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Can we define a (non-trivial) rigid pyramidoid in 3 space where every projection of a set of four edges $ (OA,OB,OC,OD)$ meeting at a vertex $O$ have the same constant Cross-Ratio by arbitrary rotations? Vertices $(A,B,C,D)$ and base sides $ (AB,BC,CD,DA)$ are not in a plane.

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Let the edges be the lines $OA$, $OB$, $OC$, $OD$. We may assume that $A$, $B$, $C$, $D$ are in the plane we project onto. If $O'$ is the projection of $O$ on their plane, then the cross-ratio of the projections is $O'A$, $O'B$, $O'C$, $O'D$.

So in plane our problem is the following: are there four points A, B, C, D in the projective plane such that for all points O' in the plane the cross-ratio of lines (O'A, O'B, O'C, O'D) is constant?

The answer is no. If we consider any points $A$, $B$, $C$, $D$, the cross-ratio $(O'A, O'B, O'C, O'D)$ equals to the cross-ratio of the intersections of these lines with any other line, for example $AB$. But considering any two points $\overline{C}$ and $\overline{D}$ in $AB$, if we choose $O':=C\overline{C}\cap D\overline{D}$, $(O'A, O'B, O'C, O'D)=(AB\overline{C}\overline{D})$. Since $\overline{C}$ and $\overline{D}$ are arbitrary points on $AB$, this cross-ratio can take any values, thus it cannot be constant.

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    Length of any edge like $OA$ can be changed making $A,B,C,D$ coplanar, right? – 2017-01-20
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    The cross-ratio depends only on the lines $(OA, OB, OC, OD)$, not the exact points; thus we can choose the points $(A, B, C, D)$ on these lines such that they are coplanar. – 2017-01-20