Lecture notes - Institut fÃ¼r Mathematik - TU Berlin

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10 ULRICH PINKALL (NOTES BY G. PAUL PETERS)for F −1m+5F m . The vanishing of the i–part is Napier’s Analogy.8. Spectral Parameter of a Lorentz Harmonic Map8.1. Spherical parallelograms. Applying Napier’s Analogy 7.1 to aspherical parallelogram with edges δ and ˜δ and inner angles ϕ and ˜ϕyieldstan˜δ2tan ϕ tan ˜ϕ = 1 + tan δ 22 21 − tan δ ˜δtan2 28.2. Associated Family and spectral parameter. So given a discreteLorentz harmonic map changing the side lengths δ and ˜δ suchthat the value of tan δ ˜δtan does not change one gets a new Lorentz2 2harmonic map with these new side lengths and the same angles. Thisamounts to scaling by λ and 1 in the coordinate tan δ , which is stereographicprojection of e iδ . This family of Lorentz harmonic maps isλ 2called the associated family of the given Lorentz harmonic map andthe parameter λ is called the spectral parameter.9. Discrete Pendulum equation9.1. sine–Gordon equation. A smooth K–surface is completely determinedby the angle between its asymptotic lines ϕ and its constantGauss curvature K < 0. The Gauss–Codazzi equations for an asymptoticline Chebyshev parametrization reduce to the sine–Gordonequationϕ uv = −K sin ϕ.9.2. Pendulum equation. The sine–Gordon equation in xt–coordinatesis ϕ xx − ϕ tt = −4K sin ϕ. For surfaces of revolution the angle ϕ is independentof x and one gets the pendulum equationϕ tt = 4K sin ϕ.9.3. Discrete pendulum equation. We may thus derive a discretependulum equation from discrete K–surfaces. The angles between theasymptotic lines are equal to the angles in the parallelogram of theGauss map of the K–surface. The angles at a vertex of the Gauss mapof a discrete K–surface satisfyϕ u + ϕ d + ϕ r + ϕ l = 2π.Because the surface and its Gauss map are rotationally symmetric alongthe x–coordinate ϕ l = ϕ r . Writing indices for the t–coordinate onlyϕ d = ϕ n−1 , ϕ u = ϕ n+1 , and writing ˜ϕ n = ϕ l = ϕ r one getsϕ n+1 − 2ϕ n + ϕ n−1 = −2(ϕ n + ˜ϕ n )modulo 2π. Since the Gauss map is Lorentz harmonic the quadrilateralwith inner angles ϕ n and ˜ϕ n are parallelograms. Thus the equationderived in 8.1 holds.□
TOPICS IN DISCRETE DIFFERENTIAL GEOMETRY AND VISUALIZATION11Writing K = tan δ 2tan ϕ n + ˜ϕ n2==tan˜δ2tanϕn21 − tan ϕn2this yields+ tan ˜ϕn2tan ˜ϕn2(1 − K) sin2ϕn2−2K sin ϕn2=tanϕn+ 1+K2 1−K1 − 1+K1−K+ (1 + K) cos2ϕn2cosϕn2(tanϕn2 )−1= 1 + K cos ϕ n−K sin ϕ nDiscrete K–surfaces thus yield the following discrete pendulum equation(DP) ϕ n+1 − 2ϕ n + ϕ n−1 = −4 arg(1 + Ke iϕn )This is similar to the discretization of the pendulum equation one getsby the Verlet–method:(VP) ϕ n+1 − 2ϕ n + ϕ n−1 = −4K sin(ϕ n ).Both methods yield symplectic integrators. That means the first ordertransformationT : R 2 → R 2 , T (ϕ n−1 , ϕ n ) = (ϕ n , ϕ n+1 )induced in the phase space satisfies det(T ′ ) = 1, or otherwise said, Tleaves the symplectic form det on R 2 invariant. This holds, becauseboth are of the form T (x, y) = (y, h(y) − x).In contrast to the Verlet–integrator (VP) the integrator (DP) obtainedfrom discrete K–surfaces is integrable in the sense that it posseses aconstant of the motion, namely the monodromy of the Gauss mapalong the x–axis. This is a time independent rotation, because the rotationalsymmetry of the initial condition is preserved by the geometricconstruction (completion of spherical parallelograms).9.4. Theorem. The function(9.1) H(ϕ n−1 , ϕ n ) = ( cos ( ϕ n− ) ( ϕ n−12 2 + K cosϕn+ )) ϕ n−1 22 2is constant under the evolution of the discrete pendulum equation (DP),i.e.,( ) ()ϕn−1ϕH ◦ T = H, where T =nϕ n 2ϕ n − 4 arg(1 + Ke iϕn ) − ϕ n−1Proof. The intrinsic geometric data of the Gauss map (the angles ϕ nand the sidelengths δ and ˜δ of the spherical parallelograms) of a surfaceof revolution is 1–periodic in x–direction. The monodromy along thex–axis is a rotation. This rotation is, in the sense of Theorem 4.8,represented by the quaternionλ = ( 1 + k tan δ 2) (1 + i cotϕ n−12) ( 1 + k tan ˜δ2) (1+ i cotϕ n)2This can be obtained with the methods introduced in the proof ofNapier’s Analogy, Theorem 7.1.
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Lecture notes - Institut fÃ¼r Mathematik - TU Berlin

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