I need some help to pull this through, first a background, then some questions. But first of all, I prefer using the following "parametric polar" formulas for an ellipse:
x = cx + a Cos[t] Cos[T] + b Sin[t] Sin[T]
y = cy - a Cos[t] Sin[T] + b Sin[t] Cos[T]
cx and cy: Coordinates of the center point of the ellipse..
a and b: Lengths of semi-axes.
T: Tilt of the ellipse.
t: That polar coordinate stuff thing 0 to 2pi, you know.
BACKGROUND
Given parameters for an ellips, I want to find the parameters for another ellipse which:
1) is smaller than the first ellipse (but same proportions between axes a/b and same tilt T and same center point cx;cy).
2) has another location, so that its a center point is higher or lower than the center of the first ellipse. Actually, more generally, this new center point would be located on the rotational axis which is common to both ellipses, so if the first ellipse is tilted, then the new ellipse center will not be straight above that of the other, but somewhere on a straight line describing that rotational axis.
3) Both smaller and with a different center point.
The parameters of the new ellipse should be determined from some observational points on its perimeter.
QUESTIONS
This is how I start out on the problem with the formulas for the new ellipse:
X = r + cx + s a Cos[t] Cos[T] + s b Sin[t] Sin[T]
Y = k*r + cy - s a Cos[t] Sin[T] + s b Sin[t] Cos[T]
Code:
Sin[-T+pi/2]
k = ---------
Cos[-T+pi/2]
s is the size of the new ellipse in relation to the first ellipse s = a2/a1 = b2/b1.
r is the difference in center position along the x-axis.
k is the slope of the line which connects the centers of the two ellipses and hence k*r is the difference in center position along the y-axis. Note that k is given by the first ellipse because the rotational axis should be pi/2 radians off from the tilt.
so it is s and r I am looking for. To my help I have some points on the perimeter of the new ellipse: x1;y1 x2,y2 (and more if needed).
I have started with putting X - x1 == 0 and so on in the Solve command of Mathematica in order to isolate s and r and substitute them until X is expressed only in the a, b, T, cx, cy of the old ellipse. I end up with a nice enough formula, but when I put real values in it, Mathematica protest about "complex infinity". I dont know where to go from here.
What should the end result look like, or how should I proceed to get there?