17 地吃薄餅？ By Devandhira Wijaya Wangsa The Remarkable Theorem Then why is Gaussian curvature important? We have to understand the “Theorema Egregrium” established by Gauss, which is Latin for “remarkable theorem”. The theorem states that the Gaussian curvature is an intrinsic measure of a surface, meaning that the Gaussian curvature at every point of the surface does not change by bending the surface, as long as you do not stretch it. We call this the “normal curvature” – the curvature at a certain point on the surface with respect to a curve passing through the point (footnote 2); in our example, the surface has a positive normal curvature with respect to the direction of the red curve, and a negative normal curvature with respect to that of the blue curve. Now imagine we take all possible normal curvatures at a point by drawing every curve with respect to all possible directions. From all such possibilities, there has to be the greatest normal curvature and the least normal curvature (footnote 3), multiplying these two curvatures gives the Gaussian curvature. Let’s look at the three examples in figure 2. The first surface has a positive Gaussian curvature at the labelled point because both the maximum and minimum normal curvatures are positive, hence their product is also positive. The other two shapes have zero and negative Gaussian curvature respectively for the same reasoning. While there are other ways to describe the curvature of a surface at a point, such as mean curvature, Gaussian curvature is the way relevant to our question. Figure 2 The Gaussian curvature is obtained by multiplying the maximum (blue) and minimum (green) normal curvatures at a point of a surface. The three surfaces have a (a) positive, (b) zero and (c) negative Gaussian curvatures respectively at the labelled point. a b c Figure 3 A flat piece of paper (left) and a cylinder resulted from rolling the paper (right). Both have a zero Gaussian curvature, with a minimum (= 0) and maximum normal curvature of the cylinder at the black dot labelled in green and blue, respectively. To demonstrate this idea, imagine rolling a piece of paper into a cylinder. Since a flat piece of paper has a zero Gaussian curvature everywhere (with normal curvatures being zero in all directions), by the Theorema Egregium, we know the cylinder also has a zero Gaussian curvature everywhere. And in fact, it is true as you can see in figure 3. The minimum normal curvature is always zero, so multiplying zero always gives zero. Properly?
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