If you trace a point on the circle rolling inside another circle with four times its radius you get a star-shaped* curve called the astroid. ✨ [Wiki bit.ly/2U0A0oW] #50FamousCurves
*I know, I know, stars aren't actually spikey (bit.ly/2EkZrw3).
Drawing an astroid with a do-nothing machine. #50FamousCurves
By the way, do-nothing machines can be used to draw ellipses as well.
Fun fact: The envelope of ellipses having the same centre and sum of major and minor axes is an astroid. #50FamousCurves
The area of an astroid is one and a half times the area of its inscribed circle. So the area of the astroid given by x⅔+y⅔=a⅔ is A=(3/2)×(a/2)²π=3πa²/8. #50FamousCurves
Fun fact: You see astroids used every time you hop on/off a bus with doors sliding along perpendicular tracks. This door mechanism saves 62.5% of floor space if compared to the usual hinged door. #50FamousCurves
Fun fact: The traces of 4×4 unitary matrices with determinant 1 form an astroid in the complex plane.
For more details, see blogs.ams.org/visualinsight/….
Small correction: the traces form a closed region that has an astroid for boundary.
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