Ask A Scientist© Mathematics Archive

name         Jeff P.
status       other
age          30s

Question -   I would like to create some displays for our school's
web site that use items distributed in circles to illustrate ideas.  I
thought I'd put together a function in Flash that would do the
distribution for me.  I'm getting lost in mathematical research which,
though interesting, is taking too much time.  I need a mathematician!
In particular, I need equations for determining a point on a circle given
the number of equidistant points, which point in the sequence I'm
looking for, and the radius.  So:
circlePoint (instances, item, radius)
would return x,y coordinates for the point in position "item" of
"instances" on the circle whose radius is "radius" from (0,0).  Any
assumed position [(0,r), (r,0), etc.] for the first point in the sequence
is OK.
I don't need program code, just the mathematics.  Our teachers and
students (and I!) thank you for any and all assistance.

  I am not sure I follow exactly where your question is leading. Nonetheless,
the following two equations tell just about anything you need to know about
circles:

1.    An especially compact and elegant equation of a circle is:
                 ((x-h)/R)^2 + ((y-k)/R)^2 = 1
where, (x,y) are the coordinates of points on the circle, R is the radius,
and (h,k) is the location of the origin of the circle. This is called the
center/radius form.

2.    The most general equation for a circle is:
                 x^2 +y^2 +D*x +E*y +F = 0
Given the coordinates of any three points: (x1,y1) ; (x2,y2) ; (x3,y3) gives
a system of linear equations in three unknowns: D, E, and F in terms of the
coordinates of the three points. Solving that system of equations gives the
equation of the circle containing the three points.

Good Luck,

Vince Calder
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Jeff,

Trigonometry is your tool for this problem.  You will be dividing the 360
degree circle into "instances" parts.  Let "angle" be (360)/"instances".
You want to 'rotate' a multiple of this "angle" for each item.  Let your x
coordinate be "radius" * Cosine("item" * "angle").  Let your y-coordinate be
"radius" * Sine("item" * "angle").  To adjust positions, you can add or
subtract any angle from your function argument.  It is just VERY IMPORTANT
that both functions have exactly the same argument.

Note:  Some programs and languages use the full words sine and cosine.  Some
use SIN and COS.

Dr. Ken Mellendorf
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