PathParameter Command

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PathParameter( <Point On Path> )
Returns the parameter (i.e. a number ranging from 0 to 1) of the point that belongs to a path.
Let f(x) = x² + x - 1 and A = (1, 1) is a point attached to this function.
PathParameter(A) yields a = 0.47.

In the following table f(x)=\frac{x}{1+|x|} is a function used to map all real numbers into interval (-1,1) and \phi(X,A,B)=\frac{\overrightarrow{AX}\cdot\overrightarrow{AB}}{|AB|^2} is a linear map from line AB to reals which sends A to 0 and B to 1.

Line AB \frac{f(\phi(X,A,B))+1}2
Ray AB f(\phi(X,A,B))
Segment AB \phi(X,A,B)
Circle with center C and radius r Point X=C+(r\cdot cos(\alpha),r\cdot sin(\alpha)), where \alpha\in(-\pi,\pi) has path parameter \frac{\alpha+\pi}{2\pi}
Ellipse with center C and semiaxes \vec{a}, \vec{b} Point X=C+\vec{a}\cdot cos(\alpha)+\vec{b}\cdot sin(\alpha), where \alpha\in(-\pi,\pi) has path parameter \frac{\alpha+\pi}{2\pi}
Hyperbola Point X = C \pm \vec{a} ·cosh(t) + \vec{b} ·sinh(t) has path parameter \frac{f(t)+1}{4} or \frac{f(t)+3}{4}
Parabola with vertex V and direction of axis \vec{v}. Point V+\frac{1}{2}p\cdot t^2\cdot \vec{v}+p\cdot t \cdot \vec{v}^{\perp} has path parameter \frac{f(t)+1}2.
Polyline A1...An If X lies on AkAk+1, path parameter of X is \frac{k-1+\phi(X,A,B)}{n-1}
Polygon A1...An If X lies on AkAk+1 (using An+1=A1), path parameter of X is \frac{k-1+\phi(X,A,B)}{n}
List of paths L={p1,...,pn} If X lies on pk and path parameter of X w.r.t. pk is t, path parameter of X w.r.t.L is \frac{k-1+t}{n}
List of points L={A1,...,An} Path parameter of Ak is \frac{k-1}{n}. Point[L,t] returns A_{\lfloor tn\rfloor+1}.
Implicit polynomial No formula available.
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