Definition
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[2G6]By polygonal curve \(φ: [0,1] → ℝ^ 2\) we will mean: a not self-intersecting (that is, injective) polygonal (that is, piecewise linear) curve in the plane. Analytically, there are points \(V_ 0,V_ 1,\ldots V_ n\) (called “vertices”) in the plane, and \(0=t_ 0{\lt}t_ 1\ldots {\lt}t_ n=1\) such that
\[ φ (t) = \frac{t-t_ i}{t_{i+1}-t_ i} V_{i+1} + \frac{t_{i+1}-t}{t_{i+1}-t_ i} V_ i ~ \text{~ when~ }~ t_ i≤ t≤ t_{i+1}~ . \]
The polygonal curve is closed if \(φ(0)=φ(1)\). (In this case we require that \(φ\) is injective when restricted to \([0,1)\)).