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CS 39, 2017

CS 39, 2017. Euler Characteristics and Genus. Carlo H. Séquin EECS Computer Science Division University of California, Berkeley. Topological Analysis. Surface Classification Theorem: All 2-manifolds embedded in Euclidean 3-space can be characterized by 3 parameters:

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CS 39, 2017

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  1. CS 39, 2017 Euler Characteristicsand Genus Carlo H. Séquin EECS Computer Science Division University of California, Berkeley

  2. Topological Analysis Surface Classification Theorem: • All 2-manifolds embedded in Euclidean 3-spacecan be characterized by 3 parameters: • Number of borders, b: # of 1D rim-lines; • Orientability, σ: single- / double- sided; • Genus, g, or: Euler Characteristic, χ,specifying “connectivity” . . .

  3. Determining the Number of Borders Run along a rim-line until you come back to the starting point; count the number of separate loops. Here, there are 4 borders:

  4. Determining the Surface Orientability A double-sided surface • Flood-fill_paint the surface without stepping across rim. • If whole surface is painted, it is a single-sided surface(“non-orientable”). • If only half is painted,it is a two-sided surface(“orientable”).The other side can then be painted a different color.

  5. Determining Surface Orientability (2) A shortcut:If you can find a path to get from “one side” to “the other” without stepping across a rim, it is a single-sided surface.

  6. Determining the Genus of a 2-Manifold Closed surfaces (e.g., handle-bodies) Surfaces with borders (e.g., disks with punctures) genus 0 genus 2 All: genus 0 The number of independent closed-loop cuts that can be made on a surface, while leaving all its pieces connected to one another. genus 4

  7. Euler Characteristic of Handle-Bodies  Genus = (2 – χ – b) / 2 for double-sided surfaces. For polyhedral surfaces: Euler Characteristic = χ = V – E + F Platonic Solids: V: 4 6 8 12 20 E: 6 12 12 30 30 F: 4 8 6 20 12 χ: 2 2 2 2 2

  8. “ EC-Mathematics” Fuse the two together: EC=1; b=1; g=0 EC=2; b=0; g=0 Punch multiple holes: the missing faces are replaced by additional borders.  Genus = (2 – χ – b) / 2 for double-sided surfaces. EC=2; g=0

  9. EC-Mathematics (2) Two or more cylindersstuck together end-to-end: At each joint, n vertices and n edges are removed;  EC does not change! Without end-faces: Closed into torus: EC=0; b=0; g=1 EC=0; b=2; g=0 EC=0; b=2; g=0  Genus = (2 – χ – b) / 2 for double-sided surfaces. EC=2; g=0

  10. Determining the Euler Characteristicof Tubular Structures Cutting a tube (“circular” cut) does not change χ: • χ = V – E + F = Euler Characteristic • How many cuts to obtain a tree-like connected graph? • A tree-like tubular graph structure has genus = 0 It is a “sphere” with (n “cuts”) = n punctures. • Each cut done adds +1 to the genus of the structure, e.g., a tubular Tetrahedral frame has genus = 3 • If the structure was single-sided (Klein bottle),multiply by 2the calculated genus

  11. Determining the Euler Characteristicof Disks with Punctures & Borders χ = V – E + F • Closing the gap • eliminates 1 edgeand 2 vertices; • EC := EC 1 • b := b +1 • Genus unchanged “Disk” • Genus = (2 – χ – b) / 2 for double-sided surfaces. E=V; F=1 V+F=E+1 V=E=F=1  Disk: χ = 1

  12. EC-Mathematics (3) A ribbon is a “disk”; EC =1 Closing the ribbon into a loop yields: EC = 1  1 = 0 Odd number of half-twists  1-sided Möbius band with 1 border; genus = 1 Genus = 2 – χ – b for non-orientable surfaces; Genus = (2 – χ – b) / 2 for double-sided surfaces. Even number of half-twists  2-sided annulus/cylinder with 2 borders; genus = 0

  13. Determining the Euler Characteristic From this: • Genus = 2 – χ – b for non-orientable surfaces; • Genus = (2 – χ – b)/2 for double-sided surfaces. Sometimes a simpler approach: Find χ • χ = V – E + F = Euler Characteristic • How many cuts(rim-to-rim) are neededto obtain a single connected disk? • Disk: χ = 1; do the inverse process:every cut re-join lowers χ by 1; thus “Tetra” ribbon frame: χ = –2

  14. “Endless Ribbon”Max Bill, 1953, stone • Single-sided (non-orientable) • Number of borders b = 1 • E.C. χ= 2–3+1= 0 • Genus g= 2 – χ – b= 1 • Independent cutting lines: 1

  15. Costa_in_CubeCarlo Séquin, 2004, bronze Double-sided (orientable) Number of borders b = 3 Euler characteristic χ = –5(make 6 cuts to turn into disk) Genus g = (2 – χ – b)/2 = 2 Independent cutting lines: 2

  16. “Möbius Shell”Brent Collins, 1993, wood • Single-sided (non-orientable) • Number of borders b = 2 (Y,R) • Euler characteristic χ = –1(need 2 cuts to turn into disk) • Genus g= 2 – χ – b= 1 • Independent cutting lines: 1blueorgreen–but not both,they would intersect!

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