Astroid as Envelope of Ellipses

Steven Dutch, Natural and Applied Sciences,University of Wisconsin - Green Bay
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The astroid (not "asteroid") is a curve that can be defined several ways. It is the envelope swept out by a line of constant length with ends that slide along the x- and y- axes. It is also the locus of a point on a circle that rolls around the inner circumference of a circle with four times the radius. That is, it's a hypocycloid.

Another interesting definition is that it's the envelope of ellipses centered at the origin whose major and minor axes sum to a constant. That's the form shown here.

The equation of the astroid is x2/3+ y2/3= 1

Number of Ellipses

Note on the Graphics

I know of two main graphics routines for Javascript. Both are kludges. They define tiny <Div> elements to create pixels. That's not a reflection on their authors but solely on the almost criminal negligence of the people who created JavaScript in not incorporating graphics capability. My definition of a serious computer manual is one that devotes Chapter 1 to the usual intro material and Chapter 2 to graphics. Anyway, the two JavaScript graphics modules are jsDraw2D.js by and wz_jsgraphics.js by the late Walter Zorn (and perpetuated on legacy sites). jsDraw2D has some annoying bugs that cause it to drop the ends of line segments, and drop portions of lines and arcs that are horizontal or vertical. wz_jsgraphics has simpler format but the very annoying feature that ellipses are defined not by centers and axes, but by the box that encloses them. Nevertheless after test-driving both I concluded Zorn's is better because it draws complete lines and arcs.

How they come out will depend on your browser. Internet Explorer 8 renders them with ugly clumps of multiple pixels in XP but they come out just fine in Windows 7, Chrome and Firefox.

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Created 29 November 2010, Last Update 05 February 2012

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