Thread: Line tangent to ellipse

I trying to find a set of equations to fidn the tangent point of a line and a rotated ellipse. Set of given parameters are; semi-major, a=320, semi-minor, b=120, angle of rotation abou tthe ellipse center is 6.4 degrees, ellipse origin is at 0,0. The line is sloping up to the left at 10 degrees relative to the x-axis. The resultant equations will be used in a CAD C++ application.
Some of the formulas I have been trying;
y = mx + b
x^2/a^2 + y^2/b^2 = 1
y = -x/m * (b/a)^2
paramteric equations for an ellipse
x = a * cos(t)
y = b * sin(t)
0 <= t <=2pi
X = a * cos(t) * cos(ra) - b * sin(t) *sin(ra)
Y = a * cos (t) * cos(ra) + b * sin(t) * cos(ra)
ra = the angle of rotation of the ellipse

But no luck in solving this one. It's been a long time since I had my math classes.
I have been trying to come up with two equations, one for the X coordinate and the Y coordinate of the tangent point. I also understand there will be two solutions, (x,y) and (-x,-y). I require equations so the application can use different values for the parameters, such the slope (angle) of the line will change, the semi-major/semi-minor/angle rotation of the ellipe will change.

Any help will be greatly appreciated!

Thanks in advanced.
Willie

You seem to be missing one thing and you may have overlooked another.

is NOT rotated.

The parametric representation is a bit misleading. That 't' is important, and it will trace the entire curve, but t = pi/6 wont give you the point on the ellipse that produces a line through the origin with an elevation of pi/6 above the x-axis.

Thank you for reply.
I agreed that the x^2/a^2 + y^2/b^2 = 1 is not rotated, but that is one of the first equation I was trying without rotation.
And I also agree with the parameter t; changes in t creates the ellipse.
I have a solution for an ellipse with an angle of rotation = 0, but once I start rotating the ellipse is where I have the problem.