Difference between revisions of "PathParameter Command"
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| − | <noinclude>{{Manual Page|version= | + | <noinclude>{{Manual Page|version=5.0}}</noinclude>{{command|function}} |
| − | {{command|function}} | + | ;PathParameter( <Point On Path> ) |
| − | ;PathParameter | + | :Returns the parameter (i.e. a number ranging from ''0'' to ''1'') of the point that belongs to a [[Geometric Objects#Paths|path]]. |
| − | :Returns the parameter (i.e. a number ranging from 0 to 1) of the point that belongs to a [[Geometric Objects#Paths|path]]. | + | :{{example| 1=<div>Let <code><nowiki>f(x) = x² + x - 1</nowiki></code> and <code><nowiki>A = (1, 1)</nowiki></code> is a point attached to this function. <br><code><nowiki>PathParameter(A)</nowiki></code> yields ''a = 0.47''.</div>}} |
| + | |||
| + | In the following table <math>f(x)=\frac{x}{1+|x|}</math> is a function used to map all real numbers into interval (-1,1) and | ||
| + | <math>\phi(X,A,B)=\frac{\overrightarrow{AX}\cdot\overrightarrow{AB}}{|AB|^2}</math> is a linear map from line AB to reals which sends A to 0 and B to 1. | ||
| + | {| class=pretty | ||
| + | |- | ||
| + | |Line AB | ||
| + | | <math>\frac{f(\phi(X,A,B))+1}2</math> | ||
| + | |- | ||
| + | |Ray AB | ||
| + | |<math>f(\phi(X,A,B))</math> | ||
| + | |- | ||
| + | |Segment AB | ||
| + | |<math>\phi(X,A,B)</math> | ||
| + | |- | ||
| + | |Circle with center ''C'' and radius ''r'' | ||
| + | |Point <math>X=C+(r\cdot cos(\alpha),r\cdot sin(\alpha))</math>, where <math>\alpha\in(-\pi,\pi)</math> has path parameter <math>\frac{\alpha+\pi}{2\pi}</math> | ||
| + | |- | ||
| + | |Ellipse with center ''C'' and semiaxes <math>\vec{a}</math>, <math>\vec{b}</math> | ||
| + | |Point <math>X=C+\vec{a}\cdot cos(\alpha)+\vec{b}\cdot sin(\alpha)</math>, where <math>\alpha\in(-\pi,\pi)</math> has path parameter <math>\frac{\alpha+\pi}{2\pi}</math> | ||
| + | |- | ||
| + | |Hyperbola | ||
| + | |Point <math>X = C \pm \vec{a} ·cosh(t) + \vec{b} ·sinh(t)</math> has path parameter <math> \frac{f(t)+1}{4}</math> or <math>\frac{f(t)+3}{4}</math> | ||
| + | |- | ||
| + | |Parabola with vertex V and direction of axis <math>\vec{v}</math>. | ||
| + | |Point <math>V+\frac{1}{2}p\cdot t^2\cdot \vec{v}+p\cdot t \cdot \vec{v}^{\perp}</math> has path parameter <math>\frac{f(t)+1}2</math>. | ||
| + | |- | ||
| + | |Polyline A<sub>1...A<sub>n</sub> | ||
| + | |If X lies on A<sub>k</sub>A<sub>k+1</sub>, path parameter of ''X'' is <math>\frac{k-1+\phi(X,A,B)}{n-1}</math> | ||
| + | |- | ||
| + | |Polygon A<sub>1...A<sub>n</sub> | ||
| + | |If X lies on A<sub>k</sub>A<sub>k+1</sub> (using A<sub>n+1</sub>=A<sub>1</sub>), path parameter of ''X'' is <math>\frac{k-1+\phi(X,A,B)}{n}</math> | ||
| + | |- | ||
| + | |List of paths L={p<sub>1</sub>,...,p<sub>n</sub>} | ||
| + | |If X lies on p<sub>k</sub> and path parameter of X w.r.t. p<sub>k</sub> is ''t'', path parameter of ''X'' w.r.t.''L'' is <math>\frac{k-1+t}{n}</math> | ||
| + | |- | ||
| + | ||List of points L={A<sub>1</sub>,...,A<sub>n</sub>} | ||
| + | |Path parameter of A<sub>k</sub> is <math>\frac{k-1}{n}</math>. Point[L,t] returns <math>A_{\lfloor tn\rfloor+1}</math>. | ||
| + | |- | ||
| + | |Locus | ||
| + | | | ||
| + | |- | ||
| + | |Implicit polynomial | ||
| + | |No formula available. | ||
| + | |} | ||
Latest revision as of 08:03, 2 June 2019
- PathParameter( <Point On Path> )
- Returns the parameter (i.e. a number ranging from 0 to 1) of the point that belongs to a path.
- Example:Let
f(x) = x² + x - 1andA = (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}. |
| Locus | |
| Implicit polynomial | No formula available. |
