Differential Geometry: Conformal Maps

Differentil Geometry: Conforml Mps

Liner Trnsformtions Definition: We sy tht liner trnsformtion M:R n R n preserves ngles if M(v) 0 for ll v 0 nd: Mv, Mw v, w Mv Mw v w for ll v nd w in R n.

Liner Trnsformtions Mv, Mw v, w Note: Mv Mw v w If we denote by e i, the vector tht hs one in the i-th plce nd zero everywhere else, then: T e Me M j i ij

Note: Liner Trnsformtions Mv, Mw v, w Mv Mw v w If we denote by e i, the vector tht hs one in the i-th plce nd zero everywhere else, then: T e Me M j Since e i,e j 0 whenever i j, this implies tht: i ij, 0 Me i Me j ( M T M ) 0 ij

So M T M must be digonl mtrix. Note: Liner Trnsformtions Mv, Mw v, w Mv Mw v w If we denote by e i, the vector tht hs one in the i-th plce nd zero everywhere else, then: T e Me M j Since e i,e j 0 whenever i j, this implies tht: i ij, 0 Me i Me j ( M T M ) 0 ij

Liner Trnsformtions Mv, Mw v, w Note: Mv Mw v w If we denote by e i, the vector tht hs one in the i-th plce nd zero everywhere else, then: T e Me M j Similrly, since e i -e j,e i +e j 0, this implies tht: M ( e e ), M ( e + e ) 0 i j ( T ) ( T M M M M ) 0 ii i i j jj ij

So the digonl entries of M T M re ll equl. Liner Trnsformtions Mv, Mw v, w Note: Mv Mw v w If we denote by e i, the vector tht hs one in the i-th plce nd zero everywhere else, then: T e Me M j Similrly, since e i -e j,e i +e j 0, this implies tht: M ( e e ), M ( e + e ) 0 i j ( T ) ( T M M M M ) 0 ii i i j jj ij

Note: Liner Trnsformtions Mv, Mw v, w Mv Mw v w Thus, if M preserves ngles, then M T M must be of the form: λ 0 M T M 0 λ

Note: Liner Trnsformtions Mv, Mw v, w Mv Mw v w Thus, if M preserves ngles, then M T M must be of the form: λ 0 M T M If we set NM/λ 1/2, then: 0 N T N Id λ

Note: Liner Trnsformtions Mv, Mw v, w Mv Mw v w Thus, if M preserves ngles, then M T M must be of the form: λ 0 M T M If we set NM/λ 1/2, then: λ So N must be rottion/reflection. 0 N T N Id

Note: Liner Trnsformtions Mv, Mw v, w Mv Mw v w Thus, if M preserves ngles, then M T M must be of the form: λ 0 M T M If we set NM/λ 1/2, then: So N must be rottion/reflection. And M must be rottion/reflection composed with scling trnsformtion. 0 N T N Id λ

Note: Liner Trnsformtions Mv, Mw v, w Mv Mw v w In prticulr, M preserves ngles, if nd only if it mps circles to circles. M

Complex Numbers A complex number z is ny number tht cn be written s: z x + iy where x nd y re rel numbers nd i is the squre root of -1: 2 i 1

Complex Numbers Given two complex numbers, z 1 x 1 +iy 1 nd z 2 x 2 +iy 2 : The sum of the numbers is: z + z x + x ) + i( y + 1 2 ( 1 2 1 y2 )

Complex Numbers Given two complex numbers, z 1 x 1 +iy 1 nd z 2 x 2 +iy 2 : The sum of the numbers is: The product of the numbers is: ) ( ) ( ) )( ( 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 2 1 1 2 1 x y y x i y y x x x iy x iy iy iy x x iy x iy x z z + + + + + + + ) ( ) ( 2 1 2 1 2 1 y y i x x z z + + + +

Complex Numbers Given complex numbers, zx+iy: The negtion of the number is: z x iy

Complex Numbers Given complex numbers, zx+iy: The negtion of the number is: z x iy The conjugte of the number is: z x iy

Complex Numbers Given complex numbers, zx+iy: The negtion of the number is: z x iy The conjugte of the number is: z x iy The squre-norm of the number is: 2 2 2 z z z x + y

Complex Numbers Given complex numbers, zx+iy: The negtion of the number is: z x iy The conjugte of the number is: z x iy The squre-norm of the number is: 2 2 2 z z z x + y The reciprocl of the number is: 1 1 z x y i 2 z z z z z 2

Complex Numbers Often, we think of the complex numbers s living in the (rel) 2D plne. (z) zx +iy Re(z) Re

Complex Numbers Often, we think of the complex numbers s living in the (rel) 2D plne. Then if c is complex number, the function: f ( z) z + c is trnsltion in the complex plne. c c Re Re

Complex Exponentils Given θ R, the vlue of the complex exponentil e iθ is: θ e i cosθ + i sinθ

Complex Exponentils Given θ R, the vlue of the complex exponentil e iθ is: θ e i cosθ + i sinθ And ny complex number, zx+iy, cn be expressed in terms of its rdius nd ngle: iθ z re where r z, nd θarctn2(y,x).

Complex Numbers If ce iθ 0 is complex exponentil, then the function: iθ i( θ + θ ) f ( z) cz re re 0 is rottion by the ngle θ 0. c θ 0 Re Re

Complex Numbers More generlly, if cr 0 e iθ 0 is ny complex number, then the function: iθ ( ) i ( θ + θ ) 0 f ( z) cz re r r0 e is rottion by the ngle θ 0 followed/preceded by scling by r 0. θ c Re Re

Complex Numbers Finlly, if we consider the reciprocl function, then the function: 1 iθ 1 iθ f ( z) re e z r is n (orienttion-preserving) inversion. Re Re

Conforml Mps Definition: Given domin Ω R 2, the mp F:Ω R 2 is conforml if it preserves oriented ngles.

