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The Trisectrix as the Locus of Points of Intersection

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A trisectrix (red) is the locus of the points of intersection (red point ) of a moving horizontal line (blue) and a rotating line (black).

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Initially, the line lies along the line and the line lies along the line . The line rotates clockwise about the origin at 1 radian per time unit, and the line moves toward the axis at a constant rate of units per time unit, so that both lines reach the axis at the same time. The line meets the axis at an angle .

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Contributed by: Roberta Grech (July 2013)
Open content licensed under CC BY-NC-SA

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The trisectrix can be used to trisect an angle. If a vertical line is dropped from to meet the axis at , the point on the line is such that , and the horizontal line meets the trisectrix at , then the angle that makes with the axis is .

Reference

[1] T. Heard, D. Martin, and B. Murphy, A2 Further Pure Mathematics, 3rd ed., London: Hodder Education 2005 p. 202, question no. 2.

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