Domain: Humanities and Social Sciences — Geography
Target concept: Transverse Mercator projection: conformal unfolding of a spherical Earth's latitude/longitude graticule onto a plane by developing it along a cylinder tangent to a chosen central meridian, which stays straight and true to scale while other meridians curve symmetrically away from it.

Video generation prompt: Display a spherical Earth model with a graticule of latitude and longitude lines. Select a central meridian. Apply the transverse Mercator projection method to unfold points on the sphere into a planar map. Use time-lapse photography to record the process of image focusing and projection, as well as the final rectangular map.

Evaluation guide:
(i) Target concept: Conformal mapping of spherical surface onto plane via transverse cylindrical projection.
Minimum mechanism: The sphere must be geometrically unwrapped along a meridian-tangent cylinder; no rotation of the sphere itself is involved, only mathematical surface parameterization and planar development.
(ii)Storyline: [Phase 1] Static spherical Earth model with visible graticule and marked central meridian appears under fixed overhead view. [Phase 2] Graticule begins to deform: central meridian locks vertically; adjacent meridians start curving outward; parallels remain horizontal. [Phase 3] Deformation intensifies symmetrically: outer meridians exhibit increasing curvature; grid density remains uniform near center, stretches near edges. [Phase 4] Cylinder unrolling completes: spherical grid fully mapped onto rectangle with central meridian centered and vertical. [Phase 5] Final stable frame shows rectangular map with transverse Mercator geometry: no residual spherical curvature, no topological error, no motion. Perspective remains strictly overhead; no occlusion, no parallax, no depth cues beyond surface texture.

Manual:
Experimental Procedure
1. Place a rotating spherical Earth model with full latitude/longitude graticule under fixed overhead view.
2. Mark the central meridian (e.g., 0° or designated prime meridian) visibly on the sphere.
3. Apply transverse Mercator projection: unfold the spherical surface onto a vertical cylinder tangent along the central meridian, then unroll into a rectangle — captured via time-lapse.
Observation of Experimental Phenomena
- Sphere remains stationary (no rotation) but its surface graticule transforms continuously.
- Central meridian stays straight, vertical, and undistorted throughout.
- Meridians curve symmetrically outward from central meridian; parallels remain horizontal and evenly spaced.
- Final output is a rectangular map with central meridian centered and vertical, flanked by smoothly curving meridians.
Summary of Experimental Principles
Transverse Mercator is a conformal cylindrical projection where the cylinder is rotated 90° to touch the sphere along a meridian (central meridian), preserving angles locally and keeping that meridian straight and scale-true.

Prompt Grounding rubric:
level_1: No spherical Earth model appears; no graticule (latitude/longitude lines) visible; no central meridian marked; no transverse Mercator projection unfolding shown; no time-lapse progression from sphere to rectangle; final frame is not a rectangular map.
level_2: A spherical object appears but lacks identifiable Earth features (e.g., no blue/white/green coloration, no continents); graticule is absent or unrecognizable as latitude/longitude; central meridian is missing; projection process is not depicted; final output is non-rectangular or lacks planar grid structure.
level_3: A spherical Earth model appears with visible graticule of latitude and longitude lines; a central meridian is marked (e.g., bolded or labeled line); time-lapse progression shows change from 3D sphere to 2D layout; final frame is a rectangular map containing a distorted but recognizable grid pattern; however, the grid does not conform to transverse Mercator characteristics (e.g., central meridian straight and vertical, symmetric distortion east/west).
level_4: All prompt elements are present: spherical Earth model with clear graticule; central meridian explicitly indicated (e.g., centered, vertical, emphasized); time-lapse shows continuous morphing/unfolding from sphere to plane; final frame is a rectangular map where the central meridian is straight and vertical, and meridians converge toward poles while parallels remain horizontal, consistent with transverse Mercator projection geometry, but minor visual flaws exist (e.g., one parallel misaligned, slight warping asymmetry, or brief flicker during transition).
level_5: Spherical Earth model appears with accurate Earth-like coloration and texture; graticule is complete, evenly spaced, and correctly labeled (latitudes horizontal on sphere, longitudes converging at poles); central meridian is precisely centered, vertically oriented, and distinctly marked; time-lapse progression shows smooth, continuous, unbroken transformation from spherical surface to planar map; final rectangular map exhibits exact transverse Mercator properties: central meridian perfectly straight and vertical, meridians symmetrically curved away from it, parallels evenly spaced and horizontal, no grid discontinuities, no stretching artifacts outside central zone.

