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I want to illustrate how high dimensional objects may have misleading projections. Examples are for instance given with HiSee software, with nD bouquets of circles.
Are there non-trivial (not a 3D segment) 3D curves in $(x,y,z)$, possibly branched, such that all of their projections onto planes $(x,y)$, $(y,z)$ and $(z,x)$ are lines or line segments?
Other curves simple projections (hyperbolae, parabolae) are interesting as well.
There are no such curves: A curve (point set) $C$ that maps to a line segment $L$ under an orthogonal projection to a plane $P$ lies in the plane orthogonal to $P$ and containing $L$. If this is true for just two coordinate planes in $\mathbf{R}^{3}$, then $C$ lies in the intersection of two transverse planes, and consequently is a portion of a line.
