Conic Constructions

Compass and Straightedge Constructions

In the classical geometric constructions, a compass and an unmarked straightedge are the only allowable tools. Any given object might be assumed though, including conic sections. In Conics Apollonius gave solutions to the tangency problems below, using methods very different from my own.

Even with a sharp eye and a steady hand, many of the constructions below are not likely to yield good results on paper. Dynamic geometry software is advisable. The Geometer's Sketchpad is my own weapon of choice. It was used for all of the illustrations in the conics article, including the Java applets. Sketchpad also has several other basic tools, such as the parallel line tool, which can be duplicated with compass and straightedge. In short, if you are not entering any measurements or calculations, you may use these tools with a clear conscience and still call it a compass and straightedge construction. This way the constructions can be executed quickly and with much greater precision than hand work. When version 5 of Sketchpad was released, it debuted a welcome new feature. It is now possible to construct intersection points on loci.

Given an ellipse only, construct its major and minor axes, center, vertices, foci, and directrices.

Given a hyperbola only, construct its transverse and conjugate axes, center, vertices, foci, directrices, and asymptotes.

Given a parabola only, construct its axis, vertex, focus, and directrix.

Construct a line of tangency through a given point on a conic section.

Given a conic section and an arbitrary point on its exterior, construct the lines through the point and tangent to the given curve.

Construct a line tangent to a conic and parallel to a given line.

Locus Constructions

Excepting circles and degenerate cases, it is not possible to construct conics with a compass and straightedge. However, computer technology has opened up a new class of construction. To construct a conic curve, a geometric relationship is constructed. As one point follows a defined circular or linear path, a dependent point can be forced to follow a conic curve. When I first began playing (yes, it is play) with this feature, I soon discovered that it actually is somewhat difficult not to construct a conic.

Given five points, construct the conic section that includes all of the given points.

Given one focus of a conic section and three lines tangent to the conic, construct the conic section.

Given the focus of a parabola and two lines tangent to the curve, construct the parabola.

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