Starting and ending caps of smooth line segments may be drawn on a computer display without complicated calculations and avoiding the use of inverse square root calculations by drawing the caps using rectangles. The direction of the line segment may be determined, and using the direction certain pixels in a four-pixel grouping may be selectively illuminated. The normalized intensity values of the illuminated pixels may be determined by computing an x-fraction and a y-fraction, representing the distance in the x-direction and y-direction between a sample origin point in a corner of the four-pixel grouping and the closest corner of a region covered by the mathematical origin or endpoint corrected for the thickness of the line segment. It is generally preferable for only two pixels to be illuminated for each cap, the pixels chosen according to a formula ensuring that there are no perception problems when two smooth line segments share a common mathematical origin or endpoint.