Coordinate grids

Some values depend on where they are rather than on other data: a distance from a centre, a colour ramp, a height field, a decision boundary drawn across a plane. For those you need arrays holding the coordinates themselves.

xx, yy = np.meshgrid(np.arange(4), np.arange(3))
xx     # [[0 1 2 3], [0 1 2 3], [0 1 2 3]]   — column number
yy     # [[0 0 0 0], [1 1 1 1], [2 2 2 2]]   — row number

Two arrays of the same shape: one holds every cell's x, the other every cell's y. After that, any function of position is ordinary vectorised arithmetic — no loops:

xx + yy
np.sqrt(xx**2 + yy**2)
np.sin(xx / 3) * np.cos(yy / 3)

Three spellings, and an ordering trap

xx, yy = np.meshgrid(x, y)      # columns first
yy, xx = np.mgrid[0:3, 0:4]     # ROWS first — note the swap
yy, xx = np.ogrid[0:3, 0:4]     # sparse: shapes (3,1) and (1,4)

meshgrid takes arrays and returns x-then-y. mgrid takes slice syntax and returns them in axis order, which for a 2-D grid means rows first. Getting that backwards gives you a transposed result that looks almost right, so it's worth reading the line carefully.

ogrid is the efficient one. Rather than two full grids it returns shapes (3, 1) and (1, 4) and lets broadcasting do the expansion — same answer, a fraction of the memory. For a 4000×4000 grid that's the difference between 256MB of coordinates and 64KB. meshgrid has a sparse=True that does the same thing.

Distance fields

The pattern you'll use most:

yy, xx = np.mgrid[0:H, 0:W]
d = np.hypot(xx - cx, yy - cy)     # distance from (cx, cy) at every pixel

mask = d < 40                      # a circular mask
falloff = np.clip(1 - d / 40, 0, 1)   # a soft radial gradient

np.hypot(a, b) is sqrt(a**2 + b**2) without the intermediate overflow. Circular crops, vignettes, spotlight effects and radial blurs all start here.

You have already used this

The sunset in the first capstone was built exactly this way — a coordinate grid, a vertical ramp for the sky, a radial falloff for the sun, a sine wave for the hills, and np.where to paint the ground in. Go back and read that setup code now; it should look completely ordinary.

np.indices((3, 4)) is a related shortcut that stacks the same coordinate arrays into one array of shape (2, 3, 4).