CHALLENGE TASK: Intersection of paramteric curve and ray

[Alibeg]

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 = cos(PHI) - cos(THETA) + D cos(PHI/2 - THETA/2 )
y = sin(PHI) - sin(THETA) + D sin(PHI/2 - THETA/2 )

Ray:
y = tan(BETA) x

BETA, THETA and PHI are angles. PHI is my parameter, THETA and BETA are some variable angles.

I need to find the coordinates of the intersection of curve and the ray as a function of THETA and BETA.
In other words, I want to know coordinates of intersection but without the parameter PHI in them.

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I find the intersection, but still I cannot eliminate parameter from my solution.
It must be solvable (well, I guess) since for each combination of angle beta and angle theta, there is exactly one and only intersection with the curve.
If I could find PHI as a function of other variables I could easily substitute it back in equation of curve.

 

[Deleted member 4993]

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 = cos(PHI) - cos(THETA) + D cos(PHI/2 - THETA/2 )
y = sin(PHI) - sin(THETA) + D sin(PHI/2 - THETA/2 )

Ray:
y = tan(BETA) x

BETA, THETA and PHI are angles. PHI is my parameter, THETA and BETA are some variable angles.

I need to find the coordinates of the intersection of curve and the ray as a function of THETA and BETA.
In other words, I want to know coordinates of intersection but without the parameter PHI in them.

----------------------------------------------------------------------

I find the intersection, but still I cannot eliminate parameter from my solution.
It must be solvable (well, I guess) since for each combination of angle beta and angle theta, there is exactly one and only intersection with the curve.
If I could find PHI as a function of other variables I could easily substitute it back in equation of curve.

x = cos(PHI) - cos(THETA) + D cos(PHI/2 - THETA/2 )
y = sin(PHI) - sin(THETA) + D sin(PHI/2 - THETA/2 )

\(\displaystyle x \ + \ cos(\theta) \ = \ cos(\phi) \ + \ Dcos\left (\frac{\phi \ - \ \theta}{2}\right )\)

\(\displaystyle y \ + \ sin(\theta) \ = \ sin(\phi) \ + \ Dsin\left (\frac{\phi \ - \ \theta}{2}\right )\)

Square both sides of both equations and then add

\(\displaystyle x^2 \ + \ y^2 \ + \ 2x \ * \ cos(\theta) \ + \ 2y \ * \ sin(\theta) \ = \ 1 \ + D^2 \ + \ 2D \ * \ cos\left (\frac{\phi \ + \ \theta}{2}\right )\)

Now only one term has ? in it - surely you know how to isolate that.....