PathParameter Command

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 and A is a point attached to this function with coordinates (1,1) (you can create such point using the point on object tool or A=Point(f), SetCoords(A,1,1) commands). Then 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 \(X=C+\vec{a}\cdot cos(\alpha)\vec{b}\cdot sin(\alpha)], where stem:[\alpha\in(-\pi,\pi)] has path parameter stem:[\frac{\alpha\pi}{2\pi}\)
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…A_n	If X lies on A_kA_k+1, path parameter of X is \(\frac{k-1+\phi(X,A,B)}{n-1}\)
Polygon A₁…A_n	If X lies on A_kA_k+1 (using A_n+1=A₁), path parameter of X is \(\frac{k-1+\phi(X,A,B)}{n}\)
List of paths L={p₁,…,p_n}	If X lies on p_k and path parameter of X w.r.t. p_k is t, path parameter of X w.r.t.L is \(\frac{k-1+t}{n}\)
List of points L={A₁,…,A_n}	Path parameter of A_k is \(\frac{k-1}{n}\). Point[L,t] returns \(A_{\lfloor tn\rfloor+1}\).
Locus
Implicit polynomial	No formula available.

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 \(X=C+\vec{a}\cdot cos(\alpha)\vec{b}\cdot sin(\alpha)], where stem:[\alpha\in(-\pi,\pi)] has path parameter stem:[\frac{\alpha\pi}{2\pi}\)

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…A_n

If X lies on A_kA_k+1, path parameter of X is \(\frac{k-1+\phi(X,A,B)}{n-1}\)

Polygon A₁…A_n

If X lies on A_kA_k+1 (using A_n+1=A₁), path parameter of X is \(\frac{k-1+\phi(X,A,B)}{n}\)

List of paths L={p₁,…,p_n}

If X lies on p_k and path parameter of X w.r.t. p_k is t, path parameter of X w.r.t.L is \(\frac{k-1+t}{n}\)

List of points L={A₁,…,A_n}

Path parameter of A_k is \(\frac{k-1}{n}\). Point[L,t] returns \(A_{\lfloor tn\rfloor+1}\).

Locus

Implicit polynomial

No formula available.