# Random Convex Hull and Extreme Value Statistics

## Full text

(1)

### Random Convex Hull and Extreme Value Statistics

Satya N. Majumdar

Laboratoire de Physique Th´eorique et Mod`eles Statistiques,CNRS, Universit´e Paris-Sud, France

Collaborators:

A. Comtet (LPTMS, Orsay, FRANCE) J. Randon-Furling (Univ. Paris-1, FRANCE)

Ref: Phys. Rev. Lett. 103, 140602 (2009) Extended Review: J. Stat. Phys. 138, 955 (2010)

(2)

### Plan

Plan:

•Random Convex Hull=⇒definition

•Convex Hull ofnplanar Brownian motions

•Motivation=⇒an ecological problem

•Cauchy’s formulae forperimeterandareaof a closed convex curve in two dimensions

=⇒applied torandomconvex polygon

•Exact results for the mean perimeter and the mean area for alln.

•Summary and Conclusions

(3)

### Plan

Plan:

•Random Convex Hull=⇒definition

•Convex Hull ofnplanar Brownian motions

•Motivation=⇒an ecological problem

•Cauchy’s formulae forperimeterandareaof a closed convex curve in two dimensions

=⇒applied torandomconvex polygon

•Exact results for the mean perimeter and the mean area for alln.

•Summary and Conclusions

(4)

### Plan

Plan:

•Random Convex Hull=⇒definition

•Convex Hull ofnplanar Brownian motions

•Motivation=⇒an ecological problem

•Cauchy’s formulae forperimeterandareaof a closed convex curve in two dimensions

=⇒applied torandomconvex polygon

•Exact results for the mean perimeter and the mean area for alln.

•Summary and Conclusions

(5)

### Plan

Plan:

•Random Convex Hull=⇒definition

•Convex Hull ofnplanar Brownian motions

•Motivation=⇒an ecological problem

•Cauchy’s formulae forperimeterandareaof a closed convex curve in two dimensions

=⇒applied torandomconvex polygon

•Exact results for the mean perimeter and the mean area for alln.

•Summary and Conclusions

(6)

### Plan

Plan:

•Random Convex Hull=⇒definition

•Convex Hull ofnplanar Brownian motions

•Motivation=⇒an ecological problem

•Cauchy’s formulae forperimeterandareaof a closed convex curve in two dimensions

=⇒applied torandomconvex polygon

•Exact results for the mean perimeter and the mean area for alln.

•Summary and Conclusions

(7)

### Plan

Plan:

•Random Convex Hull=⇒definition

•Convex Hull ofnplanar Brownian motions

•Motivation=⇒an ecological problem

•Cauchy’s formulae forperimeterandareaof a closed convex curve in two dimensions

=⇒applied torandomconvex polygon

•Exact results for the mean perimeter and the mean area for alln.

•Summary and Conclusions

(8)

### Plan

Plan:

•Random Convex Hull=⇒definition

•Convex Hull ofnplanar Brownian motions

•Motivation=⇒an ecological problem

•Cauchy’s formulae forperimeterandareaof a closed convex curve in two dimensions

=⇒applied torandomconvex polygon

•Exact results for the mean perimeter and the mean area for alln.

•Summary and Conclusions

(9)

### Plan

Plan:

•Random Convex Hull=⇒definition

•Convex Hull ofnplanar Brownian motions

•Motivation=⇒an ecological problem

•Cauchy’s formulae forperimeterandareaof a closed convex curve in two dimensions

=⇒applied torandomconvex polygon

•Exact results for the mean perimeter and the mean area for alln.

•Summary and Conclusions

(10)

## 11 00 11

(11)

convex hull

(12)

### Random Convex Hull in a Plane

00000000 00000000 00000000 00000000 00000000 0000

11111111 11111111 11111111 11111111 11111111 1111

00000 00000 11111 11111 0000000 0000000 0000000 1111111 1111111 1111111

0000 0000 0000 0000 0000 0000

1111 1111 1111 1111 1111 1111

0000000 0000000 0000000 1111111 1111111 1111111

00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000

11111 11111 11111 11111 11111 11111 11111 11111 11111 11111 11111

000000 000000 000000 000000 000000 000000 000000

111111 111111 111111 111111 111111 111111 111111

•Convex Hull=⇒ Minimalconvex polygon enclosing the set

•Theshapeof the convex hull→differentfor each sample

• Question: Statisticsof observables: perimeter,areaandno. of vertices

(13)

