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Homework 2

Problem Statement

Suppose X~(α,β) is a reparameterization of X(u,v) with first fundamental form given by:

E~dα2+2F~dαdβ+G~dβ2

Then X(u,v)=X~(α(u,v),β(u,v)). Compute E~,F~,G~ from E,F,G.

We get that:

Xu=αuX~α+βuX~βXv=αvX~α+βvX~β

Hence the dual vectors are given by:

(dαdβ)=(αuαvβuβv)(dudv)

Then since Edu2+2Fdudv+Gdv2=ds2=E~dα2+2F~dαdβ+G~dβ2 , we obtain that:

ds2=(dαdβ)(E~F~F~G~)(dαdβ)=(dudv)(αuαvβuβv)(E~F~F~G~)(αuαvβuβv)(dudv)=(dudv)(EFFG)(dudv)

Thus:

(EFFG)=(αuαvβuβv)(E~F~F~G~)(αuαvβuβv)(E~F~F~G~)=(αuαvβuβv)1(EFFG)(αuαvβuβv)1

Giving us finally:

E~=1(αvβuαuβv)2(GαvβuFβv(αv+βu)+Eβv2)F~=1(αvβuαuβv)2(Gαuαv+F(αv2+αuβv)Eαvβv)G~=1(αvβuαuβv)2(Gαu2Fαu(αv+βu)+Eαvβu)