Let's consider a regular polygon with unit edge length resting with one
face on a line. Let's rotate the polygon about a vertex on the line and
trace the paths the polygon's other vertices follow. and repeat this
process to infinity in both directions.
In the case of a triangle, one base vertex is the pivot while the other
traces out an ascending arc of radius 1 and an angle of 120-degrees as
it becomes the apex and the apex traces out a descending arc that is a
shifted mirror image. Repeted, a vertex traces out a series of identical
double arcs that resemble the outer boundary of a pair of overlapping
circles of equal raidius whose circumfrences pass through each other's
center, and the three vertices together trace out an overlapping pattern
of such.
For a square, each step of this process generatesa unit semicircle(a
quarter circle each from the vertex going from base to top and the
vertext going from top to base) and a quarter circle of radius square
root of 2 above it traced by the vertex that stays in the top of
thesquare. Repeated, a lower row of overlapping unit semicircles and an
upper row of overlapping root 2 quarter circles forms, the troughs of
the upper row touching the crests of the lower row.
A pentagon forms unit arcs of 72-degrees and golden ratio arcs of
108-degrees, a hexagon unit arcs of 60-degrees, root 3 arcs of
120-degrees, and two unit arcs of 60-degrees... and it continues in a
similar manner, with a row of arcs for each distinct length of diagonal.
Though, as n goes to infinity, the polygon approaches a circle, and the
curve traced by a single vertex approaches a cycloid, and the tracing of
all vertices approaches an infinite family of overlapping cycloids.
Of course, one could replace the polygon with a prism and trace the path
of the edges to form ruled surfaces with similar profiles... though I
wonder what the tracing of edges or vertices of a rolling anti-prism
would look like.
One might also consider replacing the line with a fixed polygon the
moving polygon rolls around, or if the roller is smaller, rolls inside,
or replacing the rolling polygon with an irregular or reuleaux form.
One might also consider tracing the paths of vertices or edges of a
deltahedron rolling on a triangular grid, a cube rolling on a square
grid, or a truncated deltahedron rolling on a hexagonal grid, or of any
polyhedron rolling around a copy of itself or one isogonal polyhedra
rolling around another isogonal polyhedron with the same face shape, or
of certain Archimedean solids rolling around certain uniform
tilings(such as a rhombicuboctahedron or rhombicosidodecahedron on on a
rhombitrihexagonal tiling or a snub cube or snub dodecahedron on a snub
hexagonal tiling).
In all cases, I'm pretty sure the vertices of the moving shape will
trace circular arcs and that edges will trace cylindrical or conical
surfaces, but the radii and pattern of intersection quickly become
difficult to imagine, though I suspect all examples involving polygons
in the plane are constructible if the polygons involved are
constructible.