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By: Jacob Desmond, Chuqing He and Adil Yousuf

The Optimal Cut. By: Jacob Desmond, Chuqing He and Adil Yousuf. Initial Problem. Which of The FOLLOWING LINE(S), If any, ACCOMPLISH THIS TASK?. (p,q). q. ?. ?. ?. p. Slope:. C. Slope:. (p,q). q. B. p. Geometric Proo f. Dissecting the graph.

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By: Jacob Desmond, Chuqing He and Adil Yousuf

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  1. The Optimal Cut By: Jacob Desmond, Chuqing He and AdilYousuf

  2. Initial Problem Which of The FOLLOWING LINE(S), If any, ACCOMPLISH THIS TASK? (p,q) q ? ? ? p

  3. Slope: C Slope: (p,q) q B p

  4. Geometric Proo f

  5. Dissecting the graph How do we know that this is the wrong line? C (p,q) q B p

  6. Dissection C 2q (p,q) q B p 2p C IV 2q II III (p,q) q I II 2p p B

  7. C IV 2q II III (p,q) q I II B p 2p Arbitrary line Vs Optimal line C 2q = II III (p,q) q I II 2p = B p

  8. Optimal cut C III (p,q) q I II B

  9. Algebraic Proof

  10. C IV II III (p,q) q I II B p C IV pq (p,q) q pq B p

  11. C IV pq (p,q) q pq B p

  12. Conclusion 2q I (p,q) q I 2p p To minimize the area, find the line Such that (p,q) is the midpoint.

  13. (0,B) Q Tilted Axis (Obtuse) (P,Q) Q P (C,0) (0,2Q) (P,Q) Q Q (2P,0) P P

  14. Tilted Axis (Acute) Optimal line Vs Arbitrary line

  15. Optimization for circles I I I I

  16. Given any shape define “open region” of that shape. Open region: Given any point in the open region of a shape, a line can be drawn to touch any point on the boundary of the shape without going outside of the shape. B A Point A is in this open region, and point B is not.

  17. Problem: How do we find the cut that minimizes the area going through a given point? Claim: Given any point in the open region; if a line is drawn with the given point bisecting the line, this line must be a candidate for the optimal cut. Zooming in… Given a point in the open region, if a line is drawn through the point there are three possibilities that can occur with the shape’s boundary.

  18. Case One : Boundaries Are Parallel Result: Adjusting the cut does not affect the area.

  19. Case Two: Boundaries That Have Slopes With Different Signs Result: Area is optimized when the point is the bisector of the cut.

  20. Case Three: Boundaries That Have Different Slopes With The Same Sign B4>B3 B3 I II B2 β B Θ B III IV B1 B4 Result: : Area is optimized when the point is the bisector of the cut.

  21. Moving to the Blob

  22. Further Inquiries: Given a point in a shape, what are the number of candidates for the optimal cuts? Allowing cuts to go through boundaries of the shape. Allowing shapes with holes. Three Dimensional Blobs that are cut by planes.

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