How the square grid in the z-plane warps into two families of parabolas in the w-plane, where u = x² − y² and v = 2xy. Drag the black point in the left panel to test conformality anywhere.
Each curve on the right is the image of the same-colored line on the left. A vertical line x = c maps to u = c² − v²/(4c²), a parabola opening toward −u with vertex (c², 0). A horizontal line y = c maps to u = v²/(4c²) − c², opening toward +u with vertex (−c², 0). Since (−z)² = z², the lines x = c and x = −c land on the same parabola — the map is 2-to-1.
The two bold curves pass through the draggable point's image w = z², drawn with their tangent lines and the measured angle. Away from z = 0 the angle is always 90°, matching the right angle between the grid lines in the input — angles are preserved. At z = 0, where f′(0) = 2·0 = 0, both tangent vectors collapse to zero and there is no angle left to preserve: the lines x = 0 and y = 0 both fold onto the u-axis, pinched at w = 0.