Class E_to_S_map

An instance of this class represents a pair of approximations to the canonical conformal maps $p,q\colon EX^*\to S^2$.

This is a table indexed by $\{1,2,3\}$. Each entry is a rational function from the $z$-plane to $\mathbb{R}$, which gives an invariant real valued function on $EX^*$. The $k$'th component of our map $p\colon EX^*\to S^2$ will be the product of $u_k$ with a standard function, recorded in the global variable s2p_core[k].

This is a table indexed by $\{1,2,3\}$. The $k$'th entry is a list of pairs $(a,b)$, where we have learned by some other means that $u_k(a)$ should be equal to $b$. We will try to adjust the coefficients of $u_k$ to make this true.

Our approximation to $p$ will be stored in this field.

This sets the fields u, samples_z_u and p based on information stored in the atlas object which is suppied as a parameter to the method.

This returns $\|p(x)|^2-1$ a function of $z$

Given a point $x0\in EX^*$, this measures the failure of $p$ near $x0$ to be conformal.

Check that $p$ has the right behaviour on $v_0$, $v_3$ and $v_6$. The return value is a list of nine scalars which should all be zero.

Check that $p$ gives a conformal map to $S^2$. The argument A can either be an instance of the class E_chart, or a point in $EX^*$. In the latter case, we generate a new chart at the specified point, using the second argument d_ to specify the degree. The return value is a list of three polynomials in $t_1$ and $t_2$, which will be zero if $p$ is exactly conformal.

Generates a plot of the function $u_k$

Generates plots of the functions $u_1$, $u_2$ and $u_3$.

Generates a plot showing the image in $S^2$ of a grid in $F_{16}\subset EX^*$.