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)
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
=⇒link toExtreme Value Statistics
•Exact results for the mean perimeter and the mean area for alln.
•Summary and Conclusions
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
=⇒link toExtreme Value Statistics
•Exact results for the mean perimeter and the mean area for alln.
•Summary and Conclusions
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
=⇒link toExtreme Value Statistics
•Exact results for the mean perimeter and the mean area for alln.
•Summary and Conclusions
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
=⇒link toExtreme Value Statistics
•Exact results for the mean perimeter and the mean area for alln.
•Summary and Conclusions
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
=⇒link toExtreme Value Statistics
•Exact results for the mean perimeter and the mean area for alln.
•Summary and Conclusions
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
=⇒link toExtreme Value Statistics
•Exact results for the mean perimeter and the mean area for alln.
•Summary and Conclusions
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
=⇒link toExtreme Value Statistics
•Exact results for the mean perimeter and the mean area for alln.
•Summary and Conclusions
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
=⇒link toExtreme Value Statistics
•Exact results for the mean perimeter and the mean area for alln.
•Summary and Conclusions
Shape of a set of Points
00 00 11 11 00 00 11 11
00 00 11 11
00 00 11 11 00 00 11 11 00 00 11 11
00 00
11 11 00 00 11 11
00 00 11 11 00 00 11 11
00 00 11 11 00 00
11 11 00 00 11 11 00 00 11 11 00 00
11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00
11 11 00 00 11 11
00 00 11 11 00 00 11 11 00 00 11 11 00 00
11 11 00 00 11 11 00 00
11 11 00 00 11 11 00 00 11 11 00 00 11 11
00 110 0 1 1 00
11 00 00 11 11 00
11 00 11 0 0 1 1 00 00
11 11
00
00 11
11 00 11
Shape of a set of Points: Convex Hull
00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 11 00 00 11 11 00 00
11 11 00 00 11 11 00 11
C
convex hull
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
•Points drawn from adistribution→Independentor Correlated
• Question: Statisticsof observables: perimeter,areaandno. of vertices
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
•Points drawn from adistribution→Independentor Correlated
• Question: Statisticsof observables: perimeter,areaandno. of vertices
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
•Points drawn from adistribution→Independentor Correlated
• Question: Statisticsof observables: perimeter,areaandno. of vertices
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
•Points drawn from adistribution→Independentor Correlated
• Question: Statisticsof observables: perimeter,areaandno. of vertices
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
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
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)..
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)
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)
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)
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=?
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=?
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).
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)
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)
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)
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?
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?
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?
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?
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)
Global Convex Hull of n Independent Brownian Paths
n = 3closed paths n = 10open paths
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 2π
0
d θ M(θ)
•Area: A = 1
2 Z 2π
0
d θ h
M2(θ) − [M0(θ)]2i
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 2π
0
d θ M(θ)
•Area: A = 1
2 Z 2π
d θ h
M2(θ) − [M0(θ)]2i
A simple physicist’s proof of Cauchy’s formula
C approximate C by a polygon
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
−φ1M2(θ) − (M0(θ))2 dθ
= R22[sin(φ2) cos(φ2) + sin(φ1) cos(φ1)]= AOP1AP2
A simple physicist’s proof of Cauchy’s formula
C approximate C by a polygon
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
−φ1M2(θ) − (M0(θ))2 dθ
= R22[sin(φ2) cos(φ2) + sin(φ1) cos(φ1)]= AOP1AP2
A simple physicist’s proof of Cauchy’s formula
C approximate C by a polygon
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
−φ1M2(θ) − (M0(θ))2 dθ
= R22[sin(φ2) cos(φ2) + sin(φ1) cos(φ1)]= AOP1AP2
A simple physicist’s proof of Cauchy’s formula
C approximate C by a polygon
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
−φ1M2(θ) − (M0(θ))2 dθ
= R22[sin(φ2) cos(φ2) + sin(φ1) cos(φ1)]= AOP1AP2
A simple physicist’s proof of Cauchy’s formula
C approximate C by a polygon
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
−φ1M2(θ) − (M0(θ))2 dθ
= R22[sin(φ2) cos(φ2) + sin(φ1) cos(φ1)]= AOP1AP2