# Is the set of points of equal distance to the surface of an ellipsoid again an ellipsoid?

Consider the hyperellipsoid $A$ in $\mathbb{R}^d$ given by the semi-major axes $a_1,a_2,\ldots,a_d$. Do points on the surface of the hyperellipsoid $A'$ with semi-major axes $a_1-\varepsilon, a_2-\varepsilon,\ldots,a_d-\varepsilon$ all have distance $\varepsilon$ to the original ellipsoid $A$? (assuming $a_i>\varepsilon$ for $i=1,\ldots,d$)

If not, how good of an approximation is this for $a_i>>\varepsilon$ in relation to $\varepsilon$?

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after computing the distance in the 2d case by just inserting into the formula of an ellipse this is apparently not true. But how could an expression for the error (distance of point on $A'$ to point exactly $\varepsilon$ from A) be derived in the general case ($d$ dimensions)? –  rngantner Nov 19 '12 at 12:16
–  Rahul Nov 19 '12 at 18:21

The answer is no. It's not even true for ellipses in 2D. In 2D, the derived shape is called an "offset curve", or, in older texts, a "parallel curve". In 3D, the term "offset surface" is used. You can find a good introduction to the topic in two papers by Farouki and Neff: Analytic properties of plane offset curves & Algebraic properties of plane offset curves, Computer Aided Geometric Design 7 (1990), 83-99 & 101-127. This sort of offsetting is important in CAD/CAM, so it appears frequently in the literature.

I'm not aware of any specific known results about the quality of the approximation. Clearly it's pretty good when the $\epsilon$ values are small and very bad if any $\epsilon$ value is larger than the minimum radius of curvature of the original ellipse. In 2D, we have two curves with known (and fairly simple) equations, so figuring out the error distance between them shouldn't be too hard. I expect a few minutes with Mathematica would yield some insight.

Edit: I found a somewhat related paper, again by Rida Farouki. Conic Approximation of Conic Offsets, J. Symbolic Computation (1997) 23, 301-313. Not exactly what you were asking about, since it deals with general conics rather than ellipses, specifically, but may be helpful.

I don't know anything about $d$ dimensions (unless $d$ happens to be 2 or 3). Offsetting problems are hard enough even in the low-dimension cases, so I've never gone beyond.

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I said "figuring out the error distance between them shouldn't be too hard". I take that back. It's troublesome because you have to define what you mean by the "error" between two curves or surfaces. This is somewhat messy. –  bubba Nov 20 '12 at 7:27
Thanks for the references.. I'll have a look at them, maybe there's something there I can use. –  rngantner Nov 20 '12 at 11:51
I just googled "offset ellipse". I found hundreds of comments from people using CAD systems who are dismayed that offsetting an ellipse doesn't produce another ellipse. So, apparently, many people expect the answer to your question to be "yes". –  bubba Nov 21 '12 at 1:59