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This is a Three Star Challenge so you won't be expecting a quick fix.
But as a hint : split the trace into chunks that mean something.All three traces have this property but the "same when upside-down" quality isn't there for all of them.
What makes that "same when upside-down" work?P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?
We have four rods of equal lengths hinged at their endpoints to form a rhombus ABCD. Keeping AB fixed we allow CD to take all possible positions in the plane. What is the locus (or path) of the point D?
An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?