Re: Locating points on an ellipse

Hi Jim,

When you ask for "find N equally-spaced points at locations {X,Y} along the circumference", you are actually asking to solve a problem that is usually solved by estimation. David posted showing dividing the arc into constant angles, but that is not the same as equal arc lengths. The easiest way to look at this is as two one dimensional problems: For the x axis we plot Cos, and for the y we plot Sin. So assuming the major axis is x, and minor is y, then the two plots are K * Cos(theta), and J* Sin(theta). Each of those plots can in fact be drawn as a circle. So if you draw the two circles, you'd read the y value of the smaller circle, and the x value of the larger one... plot where those two points meet in 2D and you have your ellipse. Now if you look at the value of the arc subtended by a small angle, saw 1 degrees, then that arc length will be bigger as you approach the x axis, and smaller as you approach Pi/2 (y axis). That is, the arc length approaches the a factor of the radius of the smaller circle when Cos(theta) approaches zero, and it approaches a factor of the radius of the larger circle when Cos(theta) approaches 1. Obviously a job for calculus, but the solution is not as simple as it may appear. Read up on the elliptic integral, e.g : http://en.wikipedia.org/wiki/Elliptical_integral, if you are really interested.

The question I'd ask is can you get away with near enough on his ? If it was an elliptical gear , then probably not, but if it is just something drawn on screen, then the degree of error, which will be always less than theta* (K - J), might not be worth the effort to fix ;)




"Jim Mack" <jmack@xxxxxxxxxxxxxxx> wrote in message news:%23QgCtTn$IHA.4616@xxxxxxxxxxxxxxxxxxxxxxx
It's been too many years since I needed to do this, and I can't find a
solid reference on the web. I know there are folks here who can help,
so...

I need to find, in Cartesian terms, the end points of N equal length
arcs around the circumference / perimeter of an ellipse.

I can do this easily with a circle, but the added complication of an
oval has me loopy.

Given an ellipse centered at A,B having a major axis of length K and a
minor axis of J, find N equally-spaced points at locations {X,Y} along
the circumference.

How?

Any pointers gratefully accepted.

--
Jim


.

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