Complex map

Choose an example or edit the map, then select Display. The left plane contains z; the right contains w = f(z). Tap a point inside the source axes or edit Re(z) and Im(z) to update both markers and their coordinates. Editing the map, its coefficients, grid density or image window removes the previous result until Display is selected again. Reset keeps the inputs.

The source window is fixed at −2.5 ≤ Re(z), Im(z) ≤ 2.5. Each direction has 4–32 grid intervals. Image ± is the half-extent of the image window, from 0.1 to 100. Both planes have equal horizontal and vertical scales within their own axes; their displayed ranges can differ. Dashed orange curves correspond to constant Im(z), solid blue curves to constant Re(z). An image outside the window remains a number; it is not replaced by a marker at the edge.

The maps are f(z) = z² and f(z) = (az + b)/(cz + d), with real coefficients a, b, c, d between −5 and 5. Möbius maps require ad − bc ≠ 0. Common rescaling of all four coefficients leaves the map unchanged. After normalization by their largest absolute value, determinants no larger in magnitude than 64·ε·(|ad| + |bc|), where ε is machine precision, are rejected as numerically poorly determined. This numerical criterion is not a proof of exact degeneracy.

For a Möbius map, the pole is −d/c when c ≠ 0 and the zero is −b/a when a ≠ 0; otherwise the respective location is ∞. Also f(∞) = a/c when c ≠ 0, and ∞ otherwise. The displayed, nearest representable pole location is recognized exactly; a neighborhood around it is not treated as infinity. A finite-value overflow is reported separately. The map z² has a zero and a critical point at 0: because f′(0) = 0, it is not conformal there. Away from critical points and poles, these holomorphic maps preserve oriented angles locally.

Riemann sphere. For w = u + iv the convention used here is S(w) = (2u, |w|² − 1, 2v)/(1 + |w|²). Thus S(0) = (0, −1, 0), S(1) = (1, 0, 0), S(i) = (0, 0, 1) and S(∞) = (0, 1, 0). The vertical coordinate is y. View angle rotates about y, then tilt about the screen’s horizontal axis. These controls change only the orthographic view, not the map. Lighter dashed curves and a hollow probe distinguish the back side. Infinity is a point on this sphere, not a finite value on the image axes.

Grid density does not set curve resolution. The plane drawing separates actual poles and clips image curves at the window; visible pieces are refined in screen coordinates. Möbius line images are circular or straight arcs; their sphere images lie on circles. Drawing refinement is an approximation without a claimed global error guarantee. The PNG contains the active view, axes, labels and a background. There is no animation or WebGL requirement.

Valid parameters, point and viewing settings can be retained by local tool-state storage. Returning within the app can preserve the computed result. A page reload restores the settings and waits for Display.

Sources: Jeremy Orloff, MIT 18.04, Topic 10, pp. 3–8, on conformal and Möbius maps; David W. Lyons, Stereographic projection, especially Exercise 2. The sphere’s vertical-axis convention here interchanges the source’s y and z coordinates.