Exploiting Symmetry

We could compute the derivative (i.e. slope) at each point and then decide to step in the x or y. But, let's explore a different approach here.


A circle exhibits a great deal of symmetry.
  • Our simple circle-drawing algorithm already exploits 2-way symmetry about the x-axis.
  • A circle has even more symmetry.
  • For example about the y axis.
Lecture 6 Slide 4 6.837 Fall '98



A circle exhibits a great deal of symmetry. We've already exploited this somewhat by plotting two pixels for each function evaluation; one for each possible sign of the square-root function. This symmetry was about the x-axis. The reason that a square-root function brings out this symmetry results from our predilection that the x-axis should be used as an independent variable in function evaluations while the y-axis is dependent. Thus, since a function can yield only one value for member of the domain, we are forced to make a choice between positive and negative square-roots. The net result is that our simple circle-drawing algorithm exploits 2-way symmetry about the x-axis.

Obviously, a circle has a great deal more symmetry. Just as every point above an x-axis drawn through a circle's center has a symmetric point an equal distance from, but on the other side of the x-axis, each point also has a symmetric point on the opposite side of a y-axis drawn through the circle's center.