By Tim Hoffmann

Discrete differential geometry investigates discrete analogs of items of

smooth differential geometry. hence, throughout the notes I seek advice from various

notions of classical differential geometry. yet whereas wisdom of uncomplicated dif-

ferential geometry is naturally precious, lots of the fabric could be under-

standable with out figuring out the sleek beginning of a few of the notions.

The fabric coated during this ebook is by means of nomeans a entire overview

of the rising box of discrete differential geometry yet i am hoping that it can

serve as an advent.

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What is left to show is that a∗ + b∗ + c∗ + d∗ = 0. Conjugating the left hand side and multiplying it with ac results in αc + β ac α α ac + αa + β = αc + β d + αa + β b = 0. b d β β This shows that we can integrate the dual edges into closing quadrilaterals. Remark. As in the smooth case dualizing is a duality: F ∗∗ = F up to translation. 2 Minimal surfaces are known to be isothermic and the dual surface is their Gauß map (it maps into the unit sphere and thus is a conformal map into S 2 ). In fact minimal surfaces can be characterized by this property and the dual of any conformal map into S 2 is minimal.

17 (and Lemma) Let F be s-isothermic with centres c and radii r. F ∗ given by c∗1 − c∗ = c1 − c , r1 r c∗2 − c∗ = − c2 − c r2 r 1 r is s-isothermic again and called a dual surface of F . r∗ = 63 64 Tim Hoﬀmann Lecture 14 Proof. we have to show, that F ∗ is well deﬁned and s-isothermic. If F is s-isothermic then its lift Fˆ solves a Moutard equation and we have 1 1 + =λ r r12 1 1 + r1 r 2 c c12 + =λ r r12 , c1 c2 + r1 r2 2 and |c|2 + r2 + |c12 |2 + r12 = λ (|c1 |2 + r12 + |c2 |2 + r22 ) which ﬁxes λ.

Proof. One can interpret the sequence γk = zk,0 as a discrete curve and γˆk = zk,1 as a μ-Darboux transform of it. Then starting with z˜0,0 we can create a λ-Darboux transform for γ. The Binachi permutability now states that γˆ˜ k = z˜k,1 is given uniquely such that γˆ˜ k = z˜k,1 is a λ-Darboux transform of γˆk = zk, 1 and a μ-Darboux transform of γ˜k . Iterating this deﬁnes the map z˜ uniquely and satisﬁes the stated cross-ratio conditions. 40 Lecture 8 Tim Hoﬀmann 41 Figure 12: A Tractrix and Darboux transform of the straight line.

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