### Random Convex Hull in a Plane

00000000 00000000 00000000 00000000 00000000 0000

11111111 11111111 11111111 11111111 11111111 1111

00000 00000 11111 11111 0000000 0000000 0000000 1111111 1111111 1111111

0000 0000 0000 0000 0000 0000

1111 1111 1111 1111 1111 1111

0000000 0000000 0000000 1111111 1111111 1111111

00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000

11111 11111 11111 11111 11111 11111 11111 11111 11111 11111 11111

000000 000000 000000 000000 000000 000000 000000

111111 111111 111111 111111 111111 111111 111111

•Convex Hull=⇒ Minimalconvex polygon enclosing the set

•Theshape of the convex hull→differentfor each sample

• Question: Statisticsof observables: perimeter,areaandno. of vertices

(14)

### Random Convex Hull in a Plane

00000000 00000000 00000000 00000000 00000000 0000

11111111 11111111 11111111 11111111 11111111 1111

00000 00000 11111 11111 0000000 0000000 0000000 1111111 1111111 1111111

0000 0000 0000 0000 0000 0000

1111 1111 1111 1111 1111 1111

0000000 0000000 0000000 1111111 1111111 1111111

00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000

11111 11111 11111 11111 11111 11111 11111 11111 11111 11111 11111

000000 000000 000000 000000 000000 000000 000000

111111 111111 111111 111111 111111 111111 111111

•Convex Hull=⇒ Minimalconvex polygon enclosing the set

•Theshape of the convex hull→differentfor each sample

• Question: Statisticsof observables: perimeter,areaandno. of vertices

(15)

### Random Convex Hull in a Plane

00000000 00000000 00000000 00000000 00000000 0000

11111111 11111111 11111111 11111111 11111111 1111

00000 00000 11111 11111 0000000 0000000 0000000 1111111 1111111 1111111

0000 0000 0000 0000 0000 0000

1111 1111 1111 1111 1111 1111

0000000 0000000 0000000 1111111 1111111 1111111

00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000

11111 11111 11111 11111 11111 11111 11111 11111 11111 11111 11111

000000 000000 000000 000000 000000 000000 000000

111111 111111 111111 111111 111111 111111 111111

•Convex Hull=⇒ Minimalconvex polygon enclosing the set

•Theshape of the convex hull→differentfor each sample

• Question: Statisticsof observables: perimeter,areaandno. of vertices

(16)

### Independent Points in a Plane

00000000 00000000 00000000 00000000 00000000 0000

11111111 11111111 11111111 11111111 11111111 1111

00000 00000 11111 11111 0000000 0000000 0000000 1111111 1111111 1111111

0000 0000 0000 0000 0000 0000

1111 1111 1111 1111 1111 1111

Each point chosenindependentlyfrom the same distribution

AssociatedRandom Convex Hull→well studied bydiverse methods P. L´evy (’48), J. Geffroy (’59), Spitzer & Widom (’59), Baxter (’59).. R´enyi & Sulanke (’63), Efron (’65), ....many others

(17)

### Independent Points in a Plane

00000000 00000000 00000000 00000000 00000000 0000

11111111 11111111 11111111 11111111 11111111 1111

00000 00000 11111 11111 0000000 0000000 0000000 1111111 1111111 1111111

0000 0000 0000 0000 0000 0000

1111 1111 1111 1111 1111 1111

Each point chosenindependentlyfrom the same distribution AssociatedRandom Convex Hull→well studied bydiverse methods

P. L´evy (’48), J. Geffroy (’59), Spitzer & Widom (’59), Baxter (’59).. R´enyi & Sulanke (’63), Efron (’65), ....many others

(18)

### Independent Points in a Plane

00000000 00000000 00000000 00000000 00000000 0000

11111111 11111111 11111111 11111111 11111111 1111

00000 00000 11111 11111 0000000 0000000 0000000 1111111 1111111 1111111

0000 0000 0000 0000 0000 0000

1111 1111 1111 1111 1111 1111

Each point chosenindependentlyfrom the same distribution AssociatedRandom Convex Hull→well studied bydiverse methods P. L´evy (’48), J. Geffroy (’59), Spitzer & Widom (’59), Baxter (’59)..

