Gaussian curvature is a measure for the "curvedness" of a surface. The basic idea is simple: take a surface with a boundary (i.e. not a closed surface like a sphere). The surface should not have any holes in it, and the boundary should be smooth (no sharp corners). Now take an arrow perpendicular to the surface (i.e. a normal) and let this arrow travel along the boundary of the surface (all the time remaining perpendicular) until it reaches its start point again.
Next imagine having a unit-sphere (i.e. a sphere with radius = 1). Have an arrow point from the centre of the sphere to the surface. While the arrow on the original surface travels around the boundary, let the arrow in the sphere mimic its movement (remaining attached to the sphere’s centre). The arrow in the sphere will now describe the boundary of a part of the sphere’s surface. The size of the surface of this sphere-segment is called the Gaussian curvature of the original surface.
The picture below illustrates this idea for a very simple surface, a cone: while the arrow travels around the cone, the corresponding arrow in the sphere traces out the yellow surface, whose size is the Gaussian curvature of the cone.
It seems this idea can be extended to surfaces whose boundary is not smooth (i.e. contains sharp angles). Take for example a simple 3-sided pyramid (or, viewed differently, the corner of a cube), whose boundary is it’s base-line (i.e. a hollow pyramid without a "bottom"). While the arrow travels along the edge of one of the (flat) surfaces, the corresponding arrow in the sphere doesn’t move (!). Only when the arrow "turns around the corner" at an edge, does the sphere-arrow rotate. After the arrow has completed its journey around the pyramid, the arrow in the sphere will have traced out a 3-sided figure on the sphere. See the picture below for an illustration.
There’s something strange going on here however. The 3 planes of the pyramid are completely flat, so their curvature is zero. The fact that the surface described on the sphere is not zero, means that the curvature of the entire pyramid-surface is, so to say, contained in the top of the pyramid. I.e. the vertex of the pyramid-surface is responsible for the curvature. Or, stating it differently: the sphere-arrows trace an actual surface, because the pyramid-arrows have to cross 3 edges: at those crossings the arrows rotate, tracing out part of a circle on the sphere. Now it’s clear that a pyramid face is projected onto the sphere as a point (the sphere arrow doesn’t move while the pyramid arrow travels along a plane), a pyramid edge is projected as an edge, and a pyramid point is projected as a surface. So yes, indeed the top of the pyramid is responsible for the sphere-surface and so it contains the entire curvature of the surface.
The examples given are rather simple (cones and pyramids aren’t exactly difficult surfaces) and merely hint at the general underlying mathematics. Gaussian curvature and the related Gauss-Bonnet theorem are concerned with general surfaces and general curves: the curves may be non-smooth, and the surface may have holes, sharp edges, or anything you like. Mathematically this tends to get rather complex very quickly, although the general idea remains valid: arrows perpendicular to a surface that travel along a closed path correspond to a piece of the unit sphere. The size of this piece is a measure for the curvedness of the surface enclosed by the path.
The Gauss-Bonnet theorem is an amazing and non-trivial piece of mathematics, which we won’t describe in detail here. One of its simpler formulations is easy to understand though, but before we can continue, you first have to know what the Euler characteristic of a (closed) surface is. That is easy, fortunately.
Take any closed surface (sphere, donut, potato, whatever). Draw a graph on it, i.e. a set of points connected by lines. It’s as if you draw a map of countries with their borders, and where two or more borders meet, you put a dot. Now every country is a face, every piece of border between two points is an edge, and every dot is a vertex. Count all faces, edges and vertices, and call their totals F, E and V respectively. The Euler characteristic (EC) of the surface is now:
EC = F - E + V
The amazing thing is that for any given surface, this number is a constant. It doesn’t matter how you draw your countries, or how many you draw. As an example, lets take a simple cube, where each side of the cube is a country, the borders are the edges and the corners of the cube are the point where the borders meet. Then F = 6, E = 12, V = 8, so EC = 6 - 12 + 8 = 2. Draw an entirely different map on the cube, and count F, E and V again, and you will find that EC = 2 once more.
Next imagine this same cube being made of rubber. It will be clear that we can stretch it anyway we want (without tearing it or making real holes in it), without influencing V, E or F, and thus without influencing the Euler characteristic EC. Since we could e.g. blow up the cube to become a sphere, or push it into a pyramid shape, we say that a sphere, cube and pyramid are all topological "the same". That in turn means that the EC for all these surfaces will be the same: 2.
