Multidimensional Lines II: Proximity and Applications

In [Inselberg and Dimsdale, SIAM J Appl. Math., 54 (1994), pp. 559–577] it was shown that in coordinates any line in $R^N $ can be represented by $N - 1$planar points indexed by a pair of distinct integers from $\{ 0,1,2, \cdots ,N \}$. Based on this representation, proximity properties and construction algorithms are obtained. They include: (i) an algorithm constructing the intersection between two lines and displaying the common point when it exists; and (ii) construction and display of the points where the minimum $L_1 $ distance between two lines occurs. From this representation, tight estimates for the analogous results with the Euclidean $( {L_2 } )$ are provided.To study line proximity, line neighborhoods for a topology are proposed. Results indicate that there are ambiguities in line detection for orthogonal coordinates that are eliminated in parallel coordinates. An application to air traffic control for an information display and an algorithm for collision avoidance are illustrated.