Jim King's Conic Macros

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Cabri Conic Macros (for Cabri II) from Jim King

Given 5 points, Cabri II constructs the conic through these points. This is a very general construction that allows one to drag a point and so drag a conic which is an ellipse through being a parabola to being a hyperbola. However, this "interface" to conics is not the one in most textbooks; one can supply alternative and more familiar methods for conic construction by defining macros.

Several macros for constructing conics are supplied in the files that come with Cabri. The files in this folder are of two kinds: some give methods of constructing conics from various givens and others are general construction of conic features given a conic.

One pair of macros uses standard definitions to construct conics. Given the points F1 and F2 which are the foci and a point P on the conic, one macro constructs the ellipse through P with these foci and another constructs the hyperbola with the same foci. (One can use this with Trace to see a family of confocal conics.)

Two other macros construct conics in limiting cases. One constructs a conic when two of the 5 points "coincide"; in other words one is given 4 points A, B, C, D and a line d through D which is to be a tangent line through the conic (so D is a double point). A second macro constructs a conic from 3 points A, B, C and lines b and c through B and C, respectively. These macros use Pascal's theorem about hexagons inscribed in conics.

Finally, there are macros which construct objects from conics, namely

I hope to write in more detail later about how these macros work, but let me make a few comments at this time.

First, I have tried to make the macros as general as I can. For example, if a point P is on the given conic, the polar macro constructs the tangent line through P. This is mathematically correct, but one has to be careful in defining the macro so that it works; it is easy to find in such special cases that two points in the construction may coincide and lead the construction to fail.

In another example, if the conic is a parabola, the macros construct only one axis (the other is the line at infinity) and one focus. This works because in some cases Cabri allows one to use points at infinity (for example, a macro which uses the intersection of two lines may work when the lines are parallel).

The foci are interesting to construct, for mathematically there are four foci, two on each axis line. However, only two at most exist in the (real) plane at one time. The macro constructs 4 points, but in fact only two (or one) appear on the screen for any given conic, but as the conic is dragged, one can see the foci shift from one axis line to the other.

Second, the methods used for the macros are a combination of projective geometry and non-Euclidean geometry (or circle geometry) of the half- plane. This uses the fact that projective transformations of a line are the "same" as the motions of non-Euclidean geometry. A key idea is the idea of projective involution; this is more or less the analog of point or line reflection in projective geometry.

It should be remarked that it is not so hard to construct the axes or foci of a conic if you know that it is, say, an ellipse (and these methods are worth doing in macros, too). But the Euclidean methods (at least those I tried) do not hold up if you allow yourself to drag the ellipse so that it becomes a hyperbola.

Third, some of these macros are rather slow, especially on older Macs. Part of this seems to be inevitable given the complexity of the constructions, but in some cases this may be due to longer and more indirect methods used for constructions so that they would work in tricky special cases.

I do intend to write more about these constructions and related topics, but since this will take some time, I prefer to make these macros available for anyone who finds them useful or entertaining.

A version of these macros was made in spring 1994 using the demo version of Cabri II; these have been redone and hopefully improved. If you try out these macros, I would appreciate your reactions in general and would especially like to hear if you find bugs or cases where they do not work, but where they should work.

James King
Department of Mathematics GN-50
University of Washington
Seattle WA 98195
December, 1994

 

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