Joseph O’Rourke

COMPUTATIONAL GEOMETRY COLUMN 15

Recent solutions of three “art gallery” problems are reported. Three related still-open problems are discussed.

ON RECONSTRUCTING POLYHEDRA FROM PARALLEL SLICES

The problem of reconstructing a three-dimensional object from parallel slices has application in computer vision and medicine. Here we explore a specific existence question: given two polygons in parallel planes, is it always possible to find a polyhedron that has those polygons as faces, and whose vertices are precisely the vertices of the two polygons? We answer this question in the negative by providing an example of two polygons that cannot be connected to form a simple polyhedron. One polygon is a triangle, the other a somewhat complicated shape with spiraling pockets.

COMPUTATIONAL GEOMETRY COLUMN 29

Past accomplishments of Computational Geometry are reviewed and future directions adumbrated.

COMPUTATIONAL GEOMETRY COLUMN 16

The papers presented at the Eighth ACM Symposium on Computational Geometry (June 1992) are summarized.

COMPUTATIONAL GEOMETRY COLUMN 27

The following new result is discussed: there is a perfect matching between the edges of any two triangulations of a finite set of points in the plane.

COMPUTATIONAL GEOMETRY COLUMN 20

Open problems from the Ninth ACM Symposium on Computational Geometry (May 1993) are listed tersely.

COMPUTATIONAL GEOMETRY COLUMN 21

Open problems from the Fifth Canadian Conference on Computational Geometry (August 1993) are listed tersely.

COMPUTATIONAL GEOMETRY COLUMN 17

The papers presented at the Fourth Canadian Conference on Computational Geometry (August 1992) are summarized.