By: Jacob Desmond, Chuqing He and Adil Yousuf

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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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Presentation Transcript

The Optimal Cut

By:

Jacob Desmond, Chuqing He

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

Dissecting the graph

How do we know that this is the wrong line?

C

(p,q)

q

B

p

Dissection

C

2q

(p,q)

q

B

p

2p

C

IV

2q

II

III

(p,q)

q

I

II

2p

p

B

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

Optimal cut

C

III

(p,q)

q

I

II

B

C

IV

II

III

(p,q)

q

I

II

B

p

C

IV

pq

(p,q)

q

pq

B

p

C

IV

pq

(p,q)

q

pq

B

p

Conclusion

2q

I

(p,q)

q

I

2p

p

To minimize the area, find the line

Such that (p,q) is the midpoint.

(0,B)

Q

Tilted Axis

(Obtuse)

(P,Q)

Q

P

(C,0)

(0,2Q)

(P,Q)

Q

Q

(2P,0)

P

P

Tilted Axis

(Acute)

Optimal line

Vs

Arbitrary line

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.

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.

Case One : Boundaries Are Parallel

Result: Adjusting the cut does not affect the area.

Case Two: Boundaries That Have Slopes With Different Signs

Result: Area is optimized when the point is the bisector of the cut.

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.

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.