(a–c) An example of the conformal map between a disc (a), a triangle (b), and a square (c) showing polar grids. (d–f) An example of the conformal map between a square (d), a disc (e), and for the exponential function w = exp(z) (f) showing Cartesian grids. In all cases, grid lines remain perpendicular to each other. (g, h) An irregular polygon (g) is mapped onto a unit disc (h). The polygon’s vertices (black dots) map to the disc boundary as shown. (i) An example of the image mapping from the wing onto the disc. (j–l) We mapped polar and Cartesian grids of the same wing for demonstrational purposes. We must define a pre-image of the origin of the wing (j, k, left), that is, the point that will be mapped onto the center of the disc (j, k, right). For the Drosophila wing, it is the black dot at the intersection of the ACV and L4 veins. (l) The unit disc has an automorphism, that is, a map onto itself. It means that we can keep the boundary the same, but we still can rotate the disc and we can move the origin of the disc, as shown.