Conforml Mps Definition: Given domin Ω R 2, the mp F:Ω R 2 is conforml if it preserves oriented ngles. Tht is, if F(x,y)(f 1 (x,y),f 2 (x,y)) then the derivtive of F preserves oriented ngles: where R is rottion. R y f x f y f x f λ 2 2 1 1

Conforml Mps Definition: Given domin Ω R 2, the mp F:Ω R 2 is conforml if it preserves oriented ngles. Ω f(z)z 2.2 f(z)z 1/2 Re Re Re

Conforml Mps Definition: Given domin Ω R 2, the mp F:Ω R 2 is conforml if it preserves oriented ngles. It does not mp circles to circles. Ω f(z)z 2.2 f(z)z 1/2 Re Re Re

Conforml Mps Definition: Given domin Ω R 2, the mp F:Ω R 2 is conforml if it preserves oriented ngles. But it does mp tiny circles to circles. Ω f(z)z 2.2 f(z)z 1/2 Re Re Re

Conforml Mps Definition: Given domin Ω R 2, the mp F:Ω R 2 is conforml if it preserves oriented ngles. Note, if F:Ω R 2 nd G:F(Ω) R 2 re conforml mps, then so is G F. Ω f(z)z 2.2 f(z)z 1/2 Re Re Re

Conforml Mps Definition: Given domin Ω R 2, the mp F:Ω R 2 is conforml if it preserves oriented ngles. And, if F:Ω R 2 is conforml, then so is F -1. Ω f(z)z 2.2 f(z)z 1/2 Re Re Re

Conforml Mps Definition: In similr mnner, we sy tht mp between two surfces is conforml if it preserves ngles. [Globl Conforml Surfce Prmeteriztion, Gu nd Yu]

The Complex Plne nd the Sphere Stereogrphic Projection: A bijective mp from the unit sphere onto R 2 + : π ( x, y, z) x, y 2 z y 2 2

The Complex Plne nd the Sphere Stereogrphic Projection: A bijective mp from the unit sphere onto R 2 + : π ( x, y, z) x, y 2 Circles/lines on the plne get mpped to circles on the sphere, so the mp is conforml. z y 2

Möbius Trnsformtions The group of invertible conforml mps from the plne into itself is clled the Möbius group.

Möbius Trnsformtions The group of invertible conforml mps from the plne into itself is clled the Möbius group. Ω f(z)z 2.2 f(z)z 1/2 Re Re Re Q: Are these mps prt of the Möbius group?

Möbius Trnsformtions The group of invertible conforml mps from the plne into itself is clled the Möbius group. Ω f(z)z 2.2 f(z)z 1/2 Re Re Re Q: Are these mps prt of the Möbius group? A: No, they ren t invertible on the whole plne.

Möbius Trnsformtions The group of invertible conforml mps from the plne into itself is clled the Möbius group. Re Re Re Re Trnsltion Scle + Rottion Inversion Q: Are these mps prt of the Möbius group?

Möbius Trnsformtions If we think of the plne s the set of complex numbers, ny Möbius trnsformtion cn be expressed s frctionl liner trnsformtion: with d-bc 0. f ( z) z + b cz + d

If c0, then: Möbius Trnsformtions f ( z) z + b cz + d z + b f ( z) z + d d b d Scle+Rottion Trnsltion

If c0, then: If 0, then: Möbius Trnsformtions f ( z) b cz + d f ( z) Scle+Rottion b z + b cz + d z + b f ( z) z + d d 1 (( cz) + d ) b d Inversion Scle+Rottion Trnsltion

Möbius Trnsformtions Otherwise, if,c 0: f ( z) c + f ( z) c z + b cz + d c cb cz + c ( cz + d ) cb d cz + d z + b cz + d ( z + b) ( cz + d ) c c ( cz + d ) + ( cz + d ) cb + d cb d 1 (( cz) + d )

Möbius Trnsformtions Otherwise, if,c 0: f ( z) z cz + + f ( z) Trnsltion b d c z + b cz + d Scle+Rottion + cb d 1 (( cz) + d ) Inversion Scle+Rottion Trnsltion

Note: Möbius Trnsformtions f ( z) z + b cz + d 1. Scling the four coefficients does not chnge the trnsformtion: z + b cz + d f f z + cz + f f b d

Note: Möbius Trnsformtions f ( z) z + b cz + d 1. Scling the four coefficients does not chnge the trnsformtion: z + b cz + d 2. If we represent the mp by the 2x2 mtrix: then composing mps is the sme s multiplying mtrices. c f f z + cz + b d f f b d

Note: Möbius Trnsformtions f ( z) z + b cz + d As result, it is not uncommon to tlk bout the Möbius group s the group of 2x2 mtrices, with complex entries nd determinnt 1.

Möbius Trnsformtions If we hve conforml mps F,G:S S 2 from surfce S onto the sphere, then the composition F G -1 :S 2 S 2 must be conforml s well. F G

Möbius Trnsformtions If we hve conforml mps F,G:S S 2 from surfce S onto the sphere, then the composition F G -1 :S 2 S 2 must be conforml s well. Since the only conforml mps from the sphere/plne into itself re the Möbius trnsformtions, this mens tht, up to Möbius trnsformtion, the mp F:S S 2 is unique.