Re: Question #1: Parametric Equation of an Ellipse


"Walter" <DaHoodWid2Pac@xxxxxxxxxxx> wrote in message news:10812714.1162959986329.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxxxxx
I'm trying to solve a problem involving ellipses and whether or not a point lies inside, on, or outside of the ellipse. I think
that I'm making a mistake right from the beginning with the angle term in the parametric equation of an ellipse:

x = a cos t
y = b sin t

where a and b are the semi-major & semi-minor axes.

From what point is the angle t measured with respect to: is it the origin (0,0), one of the focal points, or somewhere else?

I've looked around quite a bit on the web but haven't found an answer to this query. I'll describe my problems in my next post if
I can't this resolved.

Thanks,
Walt

The 'angle' t is indeed not the angle between the x-axis and the
radius from (0,0) to (x,y). It is the angle
t = atan( a/b tan(theta) )
where theta is the standard polar angle.
If a > b then t runs 'ahead' to infinity of theta in the first quadrant,
and then lags behind in the second quadrant when both run to
pi. This is repeated in Q3 and Q4 until they both reach 2 pi
again.

If you want to know whether some point is inside or
outside the ellipse, just use the equation
x^2/a^2 + y^2/b^2 = 1
and note that all points inside the ellipse satisfy
x^2/a^2 + y^2/b^2 < 1
and all points outside
x^2/a^2 + y^2/b^2 > 1

Dirk Vdm


.

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