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 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 $\alpha\in ]-\pi,\pi]$ 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…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.