Rectellipse

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By analogy with the squircle, a term first apparently used by Fernández Guasti et al. (2005), the term "rectellipse" (used here for the first time) is a natural generalization to the case of unequal vertical and horizontal dimensions.

The first definition of the rectellipse is the quartic plane curve which is special case of the superellipse with 👁 r=4
, namely

illustrated above. This curve encloses area

and has area moment of inertia tensor

The second definition of the rectellipse was given, though not explicitly named, by Fernandez Guasti (1992). This curve has quartic Cartesian equation

with squareness parameter 👁 s
, where 👁 s=0
corresponds to an ellipse with semiaxes 👁 a
and 👁 b
and 👁 s=1
to a rectangle the side lengths 👁 a
and 👁 b
. This curve is actually semialgebraic, as it must be restricted to 👁 |x|<=a
and 👁 |y|<=b
to exclude other branches. This rectellipse encloses area

where 👁 E(x,k)
is an elliptic integral of the second kind, which can be verified reduces to 👁 4ab
for 👁 s->1
and 👁 piab
for 👁 s->0
.

See also

Ellipse, Rectangle, Rounded Rectangle, Squircle, Superegg, Superellipse

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References

Fernandez Guasti, M. "Analytic Geometry of Some Rectilinear Figures." Int. J. Educ. Sci. Technol. 23, 895-901, 1992.Fernández Guasti, M.; Meléndez Cobarrubias, A.; Renero Carrillo, F. J.; and Cornejo Rodríguez, A. "LCD Pixel Shape and Far-Field Diffraction Patterns." Optik 116, 265-269, 2005.

Referenced on Wolfram|Alpha

Rectellipse

Cite this as:

Weisstein, Eric W. "Rectellipse." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Rectellipse.html

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