The visibility complex

We introduce the visibility complex of a collection<inline-equation><f><sc>O</sc></f></inline-equation> of <?Pub Fmt italic>n<?Pub Fmt /italic> pairwisedisjoint convex objects in the plane. This 2-dimensiotnal cell complexmay be considered as a generalization of the tangent visibility graph of<inline-equation><f><sc>O</sc></f></inline-equation>. Its space complexity<?Pub Fmt italic>k<?Pub Fmt /italic> is proportional to the size of thetangent visibility graph. We give an <inline-equation><f>O<fen lp="par">n<rf>log</rf>n+k<rp post="par"></fen></f></inline-equation> algorithm for its construction. Furthermore we showhow the visibility complex can be used to compute the view from a pointor a convex object with respect to <inline-equation><f><sc>O</sc></f></inline-equation> in <inline-equation><f>O<fen lp="par">m<rf>log</rf>n<rp post="par"></fen></f></inline-equation> time, where<?Pub Fmt italic>m<?Pub Caret><?Pub Fmt /italic> is the size of theview. The view from a point is a generalization of the visibilitypolygon of that point with respect to <inline-equation><f><sc>O</sc></f></inline-equation>.