hexmath.py

# copyright 2010 Eric Gradman
# free to use for any purpose, with or without attribution
# from an algorithm by James McNeill at
# http://playtechs.blogspot.com/2007/04/hex-grids.html

# the center of hex (0,0) is located at cartesian coordinates (0,0)

import numpy as np

# R ~ center of hex to edge
# S ~ edge length, also center to vertex
# T ~ "height of triangle"

real_R = 75. # in my application, a hex is 2*75 pixels wide
R = 2.
S = 2.*R/np.sqrt(3.)
T = S/2.
SCALE = real_R/R

# XM*X = I
# XM = Xinv
X = np.array([
  [ 0, R],
  [-S, S/2.]
])
XM = np.array([
  [1./(2.*R),  -1./S],
  [1./R,        0.  ]
])
# YM*Y = I
# YM = Yinv
Y = np.array([
  [R,    -R],
  [S/2.,  S/2.]
])
YM = np.array([
  [ 1./(2.*R), 1./S],
  [-1./(2.*R), 1./S],
])

def cartesian2hex(cp):
  """convert cartesian point cp to hex coord hp"""
  cp = np.multiply(cp, 1./SCALE)
  Mi = np.floor(np.dot(XM, cp))
  xi, yi = Mi
  i = np.floor((xi+yi+2.)/3.)

  Mj = np.floor(np.dot(YM, cp))
  xj, yj = Mj
  j = np.floor((xj+yj+2.)/3.)

  hp = i,j
  return hp

def hex2cartesian(hp):
  """convert hex center coordinate hp to cartesian centerpoint cp"""
  i,j = hp
  cp = np.array([
    i*(2*R) + j*R,
    j*(S+T)
  ])
  cp = np.multiply(cp, SCALE)

  return cp

Its definitely worth mentioning that the cartesian2hex method takes origin as the center of hex 0,0, not the bottom left, it took me a while to figure out this issue!
Also, the hex2cartesian gives the bottom left coord of the bounding box (not the center!).
Apart from that great job!

rarryawn,
It is true that cartesian2hex uses origin as the center of hex 0,0, but hex2cartesian works using the same system and it returns the center of the hex in cartesian coordinates, not the bottom left as you said. Make sure you use hex grid that points up.
Many thanks to James and Eric for a very helpful algorithm and code.

This is great. Thank you!