On Lines and Joints

Let L be a set of n lines in Rd , for d 3. A joint of L is a point incident to at least d lines of L, not all in a common hyperplane. Using a very simple algebraic proof technique, we show that the maximum possible number of joints of L is (nd/(d-1) ). For d = 3, this is a considerable simplification of the orignal algebraic proof of Guth and Katz [9], and of the follow-up simpler proof of Elekes et al. [6]. Some extensions, e.g., to the case of joints of algebraic curves, are also presented. Let L be a set of n lines in Rd, for d 3. A joint of L is a point incident to at least d lines of L, not all in a common hyperplane. A simple construction, using the axis-parallel lines in a k

• Type: Journal
• Year: 2010
• Where: DCG
• Authors: Haim Kaplan, Micha Sharir, Eugenii Shustin
• Keywords: DCG 2010, Joints, Maximum Possible Number, Point Incident