Some interrupted projections were designed using very specific or handcrafted principles; their detailed descriptions can be quite elaborate, making those projections generally difficult to be faithfully reproduced on a computer.
After using Hammer and August projections for designing maps emphasizing nearly-continuous seas, Atelsthan Spilhaus proposed more complicated approaches, like a modified (not equal-area) Hammer map interrupted in three lobes, or interrupting the map at shorelines instead of, as usual, graticule lines.
A very different design was the “Equidistant” published by Kent Halstead in 1953, featuring many asymmetric lobes. Unlike most projections, it is built on a specific graticule.
Edges of every graticule “cell” — i.e. 10°-wide quadrilaterals, which degenerate to triangles surrounding the poles — are mapped to straight segments of true length and constant scale. Starting from each pole, the cells are separately and sequentially projected (possibly in successive rings), with just enough shearing necessary to fit along cells previously laid. Shearing is minimized along certain privileged meridians (e.g., the Northern 100°W and 60°E meridians are mostly continuous and cross parallels at nearly right angles), but inevitably angles become more and more oblique farther from them. The juxtaposition is arbitrarily interrupted whenever accumulated shearing is excessive, all the while avoiding cuts in continents. After the graticule is laid out, every cell's interior may be projected by linear interpolation between opposite edges. Therefore, this projection is equidistant along all meridians and parallels, which are both broken at most intersections, but neither conformal nor equal-area.
The graticule spacing can of course be reduced, yielding a smoother, more curvaceous grid, with a correspondingly more complicated description. Even at 10° steps — and 648 cells — the result is a pleasant and balanced map.
The Composite World projection is a very different design proposed by Halstead (2008, personal communication). It is interrupted, but avoids the broad range of lobe orientations common in star projections, thus retaining some of the North-South familiarity of conventional maps. The result is both visually and mathematically more elegant than his Equidistant projection.
The Composite World projection is based on partial oblique aspects of Lambert's equal-area azimuthal projection loosely juxtaposed privileging continental areas: lobe boundaries avoid as much as possible splitting shorelines, while the intermediary ocean regions are arbitrarily stretched to keep joints smooth, visibly departing from ideal areal equivalence.