# length of an arm from a point along major axis of ellipse

I have a point which rotates around an offset point along the major axis of an ellipse. I am trying to solve the length of the red line at any given angle and location along the perimeter of the ellipse.

The image illustrates the different positions that a single red line could be in. I need a function to know the length of the line from this point to the perimeter from any given angle.

since the offset from center is variable, the origin of the line is not necessarily the center or the foci of the ellipse.

variables: angle, offset (from center: 0,0) of the point attached to the major axis.

information already known at all times: major and minor width, the offset point along the major axis, and the current angle of the red line (in either radian or degrees).

unknown: the length of the line.

demonstration of possible line positions based off a single offset

## 1 Answer

A point on the ellipse is $(a\cos t, b\sin t)$

The focus at $(-c,0)$

$c^2 = a^2 - b^2$

$d = \sqrt {(a\cos t + c)^2 + (b\sin t)^2}$

Multiply it out and simplify.

Also worth noting that:

$\sqrt {(a\cos t + c)^2 + (b\sin t)^2}+\sqrt {(a\cos t - c)^2 + (b\sin t)^2} = 2a$

• apologies if I'm a little obtuse but could you explain what a, b, c and t refer to? – Shawn Naquin Nov 13 '17 at 20:22
• $a$ is the length of the semi-major axis. $b$ is the length of the semi-minor axis. $c$ is the distance from the center to the focus. $\frac {x^2}{a^2 } + \frac {y^2}{b^2} = 1$ – Doug M Nov 13 '17 at 20:25