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Let $\gamma_0$ and $\gamma_1$ be two paths in $\mathbb{S}^2$ (the 2-d unit sphere in $\mathbb{R}^3$). Let both $\gamma_0$ and $\gamma_1$ start at $p \in \mathbb{S}^2$ and end in $q \in \mathbb{S}^2$. What is an explicit formula that is a homotopy from $\gamma_0$ to $\gamma_1$, using intermediate curves that must all lie on $\mathbb{S}^2$ and connect $p$ to $q$.

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stereographic projection to the plane, $t\gamma_1+(1-t)\gamma_0$ go from the plane back to the sphere. – butt Aug 9 '12 at 21:45
@butt But what if one of the curves passes through the north pole? I don't think there is a nice explicit formula unless you assume the curves aren't surjective. – MartianInvader Aug 10 '12 at 0:03
This seems a good case where the Seifert-van Kampen Theorem tells you that a homotopy exists, but does not give an explicit formula, as the proof of the theorem involves subdivisions. Why should an explicit formula be expected? – Ronnie Brown Aug 10 '12 at 8:36

If you're sure that, for every $t$,

$$ \gamma_0(t) \neq - \gamma_1(t) \ , $$

that is, for every $t$, $\gamma_0(t)$ and $\gamma_1(t)$ are not antipodal points, then you can use the straight line homotopy in disguise:

$$ H(s,t) = \dfrac{(1-s) \gamma_0(t) + s \gamma_1(t)}{\vert\vert (1-s) \gamma_0(t) + s \gamma_1(t) \vert\vert} \ . $$

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The following works when $\gamma_0$ and $\gamma_1$ both miss a point $x$ of $S^2$. Which is nearly always - if you really want to use a space-filling curve or something, then you'll have to first construct a homotopy that moves the curve away from a point you choose on the sphere. This is easy to describe but hard to construct explicitly.

First of all, let's solve the problem in $\mathbb R^2$. There, it's simple: just move along the straight line between $\gamma_0(t)$ and $\gamma_1(t)$, i.e. \[H(s,t)=s\gamma_0(t)+(1-s)\gamma_1(t)\]. Then for your paths on the sphere, use stereographic projection, with the point $x$ that both paths missed as your projection point, solve the problem in the plane, and project back. There are explicit formulae for that projection, but finding them and applying them I leave an exercise to the reader.

It's mostly unenlightening in any case to have the explicit formulae. Algebraic topologists are more often than not satisfied with descriptions or even pictures - usually you're far more interested in if a homotopy exists than what it precisely is.

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