Circloid

Copyright 2005 by Paul Niquette, all rights reserved.

he figure on the right is a circle with radius a.   Its equation is x2 + y2 = a2

Point P is located along the circumference at some angle
Φ measured counterclockwise from the x-axis.  Point Q is located exactly π radians (180 degrees) beyond P.

By inspection, we ascertain that PQ = 2a and PQ is the longest line segment that will fit within the figure.  The angle Φ can be assigned any value from 0 to 2π radians (360 degrees) and PQ will always fit within the figure.
 

Can you create a figure having a different shape such that the longest line segment that will fit within the figure can be rotated all the way around while always fitting exactly within the figure?

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