New bounds on curve tangencies and orthogonalities

Abstract

We establish new bounds on the number of tangencies and orthogonal intersections determined by an arrangement of curves. First, given a set of $n$ algebraic plane curves, we show that there are $O(n^{3/2})$ points where two or more curves are tangent. In particular, if no three curves are mutually tangent at a common point, then there are $O(n^{3/2})$ curve-curve tangencies. Second, given a family of algebraic plane curves and a set of $n$ curves from this family, we show that either there are $O(n^{3/2})$ points where two or more curves are orthogonal, or the family of curves has certain special properties. We obtain these bounds by transforming the arrangement of plane curves into an arrangement of space curves so that tangency (or orthogonality) of the original plane curves corresponds to intersection of space curves. We then bound the number of intersections of the corresponding space curves. For the case of curve-curve tangency, we use a polynomial method technique that is reminiscent of Guth and Katz's proof of the joints theorem. For the case of orthogonal curve intersections, we employ a bound of Guth and the third author to control the number of two-rich points in space curve arrangements.