## “Convexity” of sets of lines.(English)Zbl 0918.68107

Summary: J. E. Goodman and R. Pollack [Mathematika 42, No. 2, 305-328 (1995; Zbl 0835.52002)] have introduced a concept of convexity for sets of translates of linear subspaces in $$n$$-dimensional space; in particular, their definition applies to sets of lines in the plane. Recently, A. Rosenfeld [Pattern Recognition Letters 16, No. 5, 549-556 (1995; Zbl 0837.68124)] proposed two ways of definining geometric properties (such as convexity) for sets of lines – one in terms of properties of the corresponding sets of points in Hough space, and the other in terms of point/line incidence. This paper compares the three definitions, as well as a fourth definition which is a weakened version of Rosenfeld’s incidence definition. We show that the Hough definition is incomparable with the other three, and that the incidence definitions are incomparable with each other but imply the Goodman-Pollack definition; these last results also hold in 3-space.

### MSC:

 68T10 Pattern recognition, speech recognition 68U10 Computing methodologies for image processing

### Keywords:

convexity; sets of lines; Hough space

### Citations:

Zbl 0835.52002; Zbl 0837.68124
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