Velocity of object moving on inclined orbit

An object is moving on circular orbit with constant speed v. Orbit has inclination α relatively to plane A. Viewer is on the A plane in the centre of circle (well, it's projection to A).

Will the projection of object to A (moving on ellipse) have the same angular velocity as the object itself?

I'm trying to track objects on circular orbits. When orbit has zero inclination, tracking is simple: angular speed = 360°/Period. But I get errors with objects on inclined orbits (I see them only on some points like π/2 and such).

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Imagine the case when $\alpha=\frac{\pi}{2}$, then the motion would be up-down sinusoidal motion on $\mathbf{A}$, but the angle would be constant (apart from flipping from $0$ to $\pi$ every half-orbit). – Alyosha Jan 27 '13 at 21:15

1 Answer

If $\alpha=0$, the motion of the particle can is

$$x=\cos(t), y=\sin(t)$$ Imagine $\alpha$ rotates the circle clockwise in the $y-z$, so that the $x$-axis is the 'pivot'. $$x=\cos(t), y=\sin(t)\sin(\alpha), z=\sin(t)\cos(\alpha)$$

The angle on plane $\mathbf{A}$ is $\arctan(\frac{y}{x})$, which is $\arctan(\tan(t)sin(\alpha))$.

$$\frac{d}{dt}\arctan(\tan(t)\sin(\alpha))=\frac{1}{(\tan(t)\sin(\alpha))^2+1}\frac{\sin(\alpha)}{\cos^2(t)}=\frac{\sin(\alpha)}{\sin(\alpha)^2 \sin^2(t)+\cos^2(t)}$$ Which obviously depends on $\alpha$.

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