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Cross-Ratio: Z −T −1(0) T −1(1) −T −1(∞) S(Z) = Z −T −1(∞) T −1(1) −T −1(0) Which Takes the Same Points to (0,1,∞) As T

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Cross-Ratio: Z −T −1(0) T −1(1) −T −1(∞) S(Z) = Z −T −1(∞) T −1(1) −T −1(0) Which Takes the Same Points to (0,1,∞) As T Differential Geometry: Willmore Flow [Discrete Willmore Flow. Bobenko and Schröder, 2005] Quaternions Quaternions are extensions of complex numbers, with 3 imaginary values instead of 1: a + ib + jc + kd Quaternions Quaternions are extensions of complex numbers, with 3 imaginary values instead of 1: a + ib + jc + kd Like the complex numbers, we can add quaternions together by summing the individual components: (a1 + ib1 + jc 1 + kd 1 ) + (a 2 + ib2 + jc 2 + kd 2 ) = (a1 +a2 )+ i (b1 + b2 )+ j (c 1 +c 2 )+ k (d1 +d 2 ) Quaternions Quaternions are extensions of complex numbers, with 3 imaginary values instead of 1: a + ib + jc + kd Like the imaginary component of complex numbers, squaring the components gives: i 2 = j 2 = k 2 = −1 Quaternions Quaternions are extensions of complex numbers, with 3 imaginary values instead of 1: a + ib + jc + kd Like the imaginary component of complex numbers, squaring the components gives: i 2 = j 2 = k 2 = −1 However, the multiplication rules are more complex: ij = k ik = − j jk = i ji = −k ki = j kj = −i Quaternions Quaternions are extensions of complex numbers, with 3 imaginary values instead of 1: a + ib + jc + kd Like the imaginary component of complex numbers, squaring the components gives: 2 2 2 Notei that= j multiplication= k = −1 of However,quaternions the multiplicationis not commutative. rules are more complex: ij = k ik = − j jk = i ji = −k ki = j kj = −i Quaternions As with complex numbers, we define the conjugate of a quaternion q=a+ib+jc+kd as: q = a −ib − jc −kd Quaternions As with complex numbers, we define the conjugate of a quaternion q=a+ib+jc+kd as: q = a −ib − jc −kd We define the norm of a quaternion q=a+ib+jc+kd as: q = a 2 + b 2 +c 2 +d 2 = qq Quaternions As with complex numbers, we define the conjugate of a quaternion q=a+ib+jc+kd as: q = a −ib − jc −kd We define the norm of a quaternion q=a+ib+jc+kd as: q = a 2 + b 2 +c 2 +d 2 = qq And, we define the reciprocal by dividing the conjugate by the square norm: 1 q = q q 2 Quaternions One way to express a quaternion is as a pair consisting of the real value and the 3D vector consisting of the imaginary components: q = (α,w ) with α = a, w = (b,c ,d ) Quaternions One way to express a quaternion is as a pair consisting of the real value and the 3D vector consisting of the imaginary components: q = (α,w ) with α = a, w = (b,c ,d ) The advantage of this representation is that it is easier to express quaternion multiplication: q1q 2 = (α1,w 1 )(α 2 ,w 2 ) = (α1α 2 − w 1,w 2 ,α1w 2 +α 2w 1 +w 1 ×w 2 ) Möbius Transformations If we think of the plane as the set of complex numbers, any Möbius transformation can be expressed as a fractional linear transformation: az + b f (z) = cz + d with ad-bc≠0. Cross Ratio A more useful description of a conformal map is in terms of the map: z − z z − z S(z) = 3 2 4 z − z4 z2 − z3 for points z2, z3, z4 in the complex plane. Cross Ratio A more useful description of a conformal map is in terms of the map: z − z z − z S(z) = 3 2 4 z − z4 z2 − z3 for points z2, z3, z4 in the complex plane. The map is a Möbius transformation and has the property that: S(z2 ) =1 S(z3 ) = 0 S(z4 ) = ∞ Cross Ratio Claim: Every Möbius transformation can be written in this form. Cross Ratio Claim: Every Möbius transformation can be written in this form. Proof: Given a Möbius transformation T, we can define the cross-ratio: z −T −1(0) T −1(1) −T −1(∞) S(z) = z −T −1(∞) T −1(1) −T −1(0) which takes the same points to (0,1,∞) as T. Cross Ratio Claim: Every Möbius transformation can be written in this form. Proof: z −T −1(0) T −1(1) −T −1(∞) S(z) = z −T −1(∞) T −1(1) −T −1(0) But then ST-1 is a transformation that takes (0,1,∞) back to (0,1,∞). Cross Ratio Claim: Every Möbius transformation can be written in this form. Proof: But if we consider the expression for the Möbius transformation: az + b f (z) = cz + d we see that the only way