The "lozenge" shape dome is a geodesic dome, split along a vertical plane, and the two halves separated. The gap between the two quarter-spheres is filled with a cylindrical array of triangular panels.
The "gap" panels are, generally, self-similar, making that part of the dome a bit easier to build.
The end dome-sections are designed as is any geodesic dome. You either must be capable of calculating the geometry, or borrow them from an existing source (another model, for example.)
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I, myself, have been working on a greenhouse design, that employs the lozenge-shape concept.
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You point-out an interest in the lozenge shape, but refer to an "elliptical" dome. Which shape are you interested in?
-Taff