Sails use the hollows/curves in the edges to set up tension through the sail, the depth of the curve is a percentage of the edge length (the longer the edge the bigger the curve). As the corners of the sail are tensioned the edge hollows/curves try to straighten, in turn pulling the fabric tight.
The twisted shape in a sail is not just for looks, by adding the high and low points into the sails the shape of the sail also pulls tension through the fabric by the high points trying to pull the fabric up and the low points trying to pull the fabric down. By inducing this twist the set up tension in the sail is also reduced as you are no longer just relying on the edges to tension the sail.
This shape also reduces wind loading on the sails by creating a pitch which has less suction force acting over it, as opposed to a flat sail.
A triangle relies purely on the edge hollows/curves to pull tension through the sail, and thus have higher setup tensions. The other disadvantage with triangles is that by the time the hollows are put into the edges, the amount of sail left is reduced and thus not a great deal of shade is achieved, more often than not triangles are used for visual effect rather than shade.
Triangles by their very shape have narrow corner angles compared to say a rectangle thus further reducing the amount of "usable" shade.
These sails work the same as a hypar above.
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