FlatLat 2.0.0

A Package for Calculating 2D Lattice Invariants

Rootform_2d

Calculation of and with root forms in 2 Dimensions

Utility Functions

distgen(q, v_1, v_2) is a generic Minkowkski distance calculator in R^2. It takes two inputs - a value of q >= 0 and two points in R^2 given as lists. If q=0 it will return the Chebyshev (L_inf) distance between points, otherwise it will return the L_q distance.

mincyc(l) takes a list l of any length and returns the cyclic permutation that puts the smallest list member first

roundlist(l,r) takes a list of floats l and an integer value r. It returns l with r-rounded float values.

minf(a,b,c,d) calculates the quantity max(|a-b|, |c-d|, |a+b-c+d|/2) required for the root form distance

makelat(a,b,t) will return the vectors of a Niggli Reduced 2D lattice given two lengths a, b and an angle t

sb_sign(veclist) will take a list of three vectors and, assuming that they are an obtuse superbase, calculate the sign of that superbase

index_sorted(list) will take any list of quantities, and return a related list whose value at each index is the position the quantity at that index in the original list would be at in a sorted version of the list.

Lat2D

The Lat2D class should be entered as two lists [[x_1, y_1], [x_2, y_2]]. The following methods are available:

make_obsb() will iterate the reduction step detailed in the paper until an obtuse superbase is reached

sb_sign() will calculate the sign of the superbase (positive if superbase vectors are ordered anticlockwise, negative if superbase vectors are ordered clockwise)

make_cf returns the unsorted coform for the lattice

lattice_sign(tol = 10*-6)* returns the sign of the lattice as calculated from the coform - it reverses the sign of the superbase if a swap of two values is required to order the coform values list, or keeps that sign if ordering the list requires only permutation. It will the sign to 0 if any value in the coform is less than the input tolerance (10**-6 default).

make_rf(tol= 10*-6* returns the signed root form object - it will return 0 if the coform is achiral (to within a set tolerance of $10^-6$), otherwise it will return the sign passed to it by the lattice_sign function.

RF_2signed Class

RF2_signed objects consist of a pair of inputs - an ordered list of three positive numbers and a sign (0, +1, -1). If an achiral root forms is entered, the sign will default to 0 regardless of the input. If the user inputs an unordered list there will be a warning message:

rightsign() will return the unordered root form - that is, it will swap the last two values of the root form if the sign is negative

rf_chir(dtype = 0, pgroup = 2) will, return the signed L_inf or L_2 based point group chirality of the root form. Id pgroup is set to 4 or 6, the D4 or D6 chirality will instead be calculated. If dtype is set to 2, the L_2 distance will be calculated rather than L_inf

projform() will return the co-ordinates of the form in the quotient triangle, along with the sign of the root form, as a PF2 object.

coform2d() and voform2d() will return the coform and voform for the obtuse superbase represented by the root form. Vectors are in anticlockwise order as stated in the paper.

make2lat() will reconstitute the actual 2 dimensional lattice based on the input root form. Note that the output lattice will not be Niggli reduced - it will be the two vectors v_1, v_2 calculated as part of the obtuse superbase.

PF2 Class

Projected Form (PF2) objects should be entered as a list [x, y] followed by a value +/-1 or 0 indicating the sign of the lattice from which the projected form is derived, and a tolerance set by defalut at 10**-6 - the sign will set to zero if either x or y value is within the tolerance level. The following methods are available:

qs_plot() will return the co-ordinates of the projected form in the quotient square.

pf_grpchir(dtype = 0, pgroup = 2) returns the signed L_inf or L_2 chirality measure for the projected form depending on whether dtype is set to 0 or 2. Setting pgroup to 4 or 6 will calculate the D4 or D6 Chirality. Any other setting of dtype or pgroup will return an error message.

pf_chir(dtype = 0) will return the signed D2 chirality. Defaults to use of Chebyshev distance - if dtype = 2 the Euclidean distane will be used instead.

root_from_pf2 (scale = 1) will return a signed root form derived from an input projected form, which can be scaled by setting the scale factor

lattice_from_pf2 (sc=1) will return the 2D lattice reconstituted from an input projected form - setting the scale factor scales the underlying root form

sphere_proj() returns the latitude and longitude of the spherical projection.

Distance Calculations

pf2dist(pf_1, pf_2, orient = True, dtype = 2) will take two projected forms and calculate the L_2 oriented distance between them by default. It will calculate the unoriented distance if orient is set to False (or both projected forms share a sign or either are achiral), and the oriented or unoriented Chebyshev distance if dtype is set to 0

rf2dist(rf_1, rf_2, orient = True, dtype = 2) will take two root forms and calculate the L_2 oriented distance between them by default. It will calculate the unoriented distance if orient is set to False (or both projected forms share a sign or either are achiral), and the oriented or unoriented Chebyshev distance if dtype is set to 0