However, if we had taken a rubber donut (a torus, in math-speak), there’s no way we could have gotten rid of the hole: a torus can never be transformed into a sphere (or the other way around), without some serious cutting and tearing and gluing (which is not allowed). A torus therefore is a topologically different object altogether. Draw a graph on a donut, and once more count F, E and V. You will now find that EC = 0! In fact, it’s not hard to prove that the Euler characteristic of a closed surface is 2 minus the number of holes (H) in the surface:
EC = 2 - 2*H
This is a rather funny and curious result. It means that if you have some closed object and draw a map (graph) on it, then
F - E + V = 2 -2* H
no matter what kind of object it is, and no matter what kind of map you draw. The formula is extremely simple, but not obvious at all. Oh, the joy of mathematics...
One of the (easier) formulations the Gauss-Bonnet theorem is the following:
The total curvature of a closed surface TC = 2*pi*EC
where EC is the Euler characteristic of the surface.
Using the last expression we found for calculating EC, we could even make this into
TC = 2*pi*(2 - 2*H)
where H is the number of holes in the surface.
So the curvature of a cube (or sphere or potato or pyramid) is 2*pi*2 = 4*pi. For a donut this becomes 2*pi*0 = 0! Note that this doesn’t mean a donut is not curved (it obviously is). It just says that the total curvature is zero, which is not the same as saying the curvature is zero in every point. What happens here is: the outside of the donut is "curved like a sphere", which means we have positive curvature here. The inside is curved "the other way around" (a point on the inside edge for example is curved like a horse-saddle, which is entirely different from how a sphere is curved) and has thus negative curvature. The sum of these two appears to be exactly zero.
This is an incredible result: it says that you can stretch a closed surface anyway you want, add bumps and dents, without influencing the total curvature of the surface! This is true because the surface's EC isn't influenced by such stretching and squashing, as we saw earlier. So a nice round balloon, and a balloon that’s completely squashed flat (you accidentally sat on it) have the same total curvature (as long as the balloon doesn’t snap: adding holes isn’t allowed). Ditto for a potato without donut-like holes in it. Likewise a donut, the inner tire from your bike, or your wedding ring all have the same total curvature (of zero): topologically they're all torusses (tori?). Gaussian curvature is a differential-geometric concept, while the Euler characteristic is topological in nature and has nothing to do with curves. Gauss-Bonnet thus manages to link two entirely different fields of mathematics in one stunningly elegant theorem.
Now let’s tie Gauss-Bonnet to what we did before: surfaces on which a perpendicular arrow travels, tied to a sphere in which an arrow traces out a surface. Let’s take a simple one: a sphere S with any radius. Let a perpendicular arrow travel all over S, to every possible point. It’s clear that the corresponding arrow that traces a surface inside the unit sphere (US) will now have touched every point of US. I.e. the surface that’s being traced out covers the entire sphere US, and so the Gaussian curvature of the sphere S is equal to the surface of US, which is 4*pi*(radius-squared). Since it’s a unit sphere, the radius=1, and thus the Gaussian curvature of S is 4*pi.
Using Gauss-Bonnet as an alternative approach, we find the following. The Euler characteristic EC of a sphere is 2 (as we saw earlier). Then Gauss-Bonnet says that TC = 2*pi*2 = 4*pi. Hey! this is a lot simpler than thinking about arrows travelling over surfaces and inside unit spheres!
But it gets even better. Take a cube, and travel around one of its corners. Earlier we already concluded that the curvature of such a path is "contained in the vertex". Take a look again at the pyramid picture, and now consider the pyramid to be one corner of a cube. The arrows trace out a kind of triangle on the unit sphere. If you look at the sphere in the picture, it’s clear that such a triangle covers 1/8 of the entire sphere, and thus the curvature of the cube-corner is 1/8 times the surface of the unit sphere = 1/8*4*pi = 0.5*pi. Since a cube has 8 corners and "all the curvature is contained in the corners", the entire cube has curvature 8*0,5*pi = 4*pi again! That shouldn’t be surprising: earlier we already stated that "stretching" a surface doesn’t alter the total curvature, so the cube (which can be considered to be a stretched rubber sphere) is indeed supposed to have the same curvature as the sphere.
However, now take this cube-corner, and cut it open so you can lay it down flat on the table (see picture below). Each cube-face has a 90 degree corner, so you have 270 degrees in total. Or: you miss 90 degrees in order to "complete" the entire 360 degrees. Now we measure corners in radians (where 360 degrees is 2*pi radians). Our 90 degrees missing piece is thus 0.5*pi radians: the exact same number we found for the corner’s curvature! Surprisingly enough this is no coincidence: it can be proven that this method always yields the proper result.
