The Geometry of Some Families of Curves Described by Conchoids

Transcription

1 The Geometry of Some Families of Curves Described by Conchoids A conchoid is a way of deriving a new curve based on a given curve, a fixed point, and a positive constant k. Curves generated in this way are sometimes called general conchoids because this method is a generalization of the conchoid of Nicomedes. Step-by-step construction: 1. Specify a curve C, a point O not on C, and a constant k.. Draw a line l passing thru O and any point P on C. 3. Mark points Q 1 and Q on l such that distance[q 1,P] = distance[q,p] = k. 4. The locus of Q 1 and Q for all variable points P on C is the conchoid of C with respect to O and offset k. The point O is called the pole. In general, if the equation of the given curve is r = f ( θ ) in polar coordinates, then the equation of its conchoid has the form: r = f ( θ ) ± k. Examples 1. The conchoid of a sinusoid with pole at ( 3, 3 ) and offset k = is displayed in the figure below. This conchoid is the pair of continuous curves formed as the union of all the bold dots in the plane. O = ( 3, 3 ). The Conchoid of Nicomedes, as shown in the figure, refers to an entire family of curves of one parameter. Each branch curve in the family is the conchoid of a common horizontal line (which is asymptotic to the curve). 1

2 If the asymptote to the Conchoid of Nicomedes is the line with the polar equation: r = a csc ( θ ) then the polar equation for each member curve is: r = a csc ( θ ) ± k for some number k. Step-by-step construction: 1. Specify a line l, a point O not on l, and an offset distance k. Q1. Draw a line m passing through O and any point P on l. 3. Mark points Q1 and Q on m such that l distance[q1,p] = distance[q,p] = k. 4. The locus of Q1 and Q for variable points P on l is a member of the Q Conchoid of Nicomedes. 3. The Limacon of Pascal refers to another family of curves of one parameter. Each member of this family is describable as the conchoid of a common circle. m Limacon of Pascal and an artistic reflection Step-by-step construction: 1. Specify a fixed point O on a circle.. Draw a line l passing through O and P, where P is any point on the circle. 3. On line l, mark points Q 1 and Q such that distance [P, Q 1 ] = distance[p, Q ] = k, where k is the offset constant. 4. Repeat steps, 3 for all possible choices of P. 5. The locus of Q 1 and Q is the Limacon of Pascal. O O

3 The graph of a curve included in the Limacon of Pascal may have one of four distinct shapes based upon the relative magnitudes of the radii of the fixed circle and that of the offset constant k which are described in the step-by-step construction procedure. Case 1. Consider the graph of the conchoid of the circle: r = sin θ with offset value k = 1. It is called an inner loop limacon. This curve has the polar equation: r = 1 + sin θ and is displayed below. In general, the polar graph of the equation r = a + b sin θ is an inner loop limacon a whenever < 1. b 3

4 Case. The graph of the conchoid of the circle: r = sin θ with offset value, k =, is called a cardioid. It has the polar equation: r = + sin θ and is displayed below. In general, the polar graph of the equation r = a + a sin θ is called a cardioid. 4

5 Case 3. The graph of the conchoid of the circle: r = sin θ with offset value, k = dimpled limacon. 11 It has polar equation: r = + sin θ and is shown below. 4 11, is called a 4 In general, the polar graph of the equation r = a + b sin θ is a dimpled limacon whenever 1 < a <. b 5

6 Case 4. The graph of the conchoid of the circle: r = sin θ with offset value, k = 4, is called a convex limacon. It has the polar equation: r = 4 + sin θ and the graph shown below. In general, the polar graph of the equation r = a + b sin θ is a convex limacon whenever a >. b 6

7 Details on Nicomedes Conchoid A plane algebraic curve of order 4 whose equation in Cartesian rectangular coordinates has the form ( x + y )( y a) k y = 0 and in polar coordinates r = a cscθ ± k. The outer (upper) branch has asymptote: y = a and two points of inflection. The inner (lower) branch has asymptote y = a. The character of the coordinate origin depends on the values of k and a. For a < k it is a node, i.e. a double point of self-intersection; For a = k it is a cusp; the curve has the y-axis as a common semi-tangent line from both sides at the origin. For a > k it is an isolated point and, in addition, the curve has two symmetric points of inflection with respect to the y axis. The curve is a conchoid of the straight line y = a. It is named after Nicomedes (3rd century B.C.), who used it to solve the problem of trisecting an angle. 7

8 Angle Trisection The conchoid can be used to solve the Greek Angle Trisection problem. Given an acute angle ;AOB, we want to construct an angle that is ;AOB, with the help of a conchoid of Nicomedes. 1 3 rd the measure of Steps: 1. Draw a line m intersecting segment AO and perpendicular to it.. Mark the points: D = m AO and L = m BO. 3. Construct a conchoid of Nicomedes of the line m with pole at O and offset k = distance[o,l]. 4. Draw line l intersecting L and perpendicular to m. 5. Let C be an intersection of the curve and l, the one on the opposite side of the pole. Then measure(;aob )= 3 measure(;aoc ). Nichomedes constructed his conchoid for angle trisection by means of a mechanical device, thus: AB is a ruler with a slot in it parallel to its length, FE a second ruler fixed at right angles to the first with a peg (C) fixed in it. A third ruler, PC, pointed at P, has a slot in it, parallel to its length, which fits the peg C. D is a fixed peg on PC in a straight line with the slot but on the under-side, and this peg D moves freely along the slot in AB. If then the ruler PC moves so that the peg D describes the length of the slot in AB on each side of F, the extremity P of the ruler describes the curve which is called a conchoid. Nichomedes called the straight line AB the ruler, the fixed point C the pole, and the constant length PD the distance. 8