How to determine the coordinates of the body if it moves along an ellipse?

Question

Programming

There is a rectangular image with dimensions n by m. On the image you need to draw the current position of the planets. Everything is clear with circular orbits, but what about those that fly in an ellipse?
Let's say the star is in the center, using Kepler's Law and the time elapsed since the planet was at 12 o'clock, I can find out how far the planet has traveled from 12 o'clock. What next?

Answer 1

The star is not in the center, but at the focus of the ellipse. All you need to know (other than the orientation of the ellipse in space) is that the sectoral velocity of the planet is constant. You will also need an orbital period T and the moment when the planet passed perihelion.
Further simply. Let (a,b) be the semiaxes of the ellipse, the star at the focus (c,0), where c=sqrt(a^2-b^2), the planet at the point X=(a*cos(x),b*sin( x)). The area of ​​the sector "seen" by the planet is S(x)=(a*b*xc*b*sin(x))/2, and the sector it passes in a year is S(T)=a*b*pi . Therefore, at the point X the planet appears at the moment t(x)=T*(xc*sin(x)/a)/(2*pi). Solving the equation to get t(x) will have to be done numerically.

Answer 2

https://ru.wikipedia.org/wiki/%D0%AD%D0%BB%D0%BB%D...
Difference from a circle only in different radius coefficients a and b.