Please explain to me the eight way symmetry of a circle.I am learning computer graphics and it is an important concept to draw circles.I have tried searching in google but I could not grasp the concept.
I'm aware of the following symmetries of the circle:
1. Rotation about its center through any angle whatsoever. Doing this just gives your circle back again.
2. Reflection about any diameter of the circle. Again, you'll just get the circle back again.
The phrase "eight way symmetry" is therefore puzzling to me.
So, if you draw a circle by specifying, say, 80 points/pixels as pairs of (x,y) coordinates so that the circle is centred at the origin (0,0), then happily you only have to do any real work to specify the 10 of them that define one eighth of the circle - say, the arc between the x axis and the line y=x. Then for each of these ten you'll get 7 more in corresponding positions on the grid, by trading and swapping. Good illustration here, Computer Graphics : Circle drawing : 5 / 8 : 8-way symmetry algorithm.
If between x-axis and y=x the point is (x,y) then between the lines y=x and the y-axis the coordinate of a point is (y,x).Continuing in the anticlock direction in the next octant the point is (-y,x) and then (-x,y),(-x,-y),(-y,-x),(y,-x),(x,-y).I want to know how these points have been found because in basic coordinate geometry any point in the first quadrant has coordinate (x,y),2nd quad (-x,y),3rd quad (-x,-y) and fourth quad (x,-y).Is there some different rule for octants.I hope my question is clearer now.Please help me.
(x,y) changes into (y,x) by reflection in y=x, so now the x coordinate of the point is y and the y coordinate is x. For example (2,1) is transformed into (1,2) by reflection in the line y=x.