An apparatus and method for rendering a smooth circular point using polygons on a computer graphics display. The circular point is approximated by a polygon and the selection of the polygon depends on the diameter of the point. The polygon is enclosed in a bounding area that is divided into a plurality of tiles. The tiles are classified into different sections depending on their location relative to the center of the point. The tiles in each section are processed according to an algorithm to generate a mask that is used to render the circular point.