In the case of a rectangular square grid RG[a,b] with the sides a, b, the initial number of curves, obtained without internal mirrors is k = GCD(a,b) (GCD- greatest common divisor), so in order to obtain a single curve, the possible number of internal two-sided mirrors is k-1, k,..., 2ab-a-b. According to the rules for introduction of internal mirrors, we propose the following algorithm for the production of mono-linear designs: in every step, each of the first internal k-1 mirrors must be introduced in crossing points belonging to different curves. After that, when the curves are connected and transformed into a single line, we may introduce other mirrors, taking care about the number of curves, according to the rules mentioned.
Symmetry of these curves P. Cromwell (1993) used for the classification of the Celtic frieze designs, and P. Gerdes (1989, 1995) for the reconstruction of Tamil designs. From the ornamental heritage, at first glance it appears that symmetry is the mathematical basis for their construction and possible classification (Gerdes, 1989, 1990; Cromwell, 1993). But, the existence of asymmetrical curves suggests two another approaches.
The first approach that we may use is the geometrical one: two curves are equal iff there is a similarity transforming one into the other. This means that one curve can be obtained from the other by a combined action of proportionality and isometry. Instead of considering the curves, we may consider equality of mirror arrangements defined in the same way. From the algorithm for constructing perfect curves and the criterion for their equality, we may try to enumerate them: to find the number of all the different curves (i.e., mirror arrangements) which can be derived from a rectangle with the sides a, b, for a given number of mirrors m (m = k-1, k, ..., 2ab-a-b).
The second approach to the classification of our perfect curves is from knot theory view point. Every such curve can be easily transformed into an interlacing knotwork design, this means, into the projection of some alternating knot. We will return to that connection in Subsection 3.2.7 on knots links and mirror curves.