{\vec {v}}\rangle =\alpha \gamma {\partial ^{2}S \over \partial s^{2}}+(\alpha \delta +\beta \gamma ){\partial ^{2}S \over \partial s\partial t}+\beta \delta {\partial ^{2}S \over \partial t^{2}}$

higher derivatives can be constructed in a similar fashion.

First fundamental form

The first fundamental form, $I_{p}\langle {\vec {u}},{\vec {v}}\rangle$, is an inner product and it captures the metric information about the surface. It is used to calculate distances and angles. If ${\vec {u}},\ {\vec {v}}$ are tangent vectors in the plane then: $I_{p}\langle {\vec {u}},{\vec {v}}\rangle =dS\langle {\vec {u}}\rangle \cdot dS\langle {\vec {v}}\rangle .$ This form is symmetric and bilinear so $I_{p}\langle {\vec {u}},{\vec {v}}\rangle =I_{p}\langle {\vec {v}},{\vec {u}}\rangle$, $I_{p}\langle 2{\vec {u}},{\vec {v}}\rangle =2I_{p}\langle {\vec {u}},{\vec {v}}\rangle$, etc.

If c:R→R² is a curve in the plane with tangent vector ${\vec {u}}$ at p, and C(t)=S(c(t)) is its image on the surface, then $I_{p}\langle {\vec {u}},{\vec {u}}\rangle$ will be the square of the speed of C. If $I_{p}=1\,$ for all points on c then C will be a unit speed curve. The length of C can be found by integrating $I_{p}\langle {\vec {u}},{\vec {u}}\rangle .$ The angle between two curves on the surface is found from $I_{p}\langle {\vec {u}},{\vec {v}}\rangle$ where ${\vec {u}},\ {\vec {v}}$ are the tangent vectors of the two curves in the plane.

Second fundamental form

For a given parametrisation a continuous unit normal vector field, ${\vec {N}}:\mathbf {R} ^{2}\to \mathbf {R} ^{3}$ and the directional derivative $d{\vec {N}}\langle {\vec {u}},{\vec {v}}\rangle$ can be found.

The second fundamental form, $II_{p}\langle {\vec {u}},{\vec {v}}\rangle$, captures second derivative information. It is defined by:

$II_{p}\langle {\vec {u}},{\vec {v}}\rangle =d^{2}S\langle {\vec {u}},{\vec {v}}\rangle \cdot {\vec {N}}.$

Differentiation ${\vec {N}}\cdot dS\langle u\rangle =0$ in the direction ${\vec {v}}$ gives

$d{\vec {N}}\langle v\rangle \cdot dS\langle {\vec {u}}\rangle +{\vec {N}}\cdot d^{2}S\langle {\vec {u}},{\vec {v}}\rangle +{\vec {N}}\cdot dS\langle d{\vec {u}}\langle {\vec {v}}\rangle \rangle =0.$

The last term is always zero, and the equations can be rearranged to give

$II_{p}\langle {\vec {u}},{\vec {v}}\rangle =-d{\vec {N}}\langle {\vec {u}}\rangle \cdot dS\langle {\vec {v}}\rangle .$

It is also symmetric and bilinear.

Curvature

The surface curves can be analysed by examining the first and second fundamental forms, to give Gaussian curvature, mean curvature and principal curvature.

Let $E=I\langle {\vec {u}},{\vec {u}}\rangle ,\ F=I\langle {\vec {u}},{\vec {v}}\rangle ,\ G=I\langle {\vec {v}},{\vec {v}}\rangle ,l=II\langle {\vec {u}},{\vec {u}}\rangle ,\ m=II\langle {\vec {u}},{\vec {v}}\rangle ,\ n=II\langle {\vec {v}},{\vec {v}}\rangle$. The Gaussian curvature is

$K={ln-m^{2} \over EG-F^{2}}$

and the mean curvature is

$H={En-2Fm+Gl \over 2(EG-F^{2})}$.

These curvatures are independent of the parametrisation used, and hence important tools for analysing the surface.

The sign of the Gaussian curvature determines whether the surface is locally convex (G>0) or saddle shaped (G<0). The terms ellipitical and hyperbolic are used for these two cases. When the Gaussian curvature is zero the surface is parabolic. In general parabolic points form a curve on the surface called the parabolic line. The Gaussian curvature is intimately connected with the Gauss map.

The equation

$II\langle {\vec {u}},{\vec {u}}\rangle =\kappa I\langle {\vec {u}},{\vec {u}}\rangle$

generally has two eigenvectors ${\vec {p}}$ and ${\vec {q}}$ with corresponding eigenvalues $\kappa _{p},\ \kappa _{q}.$ Let ${\vec {P}}=ds\langle {\vec {p}}\rangle ,\ {\vec {Q}}=ds\langle {\vec {q}}\rangle$, choose ${\vec {p}},\ {\vec {q}}$ so that ${\vec {P}},\ {\vec {Q}}$ are of unit length. ${\vec {P}},{\vec {Q}}$ are called the principal directions and $\kappa _{p},\ \kappa _{q}$ are the principal curvatures. It can be shown that the principal directions are orthogonal, and

$I\langle {\vec {p}},{\vec {p}}\rangle =1,\ I\langle {\vec {p}},{\vec {q}}\rangle =0,\ I\langle {\vec {q}},{\vec {q}}\rangle =1,II\langle {\vec {p}},{\vec {p}}\rangle =\kappa _{p},\ II\langle {\vec {p}},{\vec {p}}\rangle =0,\ II\langle {\vec {q}},{\vec {q}}\rangle =\kappa _{q}.$

Hence, the Gaussian curvature is $G=\kappa _{p}\kappa _{q}$ and the mean curvature is $H={1 \over 2}(\kappa _{p}+\kappa _{q}).$ The principal directions are the directions of maximum and minimum curvature for curves on the surface. If both have the same sign the surface will be locally convex (ellipitical) and if they have opposite signs then the surface will be saddle shaped (hyperbolic). Parabolic points occur when one of the principal curvatures is zero.

There can be points where the eigenvector-equation is degenerate. Here all directions are principal and the principal curvatures are equal. At such points called umbilics the surface is locally spherical. Generally these occur at isolated points in the ellipitical region.

See also

• Spline (mathematics)