On the Number of Tangents to Four Triangles in Three-Dimensional Space

• Version to appear in Discrete and Computational Geometry.
• Version CCCG'04.
• Maple code available on Frank Sottile's web page
  

Abstract: We establish upper and lower bounds on the number of connected components of lines tangent to four triangles in R³. We show that four triangles in R³ may admit at least 88 tangent lines, and at most 216 isolated tangent lines, or an infinity (this may happen if the lines supporting the sides of the triangles are not in general position). In the latter case, the tangent lines may form up to 216 connected components, at most 54 of which can be infinite. The bounds are likely to be too large, but we can strengthen them with additional hypotheses: for instance, if no four lines supporting each an edge of a different triangle cannot lie on a common ruled quadric, then the number of tangents is always finite and at most 162; if the four triangles are disjoint, then this number is at most 210; and if both conditions are true, then the number of tangents is at most 156 (the lower bound 88 still applies).

Related publications

• On the number of lines tangent to four polyhedra, with Olivier Devillers, Vida Dujmovic, Hazel Everett, Marc Glisse, Xavier Goaoc, Sylvain Lazard, Hyong-Suk Na, and Sue Whitesides. CCCG'02.
• Transversals to line segments in R³, with Hazel Everett, Sylvain Lazard, Frank Sottile and Sue Whitesides. CCCG'03.
• On the number of lines tangent to arbitrary polytopes in R³, with Olivier Devillers, Vida Dujmovic, Hazel Everett, Marc Glisse, Xavier Goaoc, Sylvain Lazard, Hyong-Suk Na, and Sue Whitesides. SoCG'04.