This Moiré pattern is from two families of concentric circles. Their radii are 1.1^n where n are integers. If the families of circles were arithmetically spaced, the moiré  patterns would be hyperbolas, that is points p such  D(f1, p) - D(f2, p) = k  (k a constant) Hyperbola moiré pics. I had asked in sci.math if the Moirés are circles in these exponentially spaced  circles. Chas Brown replied with this explanation: D(f1,p)/D(f2,p) = k, k constant. Consider some point of intersection of two circles, with radii r1 and r2, respectively. Then exists some m and n such that  r1 = (1.1)^m and r2 = (1.1)^n The "next intersection" in the Moire pattern will be at  r1' = (1.1)^(m+1) = 1.1*r1 and r2' = (1.1)^(n+1) = 1.1*r2 thus r1'/r2' = r1/r2 If you take your Moire pattern "to the limit", you're looking at two circle families, centered say at (-d, 0) and (d,0). Let the family of circles have radii u^n, n an integer, and u>1.0. As u gets very close to 1, the sucessessive steps in the pattern get smaller, but we still get the same pattern noted above (imagine u = (1.1)^(1/1000)); same pattern, with 1000 intervening steps). As u approaches 1, the Moire paten becomes the family curves, each of which is the locus of points with radii to the respective generating circles in a fixed ratio, say c. Thus it is the locus of points P in the triangle below with LP/PR=c. (L being the center of the left circle, R being the center of the right circle). A little trigonometry (Pa the perpendicular through P has length y, OA length x) shows that these points P satisfy, for some constant C: (y^2 + (x+d)^2) / (y^2 + (d-x)^2) = c^2 where c is the fixed ratio of LP/PR (here I'm assuming c>1), and d is the distance of the generating circles from the origin. We can then expand this to: (c^2-1)*y^2 + (c^2-1)*x^2 - 2*(c^2+1)*d*x + (c^2-1)*d^2 = 0 since c is a constant, let k = (c&2+1)/(c^2-1); k > 1 since 1 < c < infinity, then y^2 + x^2 - 2*k*d*x + d^2 = 0 y^2 + x^2 - 2*k*d*x = k^2*d^2 + d^2 = 0 y^2 + (x - k*d)^2 - (k^2-1)*d^2 = 0 so y^2 + (x - k*d)^2 = (k^2-1)*d^2 is the equation of the family of circles you see, with 1 < k < infinity. The same follows if we choose c < 1; in that case k has range -infinity < k < -1, and the circles are on the other side of the y-axis. c=0 gives us the line x=0 Back