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Homotopy

A homotopy is a very general kind of mathematical eqivalence between functions. In the case of winding numbers, we are interested in equivalent curves, so we want to think of a curve as a function $ \gamma:[0,1] \mapsto R^2$ .

If $ \gamma$ and $ \sigma$ are two curves, we say they are homotopic if there exists a continuous function $ H: [0,1] \times [0,1] \rightarrow R^2$ with the following properties:

item $ H(0,t) = \gamma(t)$

item $ H(1,t)=\sigma(t)$

item $ H(s,0) = H(s,1)$    for all $ s \in [0,1]$ .

Note for fixed $ s_0$, $ \gamma_{s_0}(t) = H(s_0,t)$ is a curve in the plane. Because of the third condition, its ends must meet up. The other two conditions say the when $ s=0$, the curve you get is $ \gamma$ and when $ s=1$, you get $ \sigma$ .

If you think of $ s$ as a time parameter, we begin deforming $ \gamma$ at $ s=0$, and when we reach time $ s=1$, the curve has evolved into $ \sigma$ . The curve $ \gamma_{s_0}(t) = H(s_0,t)$ is the curve it has evolved into at time $ s_0$.

next up previous
Next: Mathematical Description of Winding Up: Winding Number Illustrator Previous: Intuition Behind Winding Numbers
Ross Moore
1998-07-22