Area of an Elliptical Segment
Calculate the chord, the length of the elliptical arc, and the area of the segment of the arc enclosed by two angles of an ellipse.
| Area of the segment S | |
| Chord length c | |
| Length of the elliptical arc L |
| \(\normalsize Elliptical\ Arch\\ (1)\ area:\\ \hspace{20px} S=F(\theta_1)-F(\theta_0)-{\large\frac{r_0 r_1}{2}} \sin(\theta_1-\theta_0)\\ \hspace{20px} F(\theta)= {\large\frac{ab}{2}}\left[\theta- \tan^{\small-1}\left({\large\frac{(b-a) \sin 2\theta}{b+a+(b-a) \cos 2\theta}}\right)\right]\\ \hspace{20px} r(\theta)^2={\large\frac{a^2b^2}{b^2 \cos^2\theta+a^2 \sin^2\theta}}\\ (2)\ elliptical\ arch :\\ \hspace{20px} L=aE({\large\frac{x(\theta_0)}{a}},k)-aE({\large\frac{x(\theta_1)}{a}},k)\\ \hspace{20px} x(\theta)=r(\theta) \cos \theta,\ k=\sqrt{1-({\large\frac{b}{a}})^2}, \hspace{20px} a\ge b,\hspace{10px}\frac{\pi}{2}\ge \theta\ge 0\\ \hspace{20px} E(x,k):\ 2nd\ incomplete\ elliptic\ integral\\ (3)\ elliptical\ cord :\\ \hspace{20px} c=\sqrt{r(\theta_0)^2+r(\theta_1)^2-2r(\theta_0)r(\theta_1) \cos(\theta_1-\theta_0)}\\ \) |
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