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 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 \(X=C+\vec{a}\cdot cos(\alpha)\vec{b}\cdot sin(\alpha)], where stem:[\alpha\in(-\pi,\pi)] has path parameter stem:[\frac{\alpha\pi}\{2\pi}\)
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 A_1…An~~	If X lies on A_kA_k+1, path parameter of X is \(\frac{k-1+\phi(X,A,B)}\{n-1}\)
Polygon A_1…An~~	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 A_1…An~~

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

Polygon A_1…An~~

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.