II.3 Self-intersections 37II.3 Self-intersectionsSince non-orientable compact 2-manifolds without boundary cannot be embed-ded in three-dimensional Euclidean space, all their models in that space occurwith self-intersections. A practical motivation for looking at the phenomenonof self-intersections is to repair surface models of solid shapes.Mapping into space. Let M be a 2-manifold and f : M → R3a continuousmapping. For the time being assume f is smooth meaning derivatives of allorders exist. In the case at hand we have three real-valued functions in twovariables. The matrix of partial derivatives is thereforeJ ="∂f1∂s1∂f2∂s1∂f3∂s1∂f1∂s2∂f2∂s2∂f3∂s2#,consisting of the three gradients. The rank of this matrix is at most two. Themapping f is an immersion of its derivative has full rank (rank 2) at every pointof M, and it is an embedding if f restricted to its image is a homeomorphism.For smooth mappings, there are three types of generic self-intersections, allillustrated in Figure II.9. The most interesting of the three is the branchxxxFigure II.9: From left to right: a double point, a triple point, a branch point.point, which comes in several guises. We can construct it by cutting a diskfrom two sides toward the center, folding it, and re-glueing the sides as theself-intersection, as shown in Figure II.10.Triangle meshes. Classifying the types of self-intersections is easier in thepiecewise linear case in which M is given by a triangulation K. Since M is a2-manifold, the triangles that contain a vertex form a disk, or perhaps half a38 II SurfacesFigure II.10: Constructing the Whitney umbrella from a disk.disk if M has boundary. It is not difficult to see that imposing this conditionon the vertices suffices to guarantee that K triangulates a 2-manifold.We put K into space by mapping each vertex to a point in R3. The edges andtriangles are mapped to the convex hulls of the images of their vertices. Thismapping is an embedding iff any two triangles are either disjoint or they share avertex or they share an edge. Any other intersection is improper and referred toas a crossing. It is convenient to assume that the points are in general position,that is, no three are collinear and no four are coplanar. Under this assumption,there are only three types of crossing possible between two triangles, all shownin Figure II.11. Each crossing is a line segment common to two triangles. InFigure II.11: The three ways two triangles in general position in R3can cross eachother.the first case, one of the endpoints coincides with the image of a vertex. Theother endpoint lies on a unique edge, there is a unique other triangle on theother side of that edge that continues the intersection. In the other two casesboth endpoints lie on edges of the triangulation and the intersection has uniquecontinuations in both directions.II.3 Self-intersections 39Arcs and closed curves. Starting from a single crossing, we can trace theself-intersection in one or both directions, adding a line segment at a time.Since we have only finitely many triangles and thus finitely many line segments,each curve must either close up or end. In the first case we get a closed curveof almost all double points. Its preimage in K is either a pair of loops or asingle loop that covers the closed curve twice. Such a double covering loop isnecessarily orientation reversing hence M must have been non-orientable. Toconstruct an example, sweep a line segment along a circle in R3. The linesegment remains normal to the circle but its angle with the symmetry axis ofthe circle can change. If we take the angle from 0 to π during a full revolutionthen we get the M¨obius strip. If we take it from 0 toπ2we need a second fullrevolution before the surface is complete. We thus get a M¨obius strip whosemapping to R3crosses itself along the center circle, which is covered twice.We conclude this section with two immersions of the Klein bottle in R3. Inthe first and perhaps most commonly known model, the neck of the bottleextends and bends backward, like a Flamingo, but then continues and passesthrough the surface. The closed intersection curve is the common image oftwo orientation preserving loops. To construct the second model, we sweep apair of line segments along a circle in R3. The two line segments cross eachother orthogonally at their respective midpoints and they are both orthogonalto the circle. During a full revolution we take the angle one line segment formswith the symmetry axis from 0 to π. Correspondingly, the angle formed by theother line segment goes from −π2toπ2. The two line segments thus sweep outtwo M¨obius strips crossing each other along their center circles. We can nowcomplete the Klein bottle by connecting the two boundary curves by a circulararc, which we again sweep twice around the axis. In other words, we get theKlein bottle by sweeping a figure-8 curve along the circle, rotating it half-wayso that after a full revolution the two lobes are exchanged. We now have animmersion in which intersection is a closed curve whose preimage consists oftwo orientation-reversing loops.Bibliographic notes. The way surfaces mapped into R3intersect is dis-cussed in length and with many illustrations by Carter [2]. In the genericcase such a mapping has only three types of singularities, double points, triplepoints, and branch points. Whitney proved that every d-manifold has an im-mersion in R2d−1[4]. This implies that every 2-manifold can be immersedin R3, meaning there are smooth mappings without branch points. For theprojective plane we must have a branch point or a triple point which impliesthat every immersion has a triple point [1]. Whitney also proved that everyd-manifold can be embedded in R2d[3], so every 2-manifold can be embedded40 II Surfacesin R4.[1] T. F. Banchoff. Triple points and surgery of immersed surfaces. Proc. Amer.Math. Soc. 46 (1974), 403–413.[2] J. S. Carter. How Surfaces Intersect in Space. An Introduction to Topology.Second edition, World Scientific, Singapore, 1995.[3] H. Whitney. The self-intersections of a smooth n-manifold in 2n-space. Annalsof Math. 45 (1944), 220–246.[4] H. Whitney. The singularities of a smooth n-manifold in (2n − 1)-space. Annalsof Math. 45 (1944),
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