Mathematics with Cinderella: Various Curves: Maclaurin's Conic Section Drawing Instrument
This is featured in "Encyclopedia of Curves" (edited by Masami Isoda et al.: Kyoritsu Shuppan, 2009). Unlike other conic section drawing instruments, when using drawing software, you can draw ellipses, parabolas, and hyperbolas in a single diagram simply by changing the positions of the points.
Maclaurin's conic section drawing instrument is constructed as follows. Triangle ASC is given by lines AS, SC, and AC passing through fixed points P, B, and Q on a plane, respectively. Vertices A and C move along lines AT and CT, respectively. At this time, the third point S traces a conic section (the figure shows the case of an ellipse).
This explanation likely requires some supplementation. To that end, let's also look at photos of the drawing instrument available on the web. They should be a bit easier to understand than the image above.
It states that "vertices A and C move along lines AT and CT, respectively," but looking at the drawing instrument, it is not the case that both can move freely. When point A (the left point) moves along the line passing through T, the line passing through B moves, which in turn changes the position of C. If A is the independent point, then C is the dependent point. Also, since A moves along the line passing through T, the expression "A moves along line AT" is inaccurate.
Once you are satisfied with the above, let's construct it in Cinderella.
Although the four points P, Q, B, and T are fixed points, you do not need to set them as fixed points in the inspector for the Cinderella construction. Rather, you cannot draw the three types of conic sections unless you leave them as free points. First, place these four points. The order does not matter.
Assume that the point corresponding to point T is D. Draw lines DE and DF passing through D, and place point G on DE.
Draw line GA and let the intersection be H. Draw lines BG and CH, and find their intersection.
Select the locus tool, choose G as the point to move, and K as the point to trace the locus, and an ellipse will be drawn.
The following figure shows the background set to white, points E and F hidden, and the point names changed as shown in the explanatory diagram.
The following figure is a modification that mimics the drawing instrument. (Including the title screen)
However, if you represent it with rods (line segments) like this, you can only create an ellipse drawing instrument.
The book explains the reason why conic sections are drawn this way using projective geometry. Please refer to the book for details. The following three points are presented as a summary.
The parabola in Figure 2.63 is upside down, but in short, it means that three types of conic sections can be drawn depending on the positional relationship of the four points P, Q, T, and B. Let's keep the construction as is and change the positional relationship of the four points.
Although we drew the conic section as the locus of point S, looking at Figures 2.62–2.64 (reprinted), we can see that it is a quadratic curve passing through five points. These are the five points, adding point R, the intersection of PB and CT, and point U, the intersection of QB and AT.
Since Cinderella's drawing tools include a "Conic through 5 points" tool, let's add points R and U and try drawing with this tool.
When exporting to HTML and displaying with CindyJS, the locus drawn with the locus tool is not displayed, but a curve drawn with the "Conic through 5 points" tool will be displayed. The following page is set up so that it can be operated on the web.
The method of using rods for line segments is explained on the "Conic Section Drawing Instrument Using Crossed Parallelograms" page.
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