Take for example a tetrahedron (a pyramid with 4 triangular faces, where each edge has the same length). Each corner of a triangle is 60 degrees. Since 3 such triangles meet in one pyramid vertex, the total is 180 degrees, and we thus miss 180 degrees (or pi radians) to complete the corner. There are 4 corners, so the sum of the "missing corners" is 4*pi (720 degrees), which should be the total curvature of the pyramid. Well, that’s correct, since a pyramid topologically is related to the cube and sphere, which both have a Gaussian curvature of 4*pi.
Using this idea in reverse: take a dodecahedron (a regular 12-sided surface, where each face is a regular pentagon). It has 20 vertices. The total curvature of 4*pi is equally distributed over those vertices, and so each vertex has a curvature of 4*pi/20 = pi/5. Since pi/5 radians equals 180/5 = 36 degrees, you would have a "missing corner" of 36 degrees when you cut a corner open and put it flat on the table. Now 360 - 36 = 324 degrees, and since 3 faces meet in one corner, each pentagonal face has an (inside) corner of 324/3 = 108 degrees. And this is indeed correct (as you can verify yourself , hopefully).
We conclude this short excursion with an extension of the foregoing ideas. If we flatten the corner of a cube or pyramid, the "missing corner", expressed in radians, is exactly the curvature of the corner. More precisely: the missing corner is the curvature of the enclosed surface. However, since we were concerned with surfaces consisting of flat pieces whose curvature is zero, all curvature was contained in the corner. But we need not limit ourselves to such simple surfaces at all. Take any surface (like e.g. a potato) and draw a closed path on it. The path now encloses an open surface (well, 2 actually, on both sides of the curve). If you now cut out an extremely (possibly infinitely) thin strip of the surface, containing the path, you get a ring. You can now cut it open and lay it flat on the table (if it’s really thin enough, you can flatten out any ring-like surface this way). This "cut out and flattened" path will have some "missing corner" as well. And again this missing corner is exactly the curvature of the enclosed surface. The picture below shows a bunch of cut-out paths and their missing corners - including some paths that apparently enclosed a surface with negative curvature (i.e. you don’t miss a piece of the 360 degrees full corner; instead you "have too much corner").
So, to sum it up: we already saw that the total curvature of a surface doesn’t change if you stretch or blow-up the surface (without creating new holes). Now we can conclude that the same holds for parts of a surface as well: take any of the paths pictured above, and glue it back together into a ring. You can now "close up" one side of this ring with a surface, whose shape and size doesn’t matter at all: the resulting surface will always have the curvature as dictated by the "missing corner" of the cut-out path. You only have to make sure that the surface you add connects "smoothly" to the thin ring you started with. I.e. the fact that the curvature of a closed surface is topologically invariant, is true for open surfaces as well (as long as you keep the enclosing path and it’s infinitely small surroundings intact).
There are two QuickTime animations you can download, that visualise some of the things discussed above. Below you find a few stills from these animations,with a short explanation. Below the pictures you'll find the download links for the animations.
Both animations were completely hand-coded in POV-ray - no graphical front-end was used.
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A perpendicular arrow travels around the blue six-sided surface. At the same time a corresponding arrow inside the glass sphere traces out a surface (purple). The size of this purple surface is exactly the curvature of the blue surface. |
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This part of the animation shows what happens when the blue surface becomes flatter: the purple sphere-segment becomes smaller as well, meaning the blue surface is getting less curved (which indeed corresponds to our intuition of what "curved" means). |
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The final part of the animation shows what happens on a saddle-like surface: as the moving arrow on the blue surface travels counter-clockwise, the red arrow in the sphere moves clockwise instead! Since both movements have different orientation, we say the surface has negative curvature. |
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A light-blue corner of a cube, with 3 perpendicular normals attached. The normals trace out a part of the transparent-green unit sphere. This means that the surface of the green "soap bubble" is a measure for the curvature of the light-blue corner. The blue "clock" at the left shows a flat model of the blue corner. The green clock measures the curvature (= the size of the green bubble). It is clear that both clocks complement each other, and that thus indeed the "missing corner" (see article above) measures the curvature of the blue surface. |
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As the surface flattens, the "soap bubble" gets smaller, as does the "missing corner". |
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The surface is almost flat, and the surface traced out by the arrows has almost disappeared. |
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Now the surface is completely flat: the three arrows coincide and all point to the same point of the green unit sphere. The surface traced out is thus zero, as is the curvature. Fortunately: we'd like a flat surface to have curvature=0,wouldn't we? |
(c) H.J. Veenstra 2001-2002.
