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normal-curves

normal-curves - Computational Topology(Jeff Erickson Normal...

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Computational Topology (Jeff Erickson) Normal Curves and Compression The only normal people are the ones you don’t know very well. — Alfred Adler I have captured the signal, and am presently triangulating the vectors, and compressing the data down, in order to express it as a function of my hand. [Points.] They’re over therrrrrrrrre! — Prof. John Frink, “Wild Barts Can’t be Broken”, The Simpsons (1999) 13 Normal Curves and Compression In the late 1920s, Helmuth Kneser [ 3 ] introduced the theory of normal surfaces to prove a certain decomposition theorem about three-dimensional manifolds. Normal surfaces were further developed by Wolfgang Haken [ 2 ] a few decades later to answer certain algorithmic questions about 3-manifolds. Since Haken’s work, dozens of refinements, extensions, and applications of normal surface theory have been developed; a comprehensive survey is beyond the scope of this course (and the expertise of the instructor!). In this lecture, I will describe some algorithmic results related to normal curves , which are just like normal surfaces, only one dimension lower; I will return to (one of) Haken’s algorithmic applications in the next lecture. 13.1 Normal Curves and Normal Coordinates Fix a 2-manifold M , possibly with boundary. Let T be a triangulation of M : a cellularly embedded graph in which every face has three sides. Tn particular, each boundary cycle of M is covered by a cycle in T . A simple cycle in M is the image of a continuous injective map from S 1 to M . A simple arc in M is the image of a continuous injective map α : [ 0 , 1 ] M whose endpoints α ( 0 ) and α ( 1 ) lie on the boundary of M . A curve is the union of a finite number of pairwise-disjoint simple cycles and arcs. Two curves γ and δ are isotopic (relative to M ) if γ can be continuously deformed into δ without moving any point on the boundary of M or introducing any intersections. 1 A curve is normal with respect to T if every intersection with an edge of T is transverse, and the intersection of the curve with any triangle is a finite set of elementary segments : simple paths whose endpoints lie on distinct sides of the triangle. The endpoints of the elementary segments partition the edges of T into ports . Nine elementary segments in a triangle. Every triangle in T can contain three different types of elementary segments, each ‘cutting off’ one of corners of the triangle. For any corner x of any triangle A , let γ ( A , x ) denote the number of elementary segments in γ A that separate x from the other two corners of A . Suppose T has t triangles. The vector of 3 t non-negative integers γ ( A , x ) are the normal coordinates of γ . We denote the normal coordinate vector of any normal curve γ by γ . Not every vector in N 3 t is a normal coordinate vector of a curve; for 1 More formally, an isotopy from γ to δ is a continuous function of the form h : [ 0 , 1 ] × γ M , such that (1) h ( 0 , γ ) = γ , (2) h ( 1, γ ) = δ , (3) h ( t , γ ) is a curve for all t [ 0,1 ] , and (4) h ( t , x ) = x for all t [ 0,1 ] and x γ M .

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