(19)

### Correlated Points: Vertices of an Open Random Walk

0 1 0 1 0 1

0 1 01 0

1 00 11

0 1

0 1

00000 00000 00000 00000 00000 00000 00000 00000 00000 00000

11111 11111 11111 11111 11111 11111 11111 11111 11111 11111 00000 00000 00000 00000 11111 11111 11111 11111

000000 000000 000000 000000 000000 000

111111 111111 111111 111111 111111 111

00000000 11111111 0000 0000 0000 0000 00

1111 1111 1111 1111 000011 00000000 0000 11111111 11111111

00 00 11 11

00000000 00000000 0000 11111111 11111111 1111

0000 0000 1111 1111

000000 000000 000000 000000 000000 000

111111 111111 111111 111111 111111 0000000111 0000000 1111111 1111111

00 00 00 00 00 0

11 11 11 11 11 1

00000000 0000 11111111 1111 0000 0000 1111 1111

convex hull

8−step walk O

0 1

0 1 0 1

00 11

0 1

0 1 0 1

0 1 0 1 000000

000000 111111 1111110000 111100000

11 11 100000000

11111111

00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000

11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111

00000000 0000 11111111 1111

00000 00000 00000 00000 11111 11111 11111 11111 0000

0000 1111 111100000000

11111111

0000 0000 0000 00

1111 1111 1111 11

00 00 00 11 11 11

00000000 0000 11111111 1111 00000000 11111111 0000 0000 0000 1111 1111 1111 000000 000000 000000 111111 111111 111111

convex hull

O

another 8−step walk

•Continuous-time limit: Brownian pathof durationT

• mean perimeterandmean areaof the associated Convex hull?

• mean perimeter: hL1i =√

8πT (Tak´acs, ’80)

• mean area: hA1i = π2T (El Bachir, ’83, Letac ’93)

(20)

### Correlated Points: Vertices of an Open Random Walk

0 1 0 1 0 1

0 1 01 0

1 00 11

0 1

0 1

00000 00000 00000 00000 00000 00000 00000 00000 00000 00000

11111 11111 11111 11111 11111 11111 11111 11111 11111 11111 00000 00000 00000 00000 11111 11111 11111 11111

000000 000000 000000 000000 000000 000

111111 111111 111111 111111 111111 111

00000000 11111111 0000 0000 0000 0000 00

1111 1111 1111 1111 000011 00000000 0000 11111111 11111111

00 00 11 11

00000000 00000000 0000 11111111 11111111 1111

0000 0000 1111 1111

000000 000000 000000 000000 000000 000

111111 111111 111111 111111 111111 0000000111 0000000 1111111 1111111

00 00 00 00 00 0

11 11 11 11 11 1

00000000 0000 11111111 1111 0000 0000 1111 1111

convex hull

8−step walk O

0 1

0 1 0 1

00 11

0 1

0 1 0 1

0 1 0 1 000000

000000 111111 1111110000 111100000

11 11 100000000

11111111

00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000

11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111

00000000 0000 11111111 1111

00000 00000 00000 00000 11111 11111 11111 11111 0000

0000 1111 111100000000

11111111

0000 0000 0000 00

1111 1111 1111 11

00 00 00 11 11 11

00000000 0000 11111111 1111 00000000 11111111 0000 0000 0000 1111 1111 1111 000000 000000 000000 111111 111111 111111

convex hull

O

another 8−step walk

•Continuous-time limit: Brownian pathof durationT

• mean perimeterandmean areaof the associated Convex hull?

• mean perimeter: hL1i =√

8πT (Tak´acs, ’80)

• mean area: hA1i = π2T (El Bachir, ’83, Letac ’93)

(21)

### Correlated Points: Vertices of an Open Random Walk

0 1 0 1 0 1

0 1 01 0

1 00 11

0 1

0 1

00000 00000 00000 00000 00000 00000 00000 00000 00000 00000

11111 11111 11111 11111 11111 11111 11111 11111 11111 11111 00000 00000 00000 00000 11111 11111 11111 11111

