How do I calculate the chord length of ellipse? I need to design a blade vane rotating inside an elliptical profile. The blade is supposed to be in contact with ellipse with a maximum clearance of $0.4\ mm$.
The chord length (i.e., the blade vane) will obviously vary along the ellipse.
I need to find a way to fix the minor axis to major axis ratio of the ellipse such that, when the blade is fixed, it won't exceed a clearance of $.4\ mm$.
Thanks!
Take a look at this:
http://gieseanw.wordpress.com/2013/07/19/an-analytic-solution-for-ellipse-and-line-intersection/
Basic idea: starting from writing down the formula for an ellipse:
$$(\dfrac{x}{a})^2 + (\dfrac{y}{b})^2 = 1$$
where $a$ is the semi-major axis, $b$ is the semi-minor axis. Then write down the formula for a line
$$y_1 = m \times x_1 + b$$
Where
$$m = \dfrac{y_2-y}{x_2-x}$$
(We are deriving the eqn of a line given two points).
Solve for the y-intercept, $b$ in terms of $x_1$ and $y_1$, then plug that back into the line equation. Finally, represent $y$ using $x$ in the line formula, then replace $y$ in the ellipse formula using the representation you got from line. In that case you can solve the $x$ for two intersections.