{"id":427,"date":"2013-11-15T01:00:35","date_gmt":"2013-11-15T01:00:35","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=427"},"modified":"2015-07-29T00:54:00","modified_gmt":"2015-07-29T00:54:00","slug":"astroid-as-catacaustic-of-deltoid","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2013\/11\/15\/astroid-as-catacaustic-of-deltoid\/","title":{"rendered":"Astroid as Catacaustic of Deltoid"},"content":{"rendered":"
\n
\"Astroid<\/a>

Astroid as Catacaustic of Deltoid – Xah Lee<\/p><\/div>\n<\/div>\n

This image, drawn by Xah Lee<\/a>, shows a deltoid and its catacaustic.<\/p>\n

The deltoid<\/b><\/a> is the curve traced by a point on the perimeter of a circle that is rolling inside a fixed circle whose radius is three times as big. It’s called a deltoid because it looks a bit like the Greek letter delta: $\\Delta$.<\/p>\n

The catacaustic<\/b><\/a> of a curve in the plane is the envelope of rays emitted from some source and reflected off that curve.<\/p>\n

If we shine parallel rays at one corner of the deltoid, the resulting catacaustic is called the astroid<\/b><\/a>. This is the curve traced by a point on the perimeter of a circle that is rolling inside a fixed circle whose radius is four times as big!<\/i> It’s called an astroid because it looks like a star.<\/p>\n

Does this fact generalize? If we take a circle and roll it inside a circle whose radius is $n$ times as big, and trace out the motion of one point, we get a curve called the hypocycloid with $n$ cusps<\/b><\/a>. Is the catacaustic of a hypocycloid with $n$ cusps a hypocycloid with $n+1$ cusps? <\/p>\n

No: according to Egan the catacaustic of an astroid is not a hypocycloid. For an astroid with cusps on the coordinate axes and a light source at $(-\\infty,0)$, the catacaustic has 6 cusps, and it looks like this:<\/p>\n

\n
\"Catacaustic<\/a>

Catacaustic of an Astroid – Greg Egan<\/p><\/div>\n<\/div>\n

It has this parametric form, up to scale:<\/p>\n

$$ (\\cos(3t) + 3 (4 \\cos(t)+ \\cos(5t)), 2 \\sin(2t) (\\cos(t)-9 \\cos(3t)) – 4 \\cos(2t) (\\sin(t)-3 \\sin(3t)))) $$<\/p>\n

Xah Lee has a great website devoted to curves:<\/p>\n

\u2022 Xah Lee, Visual dictionary of special plane curves<\/a>.<\/p>\n

Also see Egan’s page on catacaustics:<\/p>\n

\u2022 Greg Egan, Catacaustics, resultants and kissing conics<\/a>.<\/p>\n

For more on deltoids, astroids and other hypocycloids, see these:<\/p>\n

\u2022 John Baez, Rolling circles and balls<\/a>.<\/p>\n

\u2022 Deltoid rolling inside astroid<\/a>, Visual Insight<\/i>.<\/p>\n

\n

Visual Insight<\/i> is a place to share striking images that help explain advanced topics in mathematics. I\u2019m always looking for truly beautiful images, so if you know about one, please drop a comment here<\/a> and let me know!<\/p>\n

<\/div>","protected":false},"excerpt":{"rendered":"

This image, drawn by Xah Lee<\/a>, shows a deltoid and its catacaustic. The deltoid<\/a> is the curve traced by a point on the perimeter of a circle that is rolling inside a fixed circle whose radius is three times as big. It’s called a deltoid because it looks a bit like the Greek letter delta: $\\Delta$. The catacaustic<\/a> of a curve in the plane is the envelope of rays emitted from some source and reflected off that curve.<\/p>\n

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