circlevaries

Circlevaries is a term used in discussions of parametric families of circles in the plane, referring to a set of circles whose defining parameters change with a parameter t. In its common formal form, a circlevaries is described by a center function C(t) = (x0(t), y0(t)) and a radius function r(t) > 0 on an interval I. The circle corresponding to t is the set of points X in the plane such that |X - C(t)| = r(t). If the center is fixed, the family consists of concentric circles with varying radius.

The concept is often studied through the envelope of the circlevaries, which is the locus of points that are tangent to at least one member of the family. Formally, for F(X, t) = |X - C(t)| - r(t), the envelope is found by solving F(X, t) = 0 together with ∂F/∂t = 0. The envelope can produce a variety of curves depending on the chosen C(t) and r(t).

A simple illustrative example: let C(t) = (t, 0) and r(t) = 1 for t in R. The circles have centers along the x-axis with radius 1. Solving F(x, y, t) = (x - t)^2 + y^2 - 1 = 0 and ∂F/∂t = -2(x - t) = 0 yields t = x, which gives the envelope y = ±1. Thus the two horizontal lines y = 1 and y = -1 form the envelope of this circlevaries.

Applications of circlevaries concepts appear in geometry education, computer graphics, and CAD, where understanding how a family of circles evolves helps in modeling shapes, morphing, and locus problems. The term is not standard in formal mathematics, but it conveys the idea of a variable-radius or moving-center family of circles.