Polyhedral representation of a cusp point, calculation of its concentrated curvature.
Polyhedral representations of various surfaces.
Permutation of the cusp points of a Cross-cap.
Transformation of a "right-handed" Boy's surface into a "left-handed" Boy's surface, via the Roman surface of Steiner.
"Right-left" inversion of a Boy's surface.
Jean-Pierre Petit
Research Director at CNRS
1988–1999
Abstract:
We present some elements allowing the representation of points of concentrated curvature: "posicones," "negacones," and their polyhedral equivalents: "posicoins" and "negacoins," which enable the construction of polyhedral representations of various surfaces and the recovery of their total curvature. Thus, the polyhedral representation of the Roman surface of Steiner consists of four cubes joined along their edges, making it more comprehensible. A polyhedral representation of Boy's surface was previously given in Topologicon, 1985, Belin Editions, pages 48 and 49, as a cut-out to be assembled. Page 46 also featured polyhedral representations of the torus and the Klein bottle. Polyhedral representations of the Cross-cap are provided. The total curvature of various immersions of the projective plane into R³—Boy's surface, Cross-cap, Roman surface of Steiner—is equal to 2π. The polyhedral representation of cusp points, considered as points of concentrated curvature, allows for a very simple calculation of this curvature. The Cross-cap, Roman surface of Steiner, and Boy's surface appear as "multiple faces" of a single object: the projective plane. Since this is not immediately obvious at first glance, we construct geometric transformations that allow passage from one to another. We start from the Cross-cap and transform it into the Roman surface of Steiner by creating two additional cusp points (i.e., applying, in this direction, the generic modification "creation-destruction of cusp points"), then transform the Steiner surface into Boy's surface by merging pairs of cusp points. Incidentally, by using the fact that the standard embedding of the sphere can be transformed into its antipodal embedding (sphere eversion), we show that the two cusp points of a Cross-cap can be swapped through a sequence of immersions, illustrating the fact that these two points are equivalent.
PREAMBLE:
The reader will find here general elements also present in the introduction of GEOMETRICAL PHYSICS A (definition of posicones, negacones, etc.). If you wish to skip this section, simply
click here
.
If you draw a triangle composed of straight line segments on a plane, the sum of the angles at the vertices equals π. These straight lines on the plane can be obtained differently: by gluing strips of any adhesive tape onto a surface without folding. We then call these plane paths geodesics. You can draw geodesic curves on any surface using this method, for example on a car's wing or hood.
Figure 1: A triangle considered as a set of three geodesics of the plane.
POSICONES AND NEGACONES
Make a cut in a plane and rejoin the two edges, then draw a triangle using your tape, composed of three geodesics of this cone.
Figure 2: Construction of a posicone.
By separating the two lips of the surface along the previous cut (Figure 3), you will easily notice, using a protractor, that the sum of angles A, B, and C equals π plus the cut angle α. This deviation from the Euclidean sum we call curvature, and we say that the triangle "contains" a certain amount of angular curvature α. This deviation remains the same regardless of the triangle, provided it contains the cone's apex. If it does not contain the apex, the sum will be π. We say that the curvature is concentrated at the apex M of the cone, which is then a "point of concentrated curvature." Since the sum of angles exceeds the Euclidean sum, we say this curvature is positive. Thus, from this perspective, a plane would be a surface with zero curvature.
Figure 3: The posicone laid flat.
This curvature is additive. If you glue together several such cones corresponding to angles α, β, γ, you can draw all kinds of triangles composed of geodesic arcs. If the triangle surrounds three points corresponding to curvatures equal to α, β, γ, then the sum of its vertex angles will be:
π + α + β + γ.
A surface with positive curvature can be considered as a sphere assembled from an infinite number of "posicones." Instead of having curvature concentrated at distinct points, we have curvature uniformly distributed over the entire surface. We say the sphere is a surface with "constant curvature" (or "constant angular curvature density").
Fig. 4: A triangle composed of three geodesic arcs.
On the sphere, geodesics are "great circles." The equator and meridians are great circles, and thus geodesic arcs of the sphere. However, you will not succeed in creating a parallel with a piece of tape. Parallels are not geodesics of the sphere. The sum of the angles at the vertices of a triangle drawn on a sphere depends on the ratio between the triangle's area and the sphere's area. The sum of angles of a very small triangle will be very close to π.
A triangle whose area is one-eighth of the sphere's area will have a sum A + B + C = 2π.
A great circle of the sphere can be considered a "triangle," provided we place its three vertices... anywhere on that circle. The sum A + B + C will be 3π. It contains half the surface area of the sphere.
What is the maximum possible deviation? We cannot say "enlarge the triangle beyond this great circle," since beyond that, the lengths of the geodesic arcs forming its sides will decrease and even tend toward zero.
Once the entire sphere's surface is enclosed, we obtain A + B + C = 5π = π + 4π.
We say the total curvature of the sphere is 4π.
Fig. 5: Sum of angles. Triangle composed of geodesic arcs of the sphere.
The amount of curvature contained in a triangle corresponds to a simple rule of three:
We will now create a "negacone" by inserting an angular sector α into a plane, as shown in Figure 6.
Fig. 6: A "negacone"
When you remove the angular sector, you obtain this:
Fig. 7: The negacone laid flat.
The sum of the angles of the triangle equals A + B + C = π − α.
We say this surface is a negacone possessing a "point of concentrated negative curvature." This curvature is also additive. By combining a surface with a juxtaposition of small posicones and small negacones, you can create a surface where the local curvature value is arbitrary.
Figure 7 is a surface with distributed negative curvature. We also say this surface has, at every point, a "curvature density" (angular). The deviation (negative) from the Euclidean sum of angles depends here as well on the triangle's area. The smaller the area, the closer the sum will be to π.
Fig. 7: Surface with negative (angular) curvature density.
Figure 8 shows an example of juxtaposition of three regions with positive, negative, and zero curvature density.
Fig. 8: Surface with variable curvature density. Zero in the flat part, positive in the spherical cap, negative in the junction (shaded).
If we draw geodesics, we obtain this:
Fig. 9: Planar projection, with geodesics.
In the chosen example, the amount of curvature contained in the spherical cap is equal and opposite to that in the junction (assuming the tangent plane varies continuously). Witness Figure 10 below:
Fig. 10: Geodesic triangles, drawn in the flat part, contain all the curvature they hold. Since the sum of the three angles equals π, it is inscribed in a Euclidean surface (plane). Consequently, the "integrated curvatures" contained in the spherical cap and the shaded junction are equal and opposite.
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