to have (0,1,∞) map back to itself is to have f(z) be the identity. Cross Ratio Claim: Every Möbius transformation can be written in this form. Proof: But if we consider the expression for the Möbius transformation: az + b f (z) = cz + d we see that the only way to have (0,1,∞) map back to itself is to have f(z) be the identity. Thus ST-1 is the identity so S=T. Cross Ratio Definition: Given four (distinct) points in the complex plane z1, z2, z3, z4∈C, the cross-ratio is value: z1 − z3 z2 − z4 (z1, z2 , z3 , z4 ) = z1 − z4 z2 − z3 Cross Ratio Definition: Given four (distinct) points in the complex plane z1, z2, z3, z4∈C, the cross-ratio is value: z1 − z3 z2 − z4 (z1, z2 , z3 , z4 ) = z1 − z4 z2 − z3 That is, the cross-ratio is the image of z1 under the Möbius transformation sending (z2,z3,z4) to (0,1,∞). Cross Ratio Claim 1: The cross ratio of four points z1, z2, z3, z4∈C is invariant under Möbius transformations: (z1, z2 , z3 , z4 ) = (Tz1,Tz2 ,Tz3 ,Tz4 ) Cross Ratio Claim 1: The cross ratio of four points z1, z2, z3, z4∈C is invariant under Möbius transformations: (z1, z2 , z3 , z4 ) = (Tz1,Tz2 ,Tz3 ,Tz4 ) Proof: If S is the Möbius transformation sending z to -1 (z,z2,z3,z4) then ST takes (Tz2,Tz3,Tz4) to (0,1,∞). Cross Ratio Claim 1: The cross ratio of four points z1, z2, z3, z4∈C is invariant under Möbius transformations: (z1, z2 , z3 , z4 ) = (Tz1,Tz2 ,Tz3 ,Tz4 ) Proof: If S is the Möbius transformation sending z to -1 (z,z2,z3,z4) then ST takes (Tz2,Tz3,Tz4) to (0,1,∞). But that means that: −1 ST (Tz1) = (Tz1,Tz2 ,Tz3 ,Tz4 ) Cross Ratio Claim 1: The cross ratio of four points z1, z2, z3, z4∈C is invariant under Möbius transformations: (z1, z2 , z3 , z4 ) = (Tz1,Tz2 ,Tz3 ,Tz4 ) Proof: If S is the Möbius transformation sending z to -1 (z,z2,z3,z4) then ST takes (Tz2,Tz3,Tz4) to (0,1,∞). But that means that: −1 ST (Tz1) = (Tz1,Tz2 ,Tz3 ,Tz4 ) (z1, z2 , z3 , z4 ) = (Tz1,Tz2 ,Tz3 ,Tz4 ) Cross Ratio Claim 2: Given the four points z1, z2, z3, z4∈C, the angle of the cross ratio is π-β, where β is the angle between the circum-circles. z4 z1 z2 β z3 Cross Ratio Claim 2: Given the four points z1, z2, z3, z4∈C, the angle of the cross ratio is π-β, where β is the angle between the circum-circles. Proof: z The angle of the product of two 4 complex numbers is the sum of their angles: z1 arg(c ⋅c ) = arg(c ) + arg(c ) 1 2 1 2 z2 iθ1 iθ2 i(θ1 +θ2 ) β r1e r2e = r1r2e z3 Cross Ratio Claim 2: Given the four points z1, z2, z3, z4∈C, the angle of the cross ratio is π-β, where β is the angle between the circum-circles. Proof: z Thus, the angle of the cross-ratio: 4 z1 − z3 z2 − z4 (z1, z2 , z3 , z4 ) = z1 − z4 z2 − z3 is the sum: z1 z − z z − z = 1 3 + 2 4 z arg(z1, z2 , z3 , z4 ) arg arg 2 z1 − z4 z2 − z3 β z3 Cross Ratio Claim 2: Given the four points z1, z2, z3, z4∈C, the angle of the cross ratio is π-β, where β is the angle between the circum-circles. Proof: z − z z − z 1 3 2 4 z4 (z1, z2 , z3 , z4 ) = z1 − z4 z2 − z3 But the angle of (z1-z3)/(z1-z4) is z the difference between the 1 z angles of (z3-z1) and (z4-z1). 2 β z3 Cross Ratio Claim 2: Given the four points z1, z2, z3, z4∈C, the angle of the cross ratio is π-β, where β is the angle between the circum-circles. Proof: z − z z − z 1 3 2 4 z4 (z1, z2 , z3 , z4 ) = z1 − z4 z2 − z3 Similarly, we know that the z1 angle of (z2-z4)/(z2-z3) is… z2 β z3 Cross Ratio Claim 2: Given the four points z1, z2, z3, z4∈C, the angle of the cross ratio is π-β, where β is the angle between the circum-circles. Proof: z − z z − z 1 3 2 4 z4 (z1, z2 , z3 , z4 ) = z1 − z4 z2 − z3 So, the angle of the cross-ratio z is the sum of the angles that 1 are opposite the edge, which is z2 β exactly π-β. z3 Cross Ratio Claim 2: Given the four points z1, z2, z3, z4∈C, the angle of the cross ratio is π-β, where β is the angle between the circum-circles. Note: For a complex number c∈C, the cosine of the angle of c is just the real part of c c=a+ib divided by its length, so we have: |c| b a Re(z , z , z , z ) Re(z , z , z , z ) cos(π − β ) = 1 2 3 4 cos β = − 1 2 3 4 (z1, z2 , z3 , z4 ) (z1, z2 , z3 , z4 ) “Extended” Cross Ratio We can extend the notion of cross-ratio to 3D by using the imaginary parts of quaternions.
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