Any arbitrary ellipse in the x-y plane can be described with five parameters -- usually the center’s x and y coordinate positions, x0 and y0; the distance between focal points, d; the eccentricity, $\epsilon$; and the counterclockwise tilt angle, $\psi$, of the line through the center and foci with respect to the horizontal x axis. There are some trivially derived variants of this set of five, such as using the major and minor semi-axes or doing a simple transformation to polar coordinates, but I am trying to find analytic expressions for x0 and y0 if they are not given but rather the coordinates (x1, y1) of some particular point on the perimeter are known instead. In other words I want to know if there is an analytic closed form expression for F and G where
x0 = F(d,$\epsilon$,$\psi$,x1,y1) y0 = G(d,$\epsilon$,$\psi$,x1,y1)
If the counterclockwise angle $\theta$ of the tangent line through the perimeter point (x1, y1) with respect to the horizontal x axis is also given or known could that be sufficient to determine the tilt angle $\psi$ if it is not known explicitly?
Thank you J. M. and Barry Cipra. I see the point you are making, and think that I need to explain the context. I have a software program that, when given
(1) the coordinates of a starting perimeter point, (x1, y1);
(2) a tilt angle, $\psi$ (3) a distance between foci, d (4) an eccentricity, $\epsilon$ (5) a starting perimeter point tangential angle, $\theta$ (6) a travel angle, $\phi$, with respect to a line from a center through the starting point
it will generate an elliptical arc segment whose end point is determined by these inputs. However there are times when I want to be able to specify the coordinates of the end point, (x2, y2), explicitly as input and manipulate the shape of the arc between the two fixed points with the other input parameters. I think I could modify the program for this if there were some way to express the coordinate positions of the center of the ellipse in terms of these input parameters.
Because an ellipse has five degrees of freedom for parameters, if I specify the four position coordinates of the start and end points of an arc segment (x1, y1, x2, y2), I think that specifying one further parameter from ($\psi$, d, $\epsilon$, $\theta$, or $\phi$) ought to determine the ellipse and consequently determine the other four unspecified ones among these, as well as determining expressions for other commonly used ellipse parameters (center coordinates x0 and y0, semi-major and semi-minor axes, coordinates of foci, etc.) as functions of the five specified ones.