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The Tracing of Conics

Published online by Cambridge University Press:  03 November 2016

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It is hard to understand why the problem of tracing a conic from its equation is usually subordinated to if not identified with the problem, intrinsically more difficult, of calculating axes, or why students are not encouraged to bring their knowledge of geometrical properties of the curve to the drawingboard. The essence of tracing a curve is to render the sketching of it as accurate as may be desired, and I venture to assert that for this purpose the discovery of axes and vertices is not worth a tithe of the labour which it demands. Prom the advice in some of our text-books one would imagine that the ellipse could be drawn accurately from its four vertices alone, the hyperbola from its vertices and its asymptotes, and the parabola from its axis and vertex and one or two other points. The truth is, that in any but expert hands these details are wofully insufficient; what is wanted is a multitude of points on the curve, and it matters little whether or not the vertices are included. The student finds satisfaction in drawing the curve from its equation and observing the symmetry that appears when the axes are inserted subsequently.

Type
Research Article
Copyright
Copyright © Mathematical Association 1921

References

page 201 note * One of the best elementary books known to me contains the naive admission that when the vertices and asymptotes of a hyperbola have been marked, “it is advisable by way of corroboration to find some points on the curve”. It is not surprising that later the authors are content after a page of calculation on a particular parabola to assert that they can draw the curve “fairly accurately”, and it must be confessed that their figure would not substantiate a less modest claim.

page 203 note * Unfortunately the evidence of value in the experiment is always diminished by the students’ possession of books.

page 203 note † The usual nine points and the points H, V, F, O.

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