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Unformatted text preview: Computational Topology (Jeff Erickson) Regular Homotopy and Hexahedral Meshing Turning and turning in the widening gyre The falcon cannot hear the falconer; Things fall apart; the centre cannot hold; Mere anarchy is loosed upon the world — William Butler Yeats, “The Second Coming” (1921) A cube of cheese no larger than a die May bait the trap to catch a nibbling mie. — attributed to Chauncey Depew by Ambrose Bierce, The Devil’s Dictionary (1911) 4 Regular Homotopy and Hexahedral Meshing This lecture is concerned with closed curves in the plane that are smooth, but not necessarily. Intuitively, a regular closed curve is a closed curve with no sharp corners. Two regular closed curves are regularly homotopic if one can be continuously deformed into the other without introducing any sharp corners at any time. The turning number of a regular closed curve is the number of times its tangent vector rotates counterclockwise during a single traversal of the curve. I will prove the WhitneyGraustein theorem : two regular closed curves in the plane are regularly homotopic if and only if they have the same turning number. Then I’ll describe an application of this theorem to hexahedral meshing. 4.1 Winding Numbers But first, a warmup exercise. Recall that a loop in the plane is a continuous function α : [ 0,1 ] → R 2 such that α ( ) = α ( 1 ) . Let p be an arbitrary point that is not in the image of α , and consider an infinite ray r based at p . We say that r is generic if the set { t  α ( t ) ∈ r } is finite and excludes the values 0 and 1. An intersection point α ( t ) ∈ r is called a crossing if the points α ( t " ) and α ( t + " ) lie on opposite sides of r , for all sufficiently small " > 0. The crossing is positive if the triangle ( 0, α ( t ) , α ( t + " )) is oriented counterclockwise, and negative otherwise. The winding number of α around p , denoted wind ( α , p ) , is the number of positive crossings minus the number of negative crossings, for any (generic) ray. An argument similar to the proof of Lemma ≥ 2 (the easy half of the Jordan Curve Theorem) implies that this definition is independent of the choice of ray r . If we continuously move the ray, crossings can appear and disappear, but they always do so in matched pairs: one positive and one negative. Moreover, two points in the same connected component of R 2 \ im α define the same winding number; in particular, if p is in the unbounded component of R 2 \ im α , then wind ( α , p ) = 0. If α is simple , the winding number with respect to every interior point is either 1 or 1. 1 2 3 2 1 1 –1 –1 1 Winding numbers. Essentially the same argument implies that if we continuously deform a loop without touching a fixed point, say the origin 0, the winding number around that point is constant during the entire deformation, 1 Computational Topology (Jeff Erickson) Regular Homotopy and Hexahedral Meshing even if we allow the basepoint to move during the deformation. The type of deformation we alloweven if we allow the basepoint to move during the deformation....
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 Fall '08
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 Topology, homotopy, regular homotopy

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