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 , namely
(1)
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illustrated above. This curve encloses area
(2)
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and has area moment of inertia tensor
(3)
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The second definition of the rectellipse was given, though not explicitly named, by Fernandez Guasti (1992). This curve has quartic Cartesian equation
(4)
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with squareness parameter , where corresponds to an ellipse with semiaxes and and to a rectangle the side lengths and . This curve is actually semialgebraic, as it must be restricted to and to exclude other branches. This rectellipse encloses area
(5)
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where is an elliptic integral of the second kind, which can be verified reduces to for and for .