A new family of clipping algorithms is described. These algorithms are able to clip polygons against irregular convex plane-faced volumes in three dimensions, removing the parts of the polygon which lie outside the volume. In two dimensions the algorithms permit clipping against irregular convex windows. Polygons to be clipped are represented as an ordered sequence of vertices without repetition of first and last, in marked contrast to representation as a collection of edges as was heretofore the common procedure. Output polygons have an identical format, with new vertices introduced in sequence to describe any newly-cut edge or edges. The algorithms easily handle the particularly difficult problem of detecting that a new vertex may be required at a corner of the clipping window. The algorithms described achieve considerable simplicity by clipping separately against each clipping plane or window boundary. Code capable of clipping the polygon against a single boundary is reentered to clip against subsequent boundaries. Each such reentrant stage of clipping need store only two vertex values and may begin its processing as soon as the first output vertex from the preceeding stage is ready. Because the same code is reentered for clipping against subsequent boundaries, clipping against very complex window shapes is practical. For perspective applications in three dimensions, a six-plane truncated pyramid is chosen as the clipping volume. The two additional planes parallel to the projection screen serve to limit the range of depth preserved through the projection. A perspective projection method which provides for arbitrary view angles and depth of field in spite of simple fixed clipping planes is described. This method is ideal for subsequent hidden-surface computations.