Help needed to find out an equation.

Question

I observed that if we see a circle in tilted view it looks more of an ellipse. I am intersted in finding out the equation of this ellipse given that the equation of the circle is $x^2+y^2=a^2$ and angle of axis(inclination) with the horizontal is $\theta $. Hope this isnt very much complicated as it involves $3$D geometry.

Answer 1

Let's suppose that the circle is having the $x$ axis fixed, and angle between the circle plane and $z$ axis is $\theta$. Then you have $$x^2+y^2+z^2=a^2\\\tan\theta=\frac{y}{z}$$ Using $z=y\cot\theta$ you get$$x^2+y^2(1+\cot^2\theta)=a^2$$ or if you want$$\frac{x^2}{a^2}+\frac{y^2}{a^2\sin^2\theta}=1$$ I've used $$\frac{1}{1+\cot^2\theta}=\frac{1}{1+\frac{\cos^2\theta}{\sin^2\theta}}=\frac{\sin^2\theta}{\sin^2\theta+\cos^2\theta}=\sin^2\theta$$

Obviously, I assumed that I am looking along $z$ axis. If the angle $\theta$ is $90^\circ$ you get the equation of the circle back