000000 000000 000000 000000 000000 000

111111 111111 111111 111111 111111 111

00000000 11111111 0000 0000 0000 0000 00

1111 1111 1111 1111 000011 00000000 0000 11111111 11111111

00 00 11 11

00000000 00000000 0000 11111111 11111111 1111

0000 0000 1111 1111

000000 000000 000000 000000 000000 000

111111 111111 111111 111111 111111 0000000111 0000000 1111111 1111111

00 00 00 00 00 0

11 11 11 11 11 1

00000000 0000 11111111 1111 0000 0000 1111 1111

convex hull

8−step walk O

0 1

0 1 0 1

00 11

0 1

0 1 0 1

0 1 0 1 000000

000000 111111 1111110000 111100000

11 11 100000000

11111111

00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000

11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111

00000000 0000 11111111 1111

00000 00000 00000 00000 11111 11111 11111 11111 0000

0000 1111 111100000000

11111111

0000 0000 0000 00

1111 1111 1111 11

00 00 00 11 11 11

00000000 0000 11111111 1111 00000000 11111111 0000 0000 0000 1111 1111 1111 000000 000000 000000 111111 111111 111111

convex hull

O

another 8−step walk

•Continuous-time limit: Brownian pathof durationT

• mean perimeterandmean areaof the associated Convex hull?

• mean perimeter: hL1i =√

8πT (Tak´acs, ’80)

• mean area: hA1i =π2T (El Bachir, ’83, Letac ’93)

(22)

### Correlated Points: Vertices of a Closed Random Walk

0 1 0 0 1 1 0 1 0 1

0 1 0 1

0 1 0 00001 00000000 11111111 11110000 1111 00 00 0 11 11 1 0000 0000 00 1111 1111 11

00000000 00000000 00000000 11111111 11111111 11111111

0000 0000 00 1111 1111 11 00000000 00000000 11111111 11111111

00000000 00000000 00000000 00000000 0000

11111111 11111111 11111111 11111111 1111

0000 0000 0000 0000 0000 0000 00

1111 1111 1111 1111 1111 1111 11

000000 000000 000000 000000 000000 000000 000000 000000 000000

111111 111111 111111 111111 111111 111111 111111 111111 111111

000000 111111

000000 000000 000000 111111 111111 111111

O

8 step random bridge

0 1 0 1 0 1 0 1

0 1 0 1

0 1 01 0

1

000000 000000 111111 111111 00 00 11 11 00000 00000 00000 00000 00000 00000 00000 00000 00000

11111 11111 11111 11111 11111 11111 11111 11111 11111 000000 111111

000000 000000 000000 000000 000000 000000 000

111111 111111 111111 111111 111111 111111 111

000000 000000 000000 000000 000

111111 111111 111111 111111 00000111 00000 11111 11111

000000 000000 000 111111 111111 000111 000000 000000 000

111111 111111 111111

0000 1111

000000 000000 000000 111111 111111 111111

00 00 0 11 11 1

000000 000000 000000 000000 000000 111111 111111 111111 111111 111111 00000000

00000000 00000000 00000000

11111111 11111111 11111111 11111111

O

another 8 step bridge

•Continuous-time limit: Brownian bridge of durationT : starting atO and returning to it after timeT

• mean perimeter: hL1i =q

π3

2 T (Goldman, ’96).

• mean area: hA1i=?

(23)

### Correlated Points: Vertices of a Closed Random Walk

0 1 0 0 1 1 0 1 0 1

0 1 0 1

0 1 0 00001 00000000 11111111 11110000 1111 00 00 0 11 11 1 0000 0000 00 1111 1111 11

00000000 00000000 00000000 11111111 11111111 11111111

0000 0000 00 1111 1111 11 00000000 00000000 11111111 11111111

00000000 00000000 00000000 00000000 0000

11111111 11111111 11111111 11111111 1111

0000 0000 0000 0000 0000 0000 00

1111 1111 1111 1111 1111 1111 11

000000 000000 000000 000000 000000 000000 000000 000000 000000

111111 111111 111111 111111 111111 111111 111111 111111 111111

000000 111111

000000 000000 000000 111111 111111 111111

O

8 step random bridge

0 1 0 1 0 1 0 1

0 1 0 1

0 1 01 0

1

000000 000000 111111 111111 00 00 11 11 00000 00000 00000 00000 00000 00000 00000 00000 00000

11111 11111 11111 11111 11111 11111 11111 11111 11111 000000 111111

000000 000000 000000 000000 000000 000000 000

111111 111111 111111 111111 111111 111111 111

000000 000000 000000 000000 000

111111 111111 111111 111111 00000111 00000 11111 11111

000000 000000 000 111111 111111 000111 000000 000000 000

111111 111111 111111

0000 1111

000000 000000 000000 111111 111111 111111

00 00 0 11 11 1

000000 000000 000000 000000 000000 111111 111111 111111 111111 111111 00000000

00000000 00000000 00000000

11111111 11111111 11111111 11111111

O

another 8 step bridge

•Continuous-time limit: Brownian bridge of durationT : starting atO and returning to it after timeT

• mean perimeter: hL1i =q

π3

2 T (Goldman, ’96).

• mean area: hA1i=?