Scientific and Causal Correctness rubric:
level_1: Earth model rotates or deforms inconsistently with projection logic; graticule lines move independently of surface; central meridian shifts position during time-lapse; final map shows impossible geometry (e.g., meridians crossing, latitudes diagonal, poles duplicated); no causal link between sphere rotation/unfolding and map shape.
level_2: Graticule distorts incoherently during transition (e.g., lines snap, vanish/reappear, or reverse order); central meridian bends or disappears mid-sequence; final map has inverted latitude/longitude orientation (e.g., meridians horizontal, parallels vertical); projection process violates transverse Mercator’s defining property: central meridian must remain undistorted and straight.
level_3: Graticule remains attached to sphere surface during initial phase; central meridian stays fixed in position relative to sphere; time-lapse shows progressive flattening where longitudinal lines near central meridian stay nearly vertical and evenly spaced, while outer meridians curve outward, but curvature is inconsistent or asymmetric; final map preserves latitude/longitude topology but fails to show correct transverse Mercator symmetry (e.g., only one side curves, or parallels slope).
level_4: Graticule deforms continuously and causally: lines adjacent to central meridian remain near-vertical and minimally stretched; meridians curve smoothly and symmetrically away from central meridian; parallels stay horizontal and evenly spaced; central meridian remains straight, vertical, and unchanged in length throughout; all deformation is confined to the planar unfolding (no tearing, folding, or topological violation), but one minor deviation occurs (e.g., single parallel slightly tilted, or one meridian endpoint misaligned at edge).
level_5: Every graticule line behaves according to transverse Mercator mathematics: central meridian remains perfectly straight, vertical, and constant-length; meridians curve outward symmetrically with increasing distance from center; parallels remain strictly horizontal and uniformly spaced; no line crosses another; no point on sphere maps to two locations; no line vanishes or reappears; deformation magnitude increases monotonically with distance from central meridian, fully consistent with the conformal, cylindrical, transverse projection of a sphere onto a tangent cylinder rotated 90°.

Spatiotemporal Consistency rubric:
level_1: Sphere teleports between frames; graticule flickers in/out without continuity; central meridian appears/disappears abruptly; map snaps into existence without transition; lighting changes erratically; camera viewpoint shifts or zooms; structural breaks occur (e.g., sphere fractures, grid tears).
level_2: Sphere maintains fixed position but rotates unnaturally (jerky, non-uniform motion); graticule lines stutter or skip positions; central meridian wobbles or changes thickness mid-sequence; unfolding begins before sphere is fully visible or ends before rectangle stabilizes; background or lighting shifts between key phases.
level_3: Sphere remains stationary and upright; graticule moves cohesively with sphere surface; central meridian stays visually anchored to same location on sphere; time-lapse progression shows unbroken sequence: sphere → partial unfolding → intermediate warped grid → final rectangle; no teleportation, no frame drops, no sudden lighting or shadow changes, but minor motion jerkiness occurs at one transition point (e.g., brief freeze or micro-stutter between two frames).
level_4: Fixed overhead viewpoint maintained throughout; sphere position, scale, and orientation are identical across all frames; graticule lines move smoothly and cohesively without breakage or misalignment; central meridian remains pixel-aligned and stable in position relative to sphere and frame; unfolding proceeds with continuous, uniform motion (no speed variation, no overshoot, no reversal), but one frame exhibits minor inter-frame misregistration (e.g., <1-pixel shift in grid origin).
level_5: Zero spatiotemporal discontinuity: camera viewpoint is absolutely fixed (no pan/tilt/zoom); sphere occupies identical screen region and orientation in every frame; graticule lines maintain perfect spatial coherence: no pixel-level jitter, no aliasing jump, no interpolation blur; central meridian remains exactly centered and vertically aligned in all frames; unfolding motion is mathematically smooth (easing-free linear parameterization), with no acceleration/deceleration detectable; light/shadow remains fully consistent, including specular highlights on sphere and flat illumination on final map.