Difference between revisions of "PathParameter Command"
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;PathParameter( <Point On Path> ) | ;PathParameter( <Point On Path> ) | ||
: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 | + | :{{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 | 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 | ||
Line 24: | Line 24: | ||
|- | |- | ||
|Hyperbola | |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>. | |Parabola with vertex V and direction of axis <math>\vec{v}</math>. | ||
− | |Point <math>V+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>. | + | |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> | |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}</math> | + | |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> | |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 | + | |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>} | |List of paths L={p<sub>1</sub>,...,p<sub>n</sub>} |
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 - 1
andA = (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. |