Parametric equations for cylinder/plane intersection

I'd like to make sure my thinking is right on this problem and to see what the next step would be because I'm getting stuck.

The problem is to find the parametric equations for the ellipse which made by the intersection of a right circular cylinder of radius c with the plane which intersects the z-axis at point 'a' and the y-axis at point 'b' when t=0.

I think the equation for the cylinder would be $\displaystyle x^2+y^2=c^2$

As for the plane I am less sure about the equation. I have points and I could write an equation with one if I had a normal vector to the plane... or the cross product of two vectors in it. Two vectors would be between the points (0, b, 0) and (0, 0, a) and the points (0, b, 0) and (1, 0, a) since x will take on all values. Those vectors are <0, -b, a> and <1, -b, a>

$\displaystyle \left|\begin{matrix}\mathbf{i} & \mathbf{j} & \mathbf{k}\\ 0 & -b & a \\ 1 & -b & a\end{matrix}\right|$ Which yields $\displaystyle 0\mathbf{i} + a\mathbf{j} + b\mathbf{k}$

so using the y intercept and $\displaystyle n_1(x-x_1) + n_2(y-y_1) + n_3(z-z_1) = 0 $ the equation of the plane should be $\displaystyle a(y - b) + bz = 0 $ or $\displaystyle ay - ab +bz = 0 $

I get stuck here though... what should I do next? Any help is appreciated.

http://img9.imageshack.us/img9/1640/...00313at316.png

Re: Parametric equations for cylinder/plane intersection

Quote:

Originally Posted by

**earboth** 1. The condition t = 0 has nothing to do with the ellipse.

2. You already have a parametric equation of the ellipse:

$\displaystyle e:\left\{\begin{array}{l}x=s \\ y = \sqrt{c^2-s^2} \\ z=a-\frac ab\cdot \sqrt{c^2-s^2} \end{array} \right.$

where s is a real variable with $\displaystyle -c\leq s\leq c$

I realize this is a couple years post-post, but I should probably point out that the following expression needs to have a plus/minus in both the Y and Z parameters:

± √(c^2-s^2)

When the Y parameter is positive, the Z parameter should use the minus.

Great work getting the equations, though. It was very helpful to me.