Is there anyone who never did this with graph paper in school? Number the squares from, say, 1 to 10 along the x- and y- axes. Connect 1 and 9, 2 and 8, and so on, always keeping the sum of the intercepts equal to some constant, in this case 10. You get a graceful curve.
Actually, the curve is a parabola, and that can be easily seen if you go into negative numbers, connecting 11 and -1 and so on, and extending the lines. The equation of the parabola is x1/2 + y1/2 = 1. In standard form it becomes the less elegant x2 + y2 -2xy -2x -2y +1 = 0. The xy term is the giveaway that the axis is tilted, and the x and y terms show that the origin is not at the vertex.
Sum of Intercepts (pixels):
Interval between lines (Pixels):
Created 29 November 2010, Last Update 11 February 2012
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