Cissoid is the generalization of Cissoid of Diocles. Who generalized it? Around what era?
Cissoid is a method of deriving a new curve based on two (or one) given curves C1, C2, and a fixed point O. A curve derived this way may be called the cissoid of C1 and C2 with the pole O.
- Given two curves C1 and C2, and given a fixed point O.
- Let P1 be a point on C1. Draw a line L passing O and P1. Let the intersection of L and C2 be P2.
- Mark a point Q on line L, such that distance[O,Q]==distance[P1,P2].
- The locus of Q (as P1 moves on C1) is the cissoid of C1 and C2 with the pole O.
Note: There are two points on line L such that distance[O,Q]==distance[P1,P2]. The two points are symmetric around point O, so either one will generate the same cissoid. Also, if L and C2 have more than one intersections, then we can label additional points P3, P4,… and the cissoid may have loops.)
The cissoid method can also be used on a single curve or multiple curves. Given a curve (or curves) and a point O, let a line L passing O sweeps the plane in a complete revolution. Let P_1, P_2, …P_n be the intersections of L and the given curves. Mark points on L with lengths of Length[vector[P_i,P_j]] for all possible combinations of i ,j ≤ n. (consistently mark them on one side of L) The locus of these points is the cissoid of the given curves with pole O. If the line L never makes more than one intersection with the given curves, then we may say that there is no cissoid of such curves with respect to O. The essence of the cissoid method is that it construct a poler curve that measures the distance of (two) given curves as a line sweeps by. For example, we can have the cissoid of a circle, or a cissoid of a parabola.
Cissoids of two Lines or two Circles
The cissoid of two concentric circles with pole on center is two concentric circles centered on pole. The cissoid of two circles in general is a combinations of various oval, figure-eight, or droplet-shaped curves. analyze the number of loops, branches, nodes, cusps, …etc.
The cissoid of two parallel lines is a line. The cissoid of two non-parallel lines with pole not on the line is a hyperbola-like curve. The curve has two asymptotes parallel to the given lines. If the point is on one line, then the cissoid is a line. prove or disprove that cissoid of two lines is a hyperbola.
Cissoids of a Line and a Circle
If C1 is a circle, and C2 is a line tangent to C1 at point A, and O is the point on C1 opposite A, then the cissoid of C1, C2 and the pole O is called Cissoid of Diocles. If O is a arbitrary point on the circle, the curve is a oblique cissoid. Oblique Cissoid
If the line passes through the center of the circle, and pole on the circle, then the resulting curve is a strophoid, and if the pole is a point on the circle furthest from the line, it's a right strophoid.
The cissoid of a line and a circle, with pole on the center of circle, is any member of conchoid of Nicomedes.
General Cissoid of a Circle and a Line Cissoid of a Circle and a Line with pole on circle
Tangent at Pole
Let there be a cissoid based on given curves c1, c2, and point O. If c1, c2 intersect at point P, then line[O,P] is a tangent of the cissoid at point O.
The cissoid of a algebraic curve and a line is itself algebraic. from to Robert C Yates. Needs proof. Check J Dennis Lawrence's book.
Curve relations by cissoid
|two parallel lines||point not on line||line|
|a line and a circle||center of circle||conchoid of Nicomedes|
|a circle and a tangent||on circumsference||oblique cissoid|
|circle and a tangent||opposite point of tangency||cissoid of Diocles|
|circle and line through center||on circumsference||(oblique) strophoid|
|two concentric circles||center||two concentric circles|
|a single circle (as two arcs)||point r*Sqrt distant from center||one loop of lemniscate of Bernoulli|
Related Web Sites
See: Websites on Plane Curves, Plane Curves Books .
The MacTutor History of Mathematics archive