Unfold Function Window

This is the unfold function window. Given a word W we are mainly interested in the orbit tile O(W) consisting of all points in parameter space which correspond to triangles having a periodic billiard path of type W. The set O(W) typically is a piecewise analytic polygon, whose sides are defined by certain analytic functions. This window displays these functions.

Each function is defined by a pair (a,b) of vertices in the unfolding. The function F(p)=F(a,b;p) measures the difference in y coordinates of the vertices a and b relative to the unfolding U(W,p). Here p is a point in the parameter space.

FORMS FOR THE FUNCTIONS

No Foil : Here the formula is F= Im(P times Q-conjugate). Each of P and Q is an exponential sum. A typical term in either of these sums has the form a exp(i(bx+cy)). In this formula When we list out these functions, we just list the (b,c) pairs. The alternation for Q always starts with 1. The alternation for P starts with a 1 if and only if P is colored black. We draw this function on the red window to the right. We list the P function on the top and the Q function on the bottom. Our convention is that black numbers are negative and white ones are positive. If the formula is too long to fit on the screen you can scroll it by dragging it.

Foil We multiply together P and Q-conjugate by the foil method, collect the terms, and take the imaginary part. We can write the resulting function F as

F(x,y)= a1 sin(b1 x+c1 y) + ... aN sin(bN x + cN y)

These constants are all real, and only depend on the combinatorics of W and the vertices chosen. We list the above terms just by writing the a,b,c triples in columns going across the screen. We use the same convention about positive and negative numbers.

Second Derivative Bounds We can use the Foil form to produce a useful global bound on the second partial derivatives of F. Here are the bounds: These bounds are important in our plotting algorithm. You can learn about this by reading the documentation for our plotter.

CONTROLS

The purple window has some control buttons which let you manipulate the functions:

VALUE DISPLAY

This window displays the values of the selected function F evaluated at a selected point p in parameter space. Up to a reasonable limit, you can actually see displayed the values of all the partial derivatives F as well. The blue window has various subcomponents. Here is a list of what each one does.