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Every gold point is derived from the same immutable snapshot: P = (latitude, longitude, altitude, model time). The four chart cameras never change that event. The globe is an input: rotating it under the crosshair selects a new geographic location rather than simulating physical Earth rotation.
L = 6,371 km; c = 299,792.458 km/s (exact) r = L + h X = r cos(latitude) cos(longitude) Y = r cos(latitude) sin(longitude) Z = r sin(latitude) τ = ct/L; ξ = selected signed axis/L η, ζ = the other two axis components/L
Longitude is positive east. X points through the equator at Greenwich, Y through the equator at 90° E, and Z north. This is a non-rotating spherical Earth in Minkowski spacetime. Altitude is radial height above that sphere. Earth rotation, oblateness, gravity, proper-time corrections, and live positioning are outside this model. One unit of τ is approximately 21.251 milliseconds.
The 3D display is (ξ, τ, η); ζ is suppressed visually but retained in the state and invariant checks. The wire cylinder is a coordinate reference, not a physical spacetime boundary. Camera perspective changes the appearance, not the coordinates.
u = τ − ξ; v = τ + ξ U = atan(u); V = atan(v) T = (V + U)/π; Xc = (V − U)/π Displayed graph = (ξ, 1.8 T, 0.48 τ)
The curved surface is the graph of T(ξ, τ). Its height and depth have explicitly stated display scales. It is not an embedding of gravitational curvature and its Euclidean distances have no invariant physical meaning.
ρ = √(ξ² + η² + ζ²) Uᵣ = atan(τ − ρ); Vᵣ = atan(τ + ρ) Tᵣ = (Vᵣ + Uᵣ)/π; R = (Vᵣ − Uᵣ)/π (ξc, ηc, ζc) = (R/ρ) (ξ, η, ζ) Displayed point = (ξc, Tᵣ, ηc); ζc is retained Finite-event domain: |Tᵣ| + R < 1
The diamond has actual coordinate depth: it is a transparent double cone with circular cross-sections, not a flat diagram tilted in perspective. A vertical section through its axis is a diamond. The map compactifies the full radial coordinate and retains the angular direction. The same canonical event drives every view, although this radial time transform Tᵣ generally differs from the signed-axis transform T graphed in panel 02.
At fixed altitude and time, moving around Earth leaves R and Tᵣ unchanged but moves the compactified spatial vector. This is genuine angular motion in the coordinate volume. The three-dimensional display suppresses ζc, the third compactified spatial component, but retains it in the readout, inverse check, and export. Distinct four-dimensional events can still overlap in a three-dimensional projection.
What the geometry means. This is a coordinate-domain visualization of compactified Minkowski spacetime, not its Euclidean metric, a physical boundary, or gravitational curvature. In these normalized variables the conformal metric is proportional to −dTᵣ² + dR² + [sin(πR)/π]² dΩ². Radial null rays have slope ±1 in the (R, Tᵣ) meridional chart; arbitrary projected rays and perspective screen angles need not be 45°. The blue rays are derived by compactifying genuine null trajectories (τ + a s, q + n s), with a = ±1 and |n| = 1, including their analytically computed endpoints.
The tips i⁺ and i⁻ are future and past timelike infinity; ℐ⁺ and ℐ⁻ label the future and past null boundary sheets. i⁰* marks spatial infinity with its approach directions expanded into a ring for visualization. Those directions are not distinct spatial-infinity points in the regular conformal compactification; the angular metric collapses there. No plotted finite event reaches the shell.
β = v/c; γ = 1/√(1 − β²) τ′ = γ(τ − βξ) ξ′ = γ(ξ − βτ); η′ = η; ζ′ = ζ −τ² + ξ² + η² + ζ² = −τ′² + ξ′² + η′² + ζ′²
This is a coordinate transformation of the same event, not an optical image seen by an observer, a light-travel-time correction, or a model of an Earth-bound sensor. The blue tilted grid shows transformed reference coordinates; pale axes are the primed frame.
Demo runs a deliberately synthetic, subluminal route. Scrubbing it selects a different point on that analytic route. Grabbing Earth or changing its location switches to an exploration trace. That trace connects selected events and is not assumed to be a physical worldline. Its latest full-4D interval is classified as timelike, spacelike, or null. Time advances at 0.48 units of τ per wall-clock second while playback or drag-time is enabled; this is explicitly slowed model time. Camera movement does not advance it.
All drawn paths are continuous solid polylines through sampled canonical events. Segments approximate the exact transformed path between finite samples. End-of-range playback stops; it does not join future and past infinity. Export includes the canonical state, trace, projection axis, observer speed, and model assumptions.
Anıl Zenginoğlu, How to draw Penrose diagrams · quantitative arctangent compactification. David Tong, General Relativity §4.4.1 · radial compactification, angular metric and conformal infinity. NIST, speed of light. NASA Earth Fact Sheet · mean radius. MIT, Lorentz transform.
Earth texture: NASA Blue Marble Next Generation, distributed as bmng.jpg with Basemap data; downsampled locally to 1536×768. Historical composite, not a live image. NASA image data are used for geographic context; interface lighting and atmospheric rim are illustrative. This file includes its assets and has no runtime network dependencies, tracking, or backend.