|Structures of||Regular parametrized curves and surfaces||Various curvatures of regular parametrized surfaces|
is the Euclidean 3-dimensional space consisting of points represented by triples of real numbers . The distinguished point is called the origin. In fact, can also be regarded as a set of vectors, which are quantities that have both length and direction. In this way, can be thought of as an arrow emanating from the origin, with the tip at the point , or any other arrow having the same length and direction.
There are various operations in . Addition of two vectors is done componentwise:
Regular parametrized curves and surfaces
Note: In this subsection, we will deal with some differential calculus of functions of two variables. Given , the partial differential of with respect to , denoted by , can be obtained by differentiating with respect to as if the variable were constant. Similarly for . For instance, if , then ,
Let be given by
Example 2.1 Let be given by
Example 2.2 Let be given by
Exercise 2.4 Show that, if and are smooth parametrized curves,
Remark 2.2A In fact, is the tangent vector of the curve at --it is the instantaneous direction at of a particle moving along the curve in such a way that it is at at time . Put in another way, the assertion of the above exercise just says that any tangent vector of a smooth parametrized curve at a point is perpendicular to the position vector of that point, if the curve lies on a sphere centered at the origin.
Definition 2.2B Define the curvature of a regular parametrized curve by
Exercise 2.5 If
for all , then .
Example 2.3 The curvature of the circle of radius , as in Example 2.1, is
Exercise 2.6 Prove that the curvature of the helix as in Example 2.2 is
Exercise 2.7 Consider the smooth parametrized curve ,
. Is it a regular parametrized curve?
Can you compute the curvature at ?
Remark 2.4 Curvature is a measure of how curved a curve is. Example 2.3 shows that the greater the radius is, the smaller the curvature of the circle becomes. This coincides with our geometric intuition.
Let be given by
Definition 2.5 If and are not multiple of each other for all , then is a regular parametrized surface.
Remark 2.6 The above definition is different from the usual definition found in the literature, which requires to be a homeomorphism, i.e. can be morphed into the image of without cutting and pasting. For the ease of presentation we omit this requirement in our definition.Exercise 2.8
The above exercise shows that there exists a unique plane containing vectors and passing through . We call this plane the tangent plane of at , and use to denote the tangent plane of a surface at . Moreover we know that for all . Define
Example 2.7 The unit sphere Let , . Then is the unit sphere minus the two poles. The -coordinate curves are the longitudes and the -coordinate curves are the latitudes. Note that
Example 2.8 Torus Let , , . is a torus obtained by revolving a circle of radius centered at around the -axis. The -coordinate curves are vertical circles on the torus, whereas the -coordinate curves are horizontal circles on the torus. We have
Example 2.9 Surface of revolution In general, if is a regular parametrized curve in with , then the equation of the surface of revolution obtained by rotating around the -axis is
Example 2.10 The graph of an infinitely differentiable function of two variables, , is the parametrized surface
Exercise 2.9 Show that the following are regular parametrized surfaces, and compute their normal vectors.
Various curvatures of regular parametrized surfaces
There are several notions of curvatures of surfaces, all of which are related to a special map called the Gauss map.
Definition 2.11 The Gauss map maps to the unit normal vector of at .
As you may see, the Gauss map measures how `rough' a surface is. The image of the Gauss map of a plane consists of only one vector, whereas the Gauss map of a surface with lots of sharp bumps and troughs has a `large' image which may be thought of as a large area on the unit sphere swept out by the unit normal vectors. Moreover, the local behaviour of the Gauss map gives a measure of how curved a surface is around a certain point.
Definition 2.12 The Gauss curvature at of a surface is the limit of the following quotient
Example 2.13 The Gauss map for the unit sphere is the identity map. So the Gauss curvature is 1. On the other hand, the Gauss curvature of any plane is 0 because the image of the Gauss curvature consists of only one vector.
Example 2.14 The Gauss map for the cylinder has the equator of as its image, which has no area. So the Gauss curvature of the cylinder is 0.
Though intuitive, the definition above will not be made used of to compute the Gauss curvature of other surfaces. Instead, we will introduce the first and second fundamental forms.
Definition 2.14A The first fundamental form at maps a pair of tangent vectors of at to their inner product:
Remark 2.14B The first fundamental form reflects the intrinsic property of surfaces, meaning that it gives the local information of length and area on the surface. For a regular parametrized surface , put
Definition 2.14C The second fundamental form at is defined to be
Remark 2.14D The second fundamental form reflects the extrinsic property of surfaces, meaning that it shows how surfaces are embedded in .
For a regular parametrized surface , put
Exercise 2.10 Show that
where , the normal vector at
In fact, for which can be differentiated infinitely many times, the order of taking partial differentiations is immaterial, i.e. . By the above exercise, . If and are tangent vectors at , then and for some . It can be shown that
Theorem 2.15 The Gauss curvature is given by
Another way to compute the Gauss curvature, which may be more tedious but relevant to the next section, is to use principal curvatures and , which are defined as in
Definition 2.16A Define the mean curvature .
Now with the various tools in our hands, we are able to compute and for all the regular parametrized surfaces in the previous examples.
Example 2.18 The torus.
Example 2.19 Surface of revolution
Example 2.20 The graph of an infinitely differentiable function