# Tracing points around a curve/ellipse

Sorry if this has been asked before but my maths days are long behind me.

What I want to know is how to find out the coordinates along the circumference of an ellipse.

So supposing I am at point X,Y which lies on the circumference, and I am traveling at velocity Z. How can i work out after given time t what the new X,Y values would be.

seems so simple but just cant think where to start.

Thanks for the help

Aaron

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is the speed constant? – Santosh Linkha Jul 25 '12 at 22:06
yes it is and its z – Carry on Smiling Jul 25 '12 at 22:08
youtube.com/watch?v=iUfu9RK6w44 – Will Jagy Jul 25 '12 at 22:35
@WillJagy are you aware this link is to a song? – Carry on Smiling Jul 25 '12 at 22:48
@ChuckFernández, yes, DevilWAH wrote "my maths days are long behind me" in a rather forlorn manner. The song says no, not really. Here is a different one: youtube.com/watch?v=KHVYBiVKldU – Will Jagy Jul 25 '12 at 23:06

$$u \mapsto (a\cos u, b\sin u)$$ where $u$ is a parameter and $2a$ and $2b$ are the lengths of the ellipse's major and minor axes. Your only problem is then to find the next $u$ value at each step. If you do it with small enough steps you can simply differentiate this expression to find out how far a small change in $u$ makes the point move. You get something like $$u(t+\Delta t) = u(t) + \frac{z\Delta t}{\sqrt{a^2\sin^2(u(t))+b^2\cos^2(u(t))}}$$ where you choose the time step $\Delta t$ by trial and error to be small enough that the speed ends up being close enough to constant for your needs.