Ruled Surfaces
Study's work with dual numbers and line coordinates was noted by Heinrich Guggenheimer in 1963 in his book Differential Geometry (see pages 162–5). He cites and proves the following theorem of Study: The oriented lines in R3 are in one-to-one correspondence with the points of the dual unit sphere in D3. Later he says "A differentiable curve A(u) on the dual unit sphere, depending on a real parameter u, represents a differentiable family of straight lines in R3: a ruled surface. The lines A(u) are the generators or rulings of the surface." Guggenheimer also shows the representation of the Euclidean motions in R3 by orthogonal dual matrices.
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