points_on_ellipse.py

# This code was used to create the answer
# http://tex.stackexchange.com/a/352766/12571

from scipy.optimize import fsolve
from math import *
import numpy as np

def ellipse(a,b,t):
    return (a*cos(t), b*sin(t))

def distance(p1,p2):
    return sqrt((p1[0]-p2[0])**2 + (p1[1]-p2[1])**2)

def generate_equations(n, a, b, x_0=0):
    """Generates a function suitable to be solved by scipy.optimize.fsolve
    This functions creates the set of equations to solve to place n
    points in the edge of an ellipse of radii (a,b), so that all points
    are equi-distant. x_0 is the angle for the first point"""

    result = []
    result.append("def equations(p):")
    result.append("    {unknowns} = p"
            .format(unknowns = ",".join("x_{i}".format(i=i) for i in range(n+1)))
         )
    distance_equations = []
    for i in range(n):
        next_i = (i+1) % n
        distance_to_next = "distance(ellipse(a,b,x_{i}), ellipse(a,b,x_{next_i}))"\
                           .format(a=a, b=b, i=i, next_i=next_i)
        distance_equations.append(distance_to_next)
    for i in range(n-1):
        next_i = (i+1) % n
        result.append("    f_{i} = {eq_i} - {eq_i_next}"
                .format(i=i, eq_i=distance_equations[i],
                    eq_i_next = distance_equations[next_i])
             )
    result.append("    f_{i} = x_0 - {x_0}"
            .format(i=n-1, x_0=x_0))
    result.append("    f_{i} = {any_distance} - x_{n}"
            .format(i=n, any_distance = distance_equations[-1], n=n))
    result.append("    return({equations})"
            .format(equations = ",".join("f_{i}".format(i=i) for i in range(n+1)))
        )
    return "\n".join(result)


def generate_equations_old(n, a, b, t_0=0):
    """Generates a function suitable to be solved by scipy.optimize.fsolve
    This functions creates the set of equations to solve to place n
    points in the edge of an ellipse of diameter (a,b), so that all points
    are equi-distant. t_0 is the angle for the first point"""

    result = []
    result.append("from math import *")
    result.append("def equations(p):")
    result.append("    {unknowns} = p"
            .format(unknowns = ",".join("t_{i}".format(i=i) for i in range(n+1)))
         )
    distance_equations = []
    for i in range(n):
        next_i = (i+1) % n
        distance_to_next = "sqrt(({a}*cos(t_{i}) - {a}*cos(t_{next_i}))**2 "\
                           "+ ({b}*sin(t_{i}) - {b}*sin(t_{next_i}))**2)"\
                           .format(a=a, b=b, i=i, next_i=next_i)
        distance_equations.append(distance_to_next)
    for i in range(n-1):
        next_i = (i+1) % n
        result.append("    f_{i} = {eq_i} - {eq_i_next}"
                .format(i=i, eq_i=distance_equations[i],
                    eq_i_next = distance_equations[next_i])
             )
    result.append("    f_{i} = t_0 - {t_0}"
            .format(i=n-1, t_0=t_0))
    result.append("    f_{i} = {any_distance} - t_{n}"
            .format(i=n, any_distance = distance_equations[0], n=n))
    result.append("    return({equations})"
            .format(equations = ",".join("f_{i}".format(i=i) for i in range(n+1)))
        )
    return "\n".join(result)



def generate_tikz_list(r):
    return ",".join(["%f/%f" % (ellipse(a,b,x)) for x in r])


def print_tikzpicture(a,b,n,l):
    print(r"""
\begin{tikzpicture}
\fill[white] (-3.5, -1.5) rectangle (3.5, 1.5);
\draw[blue] (0,0) ellipse(%d and %d);
\foreach \x/\y [count=\n from 0] in {%s} { 
     \node[fill=blue,circle] (node-\n) at (\x,\y) {}; 
}
\foreach \n [remember=\n as \previous (initially 0)] in {1,...,%d} {
   \draw[thick,red] (node-\previous.center) -- (node-\n.center);
}
\node[fill=yellow, circle] at (node-0) {};
\draw[thick,red] (node-%d.center) -- (node-0.center);
\end{tikzpicture}
""" % (a, b, l, n-1, n-1))

import sys
from math import *

# Parameters for the ellipse
a = 3
b = 1
n = 7 # Number of nodes


for angle in range(0,360,2):
    # Generate the code which defines the equations
    eqs = generate_equations(n,a,b,radians(angle))
    # Execute that code to define function "equations"
    exec(eqs)
    # Use simpy to solve the equations
    print("angle=", angle, file=sys.stderr)
    r = fsolve(equations, np.append(np.linspace(radians(angle), radians(angle)+ 2*pi, n, endpoint=False), [2*a]))
    # Convert the solution to tikz syntax
    tikz_points = generate_tikz_list(r[:-1])
    print_tikzpicture(a,b,n,tikz_points)