(24)

### Correlated Points: Vertices of a Closed Random Walk

0 1 0 0 1 1 0 1 0 1

0 1 0 1

0 1 0 00001 00000000 11111111 11110000 1111 00 00 0 11 11 1 0000 0000 00 1111 1111 11

00000000 00000000 00000000 11111111 11111111 11111111

0000 0000 00 1111 1111 11 00000000 00000000 11111111 11111111

00000000 00000000 00000000 00000000 0000

11111111 11111111 11111111 11111111 1111

0000 0000 0000 0000 0000 0000 00

1111 1111 1111 1111 1111 1111 11

000000 000000 000000 000000 000000 000000 000000 000000 000000

111111 111111 111111 111111 111111 111111 111111 111111 111111

000000 111111

000000 000000 000000 111111 111111 111111

O

8 step random bridge

0 1 0 1 0 1 0 1

0 1 0 1

0 1 01 0

1

000000 000000 111111 111111 00 00 11 11 00000 00000 00000 00000 00000 00000 00000 00000 00000

11111 11111 11111 11111 11111 11111 11111 11111 11111 000000 111111

000000 000000 000000 000000 000000 000000 000

111111 111111 111111 111111 111111 111111 111

000000 000000 000000 000000 000

111111 111111 111111 111111 00000111 00000 11111 11111

000000 000000 000 111111 111111 000111 000000 000000 000

111111 111111 111111

0000 1111

000000 000000 000000 111111 111111 111111

00 00 0 11 11 1

000000 000000 000000 000000 000000 111111 111111 111111 111111 111111 00000000

00000000 00000000 00000000

11111111 11111111 11111111 11111111

O

another 8 step bridge

•Continuous-time limit: Brownian bridge of durationT : starting atO and returning to it after timeT

• mean perimeter: hL1i =q

π3

2 T (Goldman, ’96).

(25)

### Home Range Estimate via Convex Hull

Models of home range for animal movement, Worton (1987)

Integrating Scientific Methods with Habitat Conservation Planning, Murphy and Noon (1992)

Theory of home range estimation from displacement measurements of animal populations, Giuggioli et. al. (2005)

Home Range Estimates, Boyle et. al., (2009)

(26)

### Home Range Estimate via Convex Hull

Models of home range for animal movement, Worton (1987)

Integrating Scientific Methods with Habitat Conservation Planning, Murphy and Noon (1992)

Theory of home range estimation from displacement measurements of animal populations, Giuggioli et. al. (2005)

Home Range Estimates, Boyle et. al., (2009)

(27)

### Home Range Estimate via Convex Hull

Models of home range for animal movement, Worton (1987)

Integrating Scientific Methods with Habitat Conservation Planning, Murphy and Noon (1992)

Theory of home range estimation from displacement measurements of animal populations, Giuggioli et. al. (2005)

Home Range Estimates, Boyle et. al., (2009)

(28)

### Global Convex Hull of n Independent Brownian Paths

•MeanperimeterhLniand meanareahAniofnindependent Brownian paths (bridges) each of durationT?

•hLni =αn

T; hAni =βnT

•Recall α1=√

8π, β1= π/2 (openpath) α1=p

π3/2, β1=? (closedpath)

• αn, βn=? → both foropenandclosedpaths→n-dependence?

(29)

### Global Convex Hull of n Independent Brownian Paths

•MeanperimeterhLniand meanareahAniofnindependent Brownian paths (bridges) each of durationT?

•hLni =αn

T; hAni =βnT

•Recall α1=√

8π, β1= π/2 (openpath) α1=p

π3/2, β1=? (closedpath)

• αn, βn=? → both foropenandclosedpaths→n-dependence?

