Calculating a vantage point from which one shape looks like another This is my first math stackexchange question, so if it is a duplicate, I will remove it. My question is as follows:
Say I have a point at the origin, in 3D space. Now from this origin point, I have 3 rays going out, where the 3 are not coplanar. I would like to find a plane where these 3 rays form a specific nondegenerate triangle. Anyone know how I would go about this? If it is not possible for any triangle, what about a right or equilateral triangle, or a square?
 A: Let us somewhat invert the issue by working on a triangle installed in the $xy$ plane and finding the position of the summit $M(x,y,z)$ from which the rays are "emitted".
Without loss of generality, one can assume that the triangle has coordinates:
$$A(-a,0,0),\ \ B(a,0,0), \ \ C(b,c,0)$$
Let $R_1, \ R_2, \ R_3$ be the three rays with resp. cosine of angles :
$$c_{1,2}=\cos(R_1,R_2), \ \ c_{2,3}=\cos(R_2,R_3), \ \ c_{3,1}=\cos(R_3,R_1)\tag{1}$$
Let us consider the particular case, where, say,  it is rays $(D_1,D_2)$ that "see" side $AB$. Let us equate the two ways to express the following dot product
$$\vec{MA}.\vec{MB}=c_{1,2} \|\vec{MA}\| \|\vec{MB}\|$$
or more exactly its square :
$$((-a-x)(a-x)+y^2+z^2)^2=c_{1,2}^2 ((-a-x)^2+y^2+z^2)((a-x)^2+y^2+z^2)\tag{2}$$
With words: quartic surface with equation (2), known under the name "spindle torus" (more or less apple-shaped) is therefore the set of points in $\mathbb{R^3}$ from which one "sees" line segment $AB$ under a given angle.
What we have done for this pair of rays $(R_1,R_2)$ and this side $AB$ can be done of course in the same way for the two other pairs of rays and sides.
If these 3 spindle torii intersect in a common point (a single point if we impose $z>0$ ; otherwise symmetry will give another point), this point is summit $M$, hence a solution exist for the given issue.
Such a solution exists if the angles mentionned in (1) are all less than a certain threshold (is it $\pi/2$ or less ? the question remains open).
Here is a graphical representation :

Fig. 1: A Matlab representation of the intersection of two spindle torii associated with sides $AB$ and $BC$. These surfaces look like half-apples with the little seeds' region around line segments $AB$ or $AC$ which are excluded areas. The third spindle torus hasn't been represented because it would have obscured the interpretation.
