I have created an ellipse of constant thickness. What are the shapes for the inner and outer boundaries? I know they are not also ellipses. I was only able to draw the shape in geometry software, and do not know the formulas. Thanks.

ellipse of constant thickness

  • $\begingroup$ Are you looking for a formula (would parametric do?) or a name? Essentially you extend point $K$ by $\pm r$ perpendicularly to the tangent at $K$ $\endgroup$ – Henry Jul 19 '18 at 22:32
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    $\begingroup$ If this previous question similar? The accepted answer describes the constructed curve as a parallel curve. $\endgroup$ – Weather Vane Jul 19 '18 at 22:48
  • $\begingroup$ I would like a formula or formulas, yes. I will be working in POV-Ray, so I'm not sure yet whether a parametric function would be better than an implicit function. $\endgroup$ – posfan12 Jul 19 '18 at 22:50
  • $\begingroup$ @Weather Vane - Yes that does look like the same question. $\endgroup$ – posfan12 Jul 19 '18 at 23:00
  • $\begingroup$ How you define the width - along the vertical or through the origin? $\endgroup$ – Moti Jul 19 '18 at 23:38

The Wikipedia article on Parallel curve has some formulas. It also writes

the offsets of an ellipse or of a hyperbola are not rational


an ellipse has as (two-sided) offsets an algebraic curve of degree 8.

So this is going to be a pretty ugly beast of a curve, algebraically speaking, but you can probably use the parametric equation from Wikipedia for some application. As far as names go, “parallel curve to an ellipse” is likely to be as easy as it gets.


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