Search Loci: Convergence:
[T]he source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special case.
I Want to Be a Mathematician, Washington: MAA Spectrum, 1985.
An Investigation of Historical Geometric Constructions
Construction of the Quadratrix II
Hide circle A and point X. Now we want to place a point on arc BXD (point E) and construct AE. Lastly, construct the intersection of AE and B'C', and name the point of intersection F. Trace point F by selecting it and choosing Trace Intersection under the Display menu (see Figure 4).
Figure 4: Continuing the Construction of the Quadratrix: Tracing Intersection Point F
The quadratrix is formed by point F as point C' moves along CD and point E moves along arc BED. Since arc BED is longer than CD, we must calibrate the animation speed so that point C' takes exactly the same amount of time to traverse CD as point E takes to traverse arc BED. To do this calculate the ratio of the length arc BED to the length of CD. Use this ratio as the speed of the animation of point E and enter a speed of 1 for point C' by first selecting points E and C'; under the Edit menu select Action Buttons and then Animation (see Figure 5).
Figure 5: Animating Two Ponts Simultaneously with Geometer's Sketchpad.
Match the properties of each animation to those in figures 6 and 7.
Figure 6: Animation Properties of Point E
Figure 7: Animation Properties of Point C'