Domain: Engineering — Scientific Computing
Target concept: Simplex Method for Linear Programming

Video generation prompt: In a two-dimensional coordinate system, plot the feasible region (shaded in light orange) defined by the constraints x ≥ 0, y ≥ 0, and x + y ≤ 1. Dynamically demonstrate the simplex method for maximizing the linear program z = 2x + y: the red vertex starts from the initial point (0, 0), iterates sequentially to the optimal vertex, and the iteration path is indicated by red dashed lines.

Evaluation guide:
(i) Target concept: Geometric execution of the simplex algorithm on a 2D LP problem.
Minimum mechanism: The feasible region must be a closed convex polygon -- here the triangle bounded by x=0, y=0, and x+y=1 -- whose three vertices, (0,0), (0,1), and (1,0), are all directly readable from the axes; the objective z=2x+y must be non-decreasing at each vertex visited, going from 0 at (0,0) to 1 at (0,1) to 2 at (1,0); adjacency between visited vertices must correspond to sharing exactly one boundary edge of the triangle.
(ii)Storyline: Frame 1–2: Static coordinate system appears with axes and constraint lines. Frame 3: Light orange shading fills the triangular feasible region. Frame 4: Red dot appears at (0,0). Frame 5–6: Red dashed segment draws from (0,0) to (0,1); red dot appears/stabilizes at (0,1). Frame 7–8: Red dashed segment draws from (0,1) to (1,0); red dot appears/stabilizes at (1,0). Frame 9–10: Final state holds with (1,0) highlighted. Perspective: Fixed front-facing 2D orthographic view; all elements fully visible; no occlusion; labels always legible.

Manual:
Experimental Procedure
1. Draw x- and y-axes with labeled non-negative quadrants.
2. Plot line x + y = 1; shade the triangular region satisfying x ≥ 0, y ≥ 0, x + y ≤ 1 in light orange.
3. Place red vertex dot at (0,0).
4. Draw red dashed line from (0,0) to (0,1); place red dot at (0,1).
5. Draw red dashed line from (0,1) to (1,0); place red dot at (1,0) -- the optimal vertex for maximizing z = 2x + y.
Observation of Experimental Phenomena
- Light orange shaded triangular region bounded by x=0, y=0, and x+y=1.
- Red vertex starts at the origin, moves along the triangle's edges via red dashed segments, ending at (1,0).
- Each visited vertex lies on a corner of the triangle; the path never cuts through the interior or crosses outside the shaded region.
Summary of Experimental Principles
- The simplex method navigates between vertices of a bounded convex feasible region.
- The optimal solution of a linear program occurs at a vertex of that region.
- Each iteration moves along an edge to an adjacent vertex whose objective value is no lower than the previous one, ending at the vertex where z = 2x + y is largest.

Prompt Grounding rubric:
level_1: No coordinate axes are drawn, and no shaded region, red vertex marker, or dashed segment linking two vertices ever appears anywhere in the video.
level_2: Labeled x- and y-axes appear, but the light-orange shaded region is missing or not bounded by the line x+y=1, or no red vertex marker ever appears at (0,0), or no red dashed segment is ever drawn.
level_3: Labeled axes, a light-orange region bounded by x=0, y=0, and x+y=1, a red vertex marker at the starting point (0,0), and at least one red dashed segment leading from (0,0) to a second marked vertex are all present.
level_4: Everything required at level 3 is present, and in addition the red marker goes on to reach a third distinct vertex -- so three vertex dots total appear, at (0,0), (0,1), and (1,0), visited in that specific order (0,0) then (0,1) then (1,0) -- each pair of consecutively visited vertices linked by its own dashed segment.
level_5: Everything required at level 4 is present, with exactly two dashed segments and three vertex dots in total, visited strictly in the order (0,0) then (0,1) then (1,0) with no detour to (1,0) before (0,1), no extra markers or segments anywhere on or off the shaded region, and the axes, shading, markers, and dashed segments all stay clearly legible for as long as they remain on screen.

Scientific and Causal Correctness rubric:
level_1: The shaded region includes points violating x≥0, y≥0, or x+y≤1 (or is unshaded or the wrong shape entirely), or the vertex marker is ever placed at a point that is not one of the triangle's three corners (0,0), (0,1), (1,0), or a dashed segment ever crosses outside the shaded triangle.
level_2: The shaded region correctly matches x≥0, y≥0, x+y≤1, but the marker's path connects (0,0) directly to (1,0) while skipping (0,1) entirely, or the path ends at (0,1) rather than (1,0), or it revisits (0,0) after having already left it.
level_3: The shaded triangle correctly covers exactly the points satisfying x≥0, y≥0, x+y≤1; the marker starts at (0,0); and the path goes (0,0) to (0,1) to (1,0), with each step following one edge of the triangle rather than cutting across its interior.
level_4: Everything required at level 3 holds, and each vertex visited after (0,0) sits higher along the value z=2x+y than the corner visited before it -- reading z=0 at (0,0), z=1 at (0,1), and z=2 at (1,0) directly from each corner's position on the axes -- so the path never moves to a corner with an equal or lower z than the one it just left.
level_5: Everything required at level 4 holds, and the path stops permanently at (1,0) -- the one corner of the triangle where z=2x+y is highest among the three corners, as read from its axis position -- with no further movement, no extra corner ever visited, and no segment drawn beyond the two required edges.

Spatiotemporal Consistency rubric:
level_1: The coordinate plane's scale or viewpoint changes abruptly mid-video, the red vertex marker appears at a new corner with no dashed segment ever having been drawn to it, the shaded triangle appears or vanishes within a single frame with no gradual reveal, or a dashed segment shows gaps or gets redrawn out of order.
level_2: The red vertex marker relocates from one corner to another without a continuous dashed segment ever linking the two positions on screen, or the shaded triangle changes its boundary or position after first appearing, or a previously placed marker or dashed segment blinks out and back in during the video.
level_3: The coordinate plane's scale and viewpoint stay fixed for the entire video; the shaded triangle, once fully rendered, keeps the same boundary and color for the rest of the video; each dashed segment is drawn as one continuous line from its starting corner to its ending corner with no gaps; every vertex marker remains visible at every corner it has already reached.
level_4: Everything required at level 3 holds, and in addition the marker's motion along each edge, (0,0)-(0,1) and (0,1)-(1,0), is a smooth continuous slide along that edge rather than an instantaneous jump, with the marker settling at the destination corner only after the corresponding dashed segment has finished drawing.
level_5: Everything required at level 4 holds, and across the full video the triangle's fade-in, the two dashed segments' drawing, and the marker's two moves all proceed in the single order given in the prompt -- plane and shading first, then the marker at (0,0), then the edge to (0,1), then the edge to (1,0) -- with no element ever appearing out of that order, disappearing, or overlapping another element in a way that hides it.