I am rather confused on this calculation, and I can't seem to find a solution online. I need to know how to find the distance between two known points on an ellipse's outer edge following the path of the curve. For example:
Example
In that image, how would I go about using the coordinates of point A and B to find the arc length/segment of the circumference between points A and B. The current coordinates for A in this picture is (0.4, 0.98) and B is (1.4, 0.72), and the ellipse's equation is (x^2/2^2)+(y^2/1^2)=1. Thank you!
Best Answer
The arc length from $x=a$ to $x=b$ on the top half of the ellipse is $$ \int_a^b \sqrt{\frac{16 - 3x^2}{16 - 4 x^2}}\; dx $$ The antiderivative is not elementary, but can be expressed in terms of incomplete elliptic integrals: (in Maple's notation)
$$ 2 {\it EllipticE} \left(\frac{b}{2}, \frac{\sqrt{3}}{2}\right) - 2 {\it EllipticE} \left(\frac{a}{2}, \frac{\sqrt{3}}{2}\right)$$
Alternatively, you can use numerical integration.
In the case at hand, with $a = 0.4$ and $b = 1.4$, the result is approximately $1.03990644$.