Difference between revisions of "PathParameter Command"

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<noinclude>{{Manual Page|version=5.0}}</noinclude>{{command|function}}
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;PathParameter( <Point On Path> )
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:Returns the parameter (i.e. a number ranging from ''0'' to ''1'') of the point that belongs to a [[Geometric Objects#Paths|path]].
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:{{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>}}
  
test
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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
uu<noinclude>{{Manual Page|version=4.0}}<noinclude/>
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<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.
{{command|function}}
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{| class=pretty
;PathParameter[ <Point On Path> ]
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|-
:{{description}}
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|Line AB
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| <math>\frac{f(\phi(X,A,B))+1}2</math>
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|-
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|Ray AB
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|<math>f(\phi(X,A,B))</math>
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|-
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|Segment AB
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|<math>\phi(X,A,B)</math>
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|-
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|Circle with center ''C'' and radius ''r''
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|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>
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|-
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|Ellipse with center ''C'' and semiaxes <math>\vec{a}</math>, <math>\vec{b}</math>
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|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>
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|-
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|Hyperbola
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|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>
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|-
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|Parabola with vertex V and direction of axis <math>\vec{v}</math>.
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|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>.
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|-
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|Polyline A<sub>1...A<sub>n</sub>
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|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>
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|-
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|Polygon A<sub>1...A<sub>n</sub>
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|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>
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|-
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|List of paths L={p<sub>1</sub>,...,p<sub>n</sub>}
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|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>
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|-
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||List of points L={A<sub>1</sub>,...,A<sub>n</sub>}
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|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>.
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|-
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|Locus
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|
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|-
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|Implicit polynomial
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|No formula available.
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|}

Latest revision as of 07:03, 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 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 \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.
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