Angles in 3 Dimensions

Question

Consider the following points;

A(0,-4,4)

B(0,4,4)

C(0,-4,0)

D(0,4,0)

E(x,y,0)

If all of these are connected, it leaves a rectangular-based pyramid with a variable vertex (E). Is there any way to compare the angle at this vertex as x and y or are angles limited to a single plane involved in the shape? TO put this into a bit more context, the closer E is to the origin, logically, this 'angle' or measurement would be larger than if E was farther away.

Best Regards,

Yazan

Answer 1

Hint:

In the point $E$ are concurrent four sides of the pyramid $\overline{EA}$, $\overline{EB}$, $\overline{EC}$, $\overline{ED}$.

If you want find the angles between any two of them you have to find the vectors parallel to the sides, e.g. $$ \vec {EA}=(-x,-4-y,4)^T \qquad \vec {EB}=(-x,4-y,4)^T $$ than the angle $\theta$ between them is given can be found by means of the dot product:

$$ \theta=\arccos\left(\frac{\vec{EA}\cdot \vec{EB}}{|\vec {EA}||\vec{EB}|} \right) $$

Answer 2

I think what you mean is a solid angle.

If a spherical patch on a unit sphere has area $A$ then its solid angle is $A$ steradians.

A spherical cap boundary of latitude $\phi$ has area $ 2 \pi R^2 (1-\sin \phi) $ so its subtended solid angle $ 2 \pi (1-\sin \phi)$ steradians. In general also defined as integral curvature $\int KdA$ which occurs as one of the four terms in Gauss-Bonnet theorem.

Recently I asked for $\Omega$ of Circle rim at off-center location.

In other cases it is not well defined, afik. I for one tend to define it (so with caution!) as follows:

Connect every point of a closed non-intersecting 3d curve poly-line in space to E (the view-point) forming an oblique developable "cone" with green p-line base boundary. Call it PL3cone. Let the cone intersect the unit sphere along a black contour PL3 in rough sketch.

The solid angle of PL3 imo is the area enclosed by such projection/ intersection on unit sphere by the PL3cone ( extended if necessary.. i.e., when slant generators are short of unity length).

In your case the PL3 is a quadrangle with a boundary of four great circles whose internal angle sum exceeds $ 2 \pi$. In eccentric configurations the intersections would be small circles.