$2D$ shapes/rectangles viewed in $3D$ space

Question

I am interested in finding if there is some trigonometric technique to verify if the shape we see of a $2D$ object lying on a plane (for eg. viewing a page placed on a table from some angle above and left/right or away from it), is actually a rectangle?

Answer 1

Let's choose the origin the position of the observer, the equation of the plane is $z=c$, where $c$ is a constant. We denote the position of the corners by the angles as seen from the origin, $\theta_i,\phi_i$. That is, we use polar coordinates. The $x,y,z$ coordinates of a point can be written in polar coordinates as $$x_i=r_i\sin\theta_i\cos\phi_i\\y_i=r_i\sin\theta_i\sin\phi_i\\z_i=r_i\cos\theta_i$$ From the equation of the plane $z_i=c$ and $\theta_i$, we can get $r_i$. Plug in into the first two equations to get $x_i, y_i$. We can ignore $z_i$, since all the points are in the same plane.

Now you reduce the problem to finding out if a quadrilateral is a rectangle. Check the following $$x_1-x_2=x_4-x_3\\y_1-y_2=y_4-y_3$$ This will tell you that the sides are parallel and equal. Also check that the angle between adjacent sides is $90^\circ$ by verifying that $$(x_1-x_2)(x_1-x_4)+(y_1-y_2)(y_1-y_4)=0$$ That's the scalar product.

Answer 2

Yes, it is the projection of the plane on another imagined plane normal to the paper-eye radial rays.

If the angle between two planes is near zero, the projection is faithful. Else projection gets narrow and distorted. If distance is too much compared to 2D dimensions we have reduction by perspective vision.These are dealt with in Projective geometry.