Solving trigonometric system of equations

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Discussion Overview

The discussion revolves around solving a trigonometric system of equations to find the intersection of a parametric curve with a ray originating from the coordinate system's origin. The participants explore the mathematical formulation of the problem, focusing on how to express the intersection coordinates as a function of specific angles while eliminating a parameter.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents a parametric equation for a curve and a ray equation, seeking to find their intersection coordinates as a function of angles θ and β, excluding the parameter φ.
  • Another participant suggests a numerical approach to find the intersection, proposing an algorithm to approximate roots without manual selection of starting values.
  • A different participant expresses the need for the intersection distance from the origin as a function of θ to determine its minimal value, indicating a potential use of derivatives.
  • One participant questions the feasibility of a numeric solution, stating it would not allow for further analysis of the data and hints at a possible misunderstanding of the approach.
  • Another participant describes the movement of the curve with varying θ and expresses a desire to find the equation for the boundary of the area traced by the curve.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best approach to solve the problem. There are competing views on whether a numerical solution is acceptable or if a more analytical method is necessary.

Contextual Notes

Participants express uncertainty regarding the implications of using numerical methods versus analytical solutions, and there are unresolved questions about the relationship between the angles and the resulting geometric properties of the intersection.

Alibeg
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Hi. I'd like to know whether is it possible to do the following, and if so, how...(and also, whether is it possible to solve similar problems)

I have parametric equation of a curve and I need to find its intersection with a ray that starts at the origin of the coordinate system and makes known angle with the positive x-axis.

Curve:
x = ( R-r )( cos([tex]\varphi[/tex]) - cos([tex]\theta[/tex]) ) + D cos([tex]\varphi[/tex] (R-r)/r - [tex]\theta[/tex] (R-r)/r )
y = ( R-r )( sin([tex]\varphi[/tex]) - sin([tex]\theta[/tex]) ) + D sin([tex]\varphi[/tex] (R-r)/r - [tex]\theta[/tex] (R-r)/r )

R, r and D are known constants.

Ray:
y = tan([tex]\beta[/tex]) x

[tex]\beta[/tex], [tex]\theta[/tex] and [tex]\varphi[/tex] are angles, [tex]\varphi[/tex] is my parameter, [tex]\theta[/tex] and [tex]\beta[/tex] are some variable angles.

I need to find the coordinates of the intersection of curve and the ray as a function of [tex]\theta[/tex] and [tex]\beta[/tex].

In short I want to know coordinates of intersection but without the parameter [tex]\varphi[/tex] in them.
 
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Mach18Image1.png


This is the picture of my problem for r = 5, R= 6 and D = 7.
Note that [tex]\theta[/tex] is not visible on this picture, but it had a fixed value while I was taking the picture.
Making [tex]\theta[/tex] change would cause curve to change position and orientation.

Thanks :D
 
How about numerically? It's not too hard to design an algorithm that zeros into the roots without having to manually select starting values.
 
Hmmm. I need distance of that intersection from the origin of the coordinate system. But I need it as a function of a [tex]\theta[/tex] so I can find its minimal value for different [tex]\theta[/tex]s . (i would have to use derivatives and other tricks later)
 
Well what, that ain't no hill neither. But first you need to make clear if you are wiling to settle for a numeric approximation. If not, then I can't do it.
 
Last edited:
I am sorry, but numeric solution doesn't help me, since I cannot analyze the data further that way, and I also don't think it's an easy task.

Maybe I am using a wrong approach.Here more description of my problem:
I have to change [tex]\theta[/tex] from 0 to some value. While changing it my (green) curve moves (the origin of the coordinate system is always inside of it).
If I trace my curve, I get a unusual white shaped area. I want to find the equation for the boundary line of that area.

mach18image2.png
 

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