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How to visualize the order-type of a set of points (which triples are clockwise, which counterclockwise, and which collinear)?
In "Minimal Representations of Order Types by Geometric Graphs" (https://arxiv.org/abs/1908.05124), Aichholzer et al consider points without collinear triples, and draw a graph over the points so that as points move the order-type changes only as points cross edges. Their graph's embedding determines the order type and has at most 1/3 as many edges as the complete graph.
arxiv.orgMinimal Representations of Order Types by Geometric GraphsIn order to have a compact visualization of the order type of a given point
set S, we are interested in geometric graphs on S with few edges that
unequivocally display the order type of S. We introduce the concept of exit
edges, which prevent the order type from changing under continuous motion of
vertices. Exit edges have a natural dual characterization, which allows us to
efficiently compute them and to bound their number.