1

The included drawing shows what I am interested in determining. I'm trying to get the equation for the points on the ellipse when the origin is at P. P should not be assumed to be a focus of the ellipse. So, I'm not sure that using equations for an ellipse where the center is shifted away from $0,0$ apply. This isn't about shifting an origin which is the center. Its about changing the origin from the center to another point along the x axis.

enter image description here

rdemo
  • 173
  • Can you explain the difference between "shifting" and "changing" the origin? – Lee Mosher Feb 21 '24 at 23:08
  • 1
    Changing the origin to the left iss the same as shifting the midpoint to the right? So what is the problem? – trula Feb 21 '24 at 23:08
  • 1
    Take the equation of an ellipse and replace $x$ by $(x-t)$ where $t$ is the amount the ellipse moves to the right – Vincent Batens Feb 21 '24 at 23:09
  • @trula: I'm sure Vincent Batens is right about how to construct the equation. I guess I'm confused because the angle changes from $\alpha$ to $\phi$. When you say midpoint I assume you mean center of the ellipse. It seems to me that no matter what kind of shifting occurs, the points O and P will always be different. I guess I'm just not understanding the concept here. – rdemo Feb 21 '24 at 23:14
  • Does the changing of the origin from O to P mean that $0,0$ is now at point P and not O. That might explain why I am confused although I haven't fully thought it through. – rdemo Feb 21 '24 at 23:17
  • Moving the origin to the left is equivalent to moving the ellipse to the right. Moving the ellipse (or any set of points) to the right is done by replacing $x$ with $x-a$. – HappyDay Feb 21 '24 at 23:30
  • There are several ways to use equations to describe an ellipse. If you have a particular way in mind, show it to us. – David K Feb 21 '24 at 23:39
  • At HappyDay and Vincent. Let's say the point M is at coordinates (3,4) based on O as the origin. If I now shift the ellipse to the right and place P at the origin of the x-y axis (which I am assuming is still $0,0$), why isn't the x coordinate of point M now at 3 +b (or t or a per notation of Vincent and HappyDay respectively). If the ellipse moves to the right, so does point M. – rdemo Feb 21 '24 at 23:55
  • You don't understand why given a translation $(x,y)\to(x+a,y+b)$ then the equation $F(x,y)=0$ describing some locus is transformed with the inverse mapping, to $F(x-a,y-b)=0$. Is it so? – Intelligenti pauca Feb 22 '24 at 14:45
  • The equation of an ellipse centered at the point $(h,k)$ but still having major and minor axes parallel to the $x$ and $y$ axes is $$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$$ As has been mentioned, it is just a translation of the equation about the origin. If the origin is not some significant point of the ellipse, then angles at the origin are immaterial to its equation, just as angles at any other random point on the plane would be immaterial. – Paul Sinclair Feb 22 '24 at 20:28

0 Answers0