9.1: Total Curvature of a Surface
The basis of this remarkable global theorem is the total Gaussian
curvature of a region of a surface. Over a region R,
it is defined to be
 RΚdA = Κ(u,v)(Xu(u,v) × Xv(u,v)) · N(u,v)dudv;
where dA is just the regular element of surface area.
The total Gaussian curvature is actually equivalent to the algebraic area of the Gauss map of the region. To see why this is true, we return to the definition of Gaussian curvature:
Nu(u,v) × Nv(u,v)
= Κ(u,v)(Xu(u,v) × Xv(u,v))
Substituting, we notice that
(Nu × Nv) · N is
the area element for
the normal image. It is called the algebraic area because regions of
negative curvature subtract from the total (we do not integrate the
absolute value of the area element). As we will see, this is
exactly what we want in order to do some topology.
Tube Around a Plane Curve (Hotdog Demo)
In this demonstration, we form an analogy between the total curvature of a space curve and the total Gaussian curvature of a surface. We begin in two dimensions by drawing a yellow plane curve bounded by two endpoints, which can be parametrized as X(t), where a ≤ t ≤ b. We also draw the closed parallel curve at a distance r from the yellow curve, which is written as:
Y(t) = X(t) + rPx(t)
Note that Px(t) refers to the principal normal of the curve X(t). To avoid confusion, we let Py(t) be the principal normal of the closed parallel curve Y(t). Since Y(t) is a closed parallel curve, we require at the endpoints (t = a and t = b), where P(t) is not defined, that Y(a) and Y(b) be the two semicircles of radius r connecting the two curves parallel strictly to the interior of X(t). The total curvature of the parallel curve Y(t) is given by:
κy(t)sy'(t)dt
Since Y(t) is a simple (not self-intersecting), closed plane curve having winding number 1, its total curvature is 2 . To see why this is true, we modify our expression for total curvature using the Frenet-Serret equations for 2-space. Recall that in the plane, Py'(t) = -κy(t)sy'(t)Ty(t). So,
κy(t)sy'(t)dt = -Py'(t) · Ty(t)dt
In the demo, the curve Y(t) is traversed in a clockwise direction. As we travel along this parallel curve, the vector Py(t) traces out a path on the unit circle. P(t) circulates clockwise along the green and magenta sections of Y(t) and counterclockwise along the dark puple section. A second window shows the color-coded path of P(t). If we associate positve length with the clockwise direction and negative length with the counterclockwise direction, then -Py'(t) · Ty(t)dt gives us the total length of path traced out by P(t), which is called the principal normal indicatrix for Y(t). Using the demo, if we add up the lengths of the green and magenta sections and subtract the length of the dark purple section, we observe that P(t) goes around the unit circle exactly once, giving it a length of 2.
We now move from the plane into 3-space by considering the closed parallel surface at a distance r around the plane curve X(t). This tube surface can be parametrized as:
Zr(t,v) = X(t) + rcos(v)P(t) + rsin(v)B(t)
At the endpoints, X(a) and X(b), the closed parallel surface consists of two spherical caps with radius r. In the demonstration, X(t) is the parabolic curve X(t) = (t, t2 - 1), which causes the closed parallel surface Zr(t,v) to resemble a hotdog. The "hotdog surface" is topologically equivalent to a sphere, and we would, therefore, expect its total Gaussian curvature to equal 4. In the surface window, the hotdog is divided up into two parts: the tube surface, which is colored purple, and the two spherical caps, which are colored green. Regions of positive and negative Gaussian curvature are indicated by the lighter and darker shades respectively. To calculate the total Gaussian curvature of this surface, we use the definition stated at the beginning of this section:
RΚdA = (Nu(u,v) × Nv(u,v)) · N(u,v)dudv
In short, what this tells us is that the total Gaussian curvature is equal to the algebraic area of the Gauss Map. In the demo, one of the windows shows the Gauss Map of the hotdog surface, in which we can see after adding up the positive and negative areas that the Gauss Map covers the sphere once. So, the area of the Gauss Map and the total curvature of the surface are equal to 4.
To summarize, we began in two dimensions and found that the total curvature of the plane curve X(t) was equal to the algebraic length of its principal normal indicatrix (the path traced out by P(t)). We then made the transition into three dimensional space and showed that the total Gaussian curvature of a surface could be found in a similar way by calculating the algebraic area of its normal image.
Surface of Revolution around a Plane Curve (Bowl Demo)
Exercises
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