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© Copyright 2001, Jim Loy
The
figure on the left is called an astroid. It is drawn by drawing several
straight line segments, each of equal length, and each with one end point on
the x-axis, and one end point on the y-axis. Such a figure (a curve that is the
outside edge of a family of lines or curves) is called an envelope.
The equation is x^(2/3)+y^(2/3)=a^(2/3), where x^(2/3) means x to the 2/3 power.
The astroid is also a hypocycloid, the path of a point on a small
circle that is rolling along the inside of a larger circle. Here we trace the
path of the green point on the smaller rolling circle. The radius (and
circumference) of the smaller circle is 1/4 the radius (and circumference) of
the larger circle.
The astroid is also related to ellipses in a couple of
ways. Here is a family of ellipses. The diagonal line is of fixed length, with
the two endpoints restrained to the axes. Various points on this line trace out
various ellipses. Again this is an envelope which contains the family of
ellipses. See Ellipse #9 and
Spirographs of Period Four (which require
Java)
In string art, we often see the figure on the right. It too is an
envelope, as is any curve drawn with straight lines. It is drawn with straight
line segments with equally spaced end points on the x and y axes. Some books on
string art have said that this is made up of four quarter circles, but it turns
out that it is made up of four parabolas.
Speaking of string art, I once designed
this string art picture of the Southern Cross (Crux), the southernmost
constellation, found on the flags of Australia, New Zealand (without the
dimmest star), and Papua New Guinea. I saw the Southern Cross when I was in
Vietnam. String art of other constellations should look very nice, too.