Get the points of a moving body on a curve each T seconds

Question

There is a body moves along an eight figure path and there is a laser beam at the origin that rotating with rotation rate $w$.
How to get the points that originate each rotation from the intersection between the moving body path (eight figure) and the rotating beam.

Note that the laser beam is used for nothing but only as time reference.

Given:

1. the equation that describes that path:
   $(x-h)^4 = a^2((x-h)^2-(y-k)^2)$
   where $(h,k)$ = (50,70)

2. The start point of the curve $(X_S,Y_S)$
   where $(X_S,Y_S)$ = (30,70).

3. The speed of the moving body ($V$=150 m/s).

4. The speed of rotation of the laser beam $w= 36$ deg/sec.

5. The value of a = 10.

Required:

The points on the curve that came from the intersection between the beam line and the motion path (yellow points on the attached figure).

Please note that:
I do not know any information about how many points came from the intersection between the body path and the beam, for sure it depends on the body speed and the rotation rate of the beam.

Here is a figure for more illustration:

Answer 1

I haven't solved the question, but I have an approach you can try.

You have the starting point and the speed and the curve. So you should be able to get the position of the body as a function of time.

Solve speed = 150 m/s = $ \sqrt{ \frac{dx}{dt}^2 + \frac{dy}{dt}^2 }$ and $ 4(x-h)^3 \frac{dx}{dt} = a^2[ 2(x-h)\frac{dx}{dt} - 2(y-k)\frac{dy}{dt}] $

to get $\frac{dx}{dt}$ and $\frac{dy}{dt}$. You also have the initial position $(x_0,y_0)$.

So now you have the position of the body as a function of time as $x = x_0 + \int x(t)$ and $y = y_0 + \int y(t)$.

For the next part, try converting to polar coordinates by using $x = r cos \theta$ and $y = r sin \theta $.
This way, you will have the angle the position vector of the object makes with the x-axis at the origin as a function of time.
Equate that to the laser's angle and you get your theta values. This can further give you the times and therefore the position of the body.

Hope this helps.