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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
slide1

The Optimal Cut

By:

Jacob Desmond, Chuqing He

and AdilYousuf

initial problem
Initial Problem

Which of The FOLLOWING LINE(S), If any, ACCOMPLISH THIS TASK?

(p,q)

q

?

?

?

p

slide3

Slope:

C

Slope:

(p,q)

q

B

p

dissecting the graph
Dissecting the graph

How do we know that this is the wrong line?

C

(p,q)

q

B

p

dissection
Dissection

C

2q

(p,q)

q

B

p

2p

C

IV

2q

II

III

(p,q)

q

I

II

2p

p

B

slide7

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
Optimal cut

C

III

(p,q)

q

I

II

B

slide10

C

IV

II

III

(p,q)

q

I

II

B

p

C

IV

pq

(p,q)

q

pq

B

p

slide11

C

IV

pq

(p,q)

q

pq

B

p

slide12

Conclusion

2q

I

(p,q)

q

I

2p

p

To minimize the area, find the line

Such that (p,q) is the midpoint.

slide13

(0,B)

Q

Tilted Axis

(Obtuse)

(P,Q)

Q

P

(C,0)

(0,2Q)

(P,Q)

Q

Q

(2P,0)

P

P

slide14

Tilted Axis

(Acute)

Optimal line

Vs

Arbitrary line

slide16

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.

slide17

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.

slide18

Case One : Boundaries Are Parallel

Result: Adjusting the cut does not affect the area.

slide19

Case Two: Boundaries That Have Slopes With Different Signs

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

slide20

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.

slide22

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.