## On the number of line tangents to four triangles in three-dimensional space

### Sylvain Lazard (LORIA, INRIA)

Résumé:

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, each supporting an edge of a different triangle,
lie on a common ruled quadric (possibly degenerate to a plane), 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).

Joint work with H. Brönnimann, F. Sottile, and O. Devillers.