(30)

### Global Convex Hull of n Independent Brownian Paths

•MeanperimeterhLniand meanareahAniofnindependent Brownian paths (bridges) each of durationT?

•hLni =αn

T; hAni =βnT

•Recall α1=√

8π, β1= π/2 (openpath) α1=p

π3/2, β1=? (closedpath)

• αn, βn=? → both foropenandclosedpaths→n-dependence?

(31)

### Global Convex Hull of n Independent Brownian Paths

•MeanperimeterhLniand meanareahAniofnindependent Brownian paths (bridges) each of durationT?

•hLni =αn

T; hAni =βnT

•Recall α1=√

8π, β1= π/2 (openpath) α1=p

π3/2, β1=? (closedpath)

• αn, βn=? → both foropenandclosedpaths→n-dependence?

(32)

### Global Convex Hull of n Independent Brownian Paths

•MeanperimeterhLniand meanareahAniofnindependent Brownian paths (bridges) each of durationT?

•hLni =αn

T; hAni =βnT

•Recall α1=√

8π, β1= π/2 (openpath) α1=p

π3/2, β1=? (closedpath)

(33)

### Global Convex Hull of n Independent Brownian Paths

n = 3closed paths n = 10open paths

(34)

### Cauchy’s Formulae for a Closed Convex Curve

θ Ο

M(θ)

C

CLOSED CONVEX CURVE C :

•For any point[X (s), Y (s)]onC define:

Support function: M(θ) = max

s∈C [X (s) cos(θ) + Y (s) sin(θ)]

•Perimeter: L = Z

0

d θ M(θ)

•Area: A = 1

2 Z

0

d θ h

M2(θ) − [M0(θ)]2i

(35)

### Cauchy’s Formulae for a Closed Convex Curve

θ Ο

M(θ)

C

CLOSED CONVEX CURVE C :

•For any point[X (s), Y (s)]onC define:

Support function: M(θ) = max

s∈C [X (s) cos(θ) + Y (s) sin(θ)]

•Perimeter: L = Z

0

d θ M(θ)

•Area: A = 1

2 Z

d θ h

M2(θ) − [M0(θ)]2i

(36)

### A

O R

M

A

P1

P2

θ ϕ

ϕ1

2

M(θ) = R cos θ Perimeter: Rφ2

−φ1M(θ)d θ = R [sin(φ1) + sin(φ2)]= LP1AP2 Area: 12Rφ2

−φ1M2(θ) − (M0(θ))2 dθ

= R22[sin(φ2) cos(φ2) + sin(φ1) cos(φ1)]= AOP1AP2

(37)

### A

O R

M

A

P1

P2

θ ϕ

ϕ1

2

M(θ) = R cos θ

Perimeter: Rφ2

−φ1M(θ)d θ = R [sin(φ1) + sin(φ2)]= LP1AP2 Area: 12Rφ2

−φ1M2(θ) − (M0(θ))2 dθ

= R22[sin(φ2) cos(φ2) + sin(φ1) cos(φ1)]= AOP1AP2

(38)

### A

O R

M

A

P1

P2

θ ϕ

ϕ1

2

M(θ) = R cos θ Perimeter: Rφ2

−φ1M(θ)d θ = R [sin(φ1) + sin(φ2)]= LP1AP2

Area: 12Rφ2

−φ1M2(θ) − (M0(θ))2 dθ

= R22[sin(φ2) cos(φ2) + sin(φ1) cos(φ1)]= AOP1AP2

(39)

### A

O R

M

A

P1

P2

θ ϕ

ϕ1

2

M(θ) = R cos θ Perimeter: Rφ2

−φ1M(θ)d θ = R [sin(φ1) + sin(φ2)]= LP1AP2 Area: 12Rφ2

−φ1M2(θ) − (M0(θ))2 dθ

= R22[sin(φ2) cos(φ2) + sin(φ1) cos(φ1)]= AOP1AP2

(40)

### A

O R

M

A

P1

P2

θ ϕ

ϕ1

2

M(θ) = R cos θ Perimeter: Rφ2

−φ1M(θ)d θ = R [sin(φ1) + sin(φ2)]= LP1AP2 Area: 12Rφ2

−φ1M2(θ) − (M0(θ))2 dθ

= R22[sin(φ2) cos(φ2) + sin(φ1) cos(φ1)]= AOP1AP2

Updating...

## References

Related subjects :