The Conchoid of Nicomedes was discovered by a greek mathematician named NIcomedes in 200 B.C.E, and has been used ever since to trisect angles, cube duplication, and square circles. This conchoid is a set of curves the hold two points that are equidistant from the directrix (labeled in sketch), and line up through a point called the pole. These curves are classified as higher curves because they cannot be created by simply using a straightedge and a ruler. The pole in this sketch is represented by the point O and the two points are represented as G and H. Z is the constant midpoint of the points G and H. These two points remain equidistant to the directrix by guidance of the line through the pole. If the radius of the doted circle increases, the two points will remain equidistant from the directrix, and still line up through the pole, but will defined by a different location on the circle. If the distance from the pole to the directrix increases, the points will again remain equidistant from the directrix, and still line up through the pole, and be defined by different points on the circle. The midpoint of the two points can be placed anywhere along the directrix, and still uphold the principles of the conchoid. GZ is congruent to ZH-- definition of a midpoint The Cartesian coordinate formula is (a – b – x)*(a + b –x)x^2 + (a – x)^2 * y^2 = 0 . (Variables not defined in sketch) Move the slider to adjust the radius of the dotted circle. Drag the point O to adjust the placement of the pole. Hit the reset button to return to the original Sketch. Works Cited: Sonom. "Conchoid of Nicomedes - GeoGebra." Conchoid of Nicomedes - GeoGebra. N.p., n.d. Web. 23 Feb. 2015.
If the radius of the dotted circle was larger than the distance from the pole to the directrix, what would the two curves look like? If the radius of the dotted circle was shorter than the distance from the pole to the directrix, what would the two curves look like? If the radius of the dotted circle was equal to the distance from the pole to the directrix, what would the two curves look like? Can the two curves ever be parallel? Why or why not? What would the curves look like if the pole was placed on the directrix?