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Topic: Ellipse and line intersection points
Replies: 22   Last Post: Jun 30, 2014 12:01 PM

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Ken.Pledger@vuw.ac.nz

Posts: 1,377
Registered: 12/3/04
Re: Ellipse and line intersection points
Posted: Jun 24, 2014 5:44 PM
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In article <loa82d$p7a$1@dont-email.me>,
Cristiano <cristiapi@NSgmail.com> wrote:

> I have the ellipse:
> x(E)= a * (cos E - e)
> y(E)= b * sen(E)
> where a, b and e are constants.
>
> Then I rotate the ellipse onto another plane:
> xr= k1 * x(E) + k2 * y(E)
> yr= k3 * x(E) + k4 * y(E)
> k's are properly chosen constants.
>
> In this new plane I have the vertical line x= k.
>
> Is there any way to find the intersection points between the ellipse and
> the line or it is impossible? ....



It's certainly possible, but writing the answer in a single formula
is messy. Here are the steps of a calculation.

Solve the last pair of equations for x(E) and y(E) in terms of xr and yr.

Solve the first pair of equations for cos(E) and sin(E) in terms of x(E)
and y(E) and hence in terms of xr and yr.

Use (cos(E))^2 + (sin(E))^2 = 1 to get a second-degree equation in xr
and yr.

Put xr = k and solve the resulting quadratic equation for yr.


Ken Pledger.



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