Domain: Engineering — Scientific Computing
Target concept: Visualization of the Feasible Region in Two-Dimensional Nonlinear Constrained Optimization

Video generation prompt: The screen displays a two-dimensional coordinate system, illustrating the circular feasible region (light blue filled) constrained by nonlinear conditions (x² + y² ≤ 4, x ≥ 0, y ≥ 0), with black solid lines as boundaries and labeled vertices; red contour lines of the objective function z = x² + 2y dynamically move to demonstrate the process of nonlinear optimization.

Evaluation guide:
(i) Target concept: Visual identification of the feasible region and objective function level-set dynamics in constrained nonlinear optimization.
Minimum mechanism: The quarter-disk must be the intersection of the disk x² + y² ≤ 4 with the first quadrant (x ≥ 0, y ≥ 0); each red contour line shown is the portion of the parabola x² + 2y = z (for one fixed z) that falls inside this quarter-disk, so on screen it appears as an open arc running between two edges of the region rather than a closed loop; because z = x² + 2y grows whenever x or y grows within the region, its smallest value is reached only at (0,0), so the animation must show z decreasing toward 0 while the contour lines shrink and migrate toward the origin.
(ii) Storyline: Frame 0–1: Coordinate system appears static and fully labeled. Frame 2–3: Black boundary lines draw sequentially (x-axis segment, y-axis segment, then the arc), all solid and connected. Frame 4: Light blue fill appears within the bounded region. Frame 5–6: Vertex labels (0,0), (2,0), (0,2) appear adjacent to their respective points. Frame 7–10: Red contour lines first appear near the outer arc boundary (large z), then shrink and migrate toward (0,0) as z decreases, with every contour line staying fully inside the quarter-disk and never crossing x<0 or y<0. Viewpoint is fixed frontal orthographic; no camera motion; all elements coexist in the same scene without layering violations.

Manual:
Experimental Procedure
1. Display a static 2D Cartesian coordinate system with labeled x- and y-axes.
2. Draw black solid boundary lines: the positive x-axis segment (y=0, x≥0), positive y-axis segment (x=0, y≥0), and the quarter-circle arc x² + y² = 4 in the first quadrant.
3. Fill the enclosed quarter-disk region with light blue color.
4. Label the three vertices: (0,0), (2,0), and (0,2).
5. Animate red contour lines of z = x² + 2y, starting from a large z near the outer arc and progressing monotonically downward, shrinking and converging toward (0,0) as z approaches its minimum value of 0.
Observation of Experimental Phenomena
- A fixed first-quadrant coordinate system shows a light blue quarter-disk bounded by two perpendicular black axes and a curved black arc.
- Three vertices at the origin and the two axis intercepts are clearly labeled with their coordinates.
- Red contour lines — open arcs that dip from a higher point down toward the x-axis — appear sequentially and shrink smoothly inward toward (0,0), staying fully within the light blue region.
Summary of Experimental Principles
The feasible region for the nonlinear constraints x² + y² ≤ 4, x ≥ 0, y ≥ 0 is a quarter-disk in ℝ². The objective function z = x² + 2y has level sets y = (z − x²)/2, which are parabolas; within the quarter-disk, each level set appears as an open arc rather than a closed curve. Since z increases whenever x or y increases within this region, the smallest z whose level set still touches the feasible region occurs at (0,0), so optimization is visualized as contour lines shrinking toward that point as z decreases.

Prompt Grounding rubric:
level_1: No coordinate system is visible; no quarter-disk region is shown; no black boundary lines are drawn; no vertex labels appear; no red contour lines appear anywhere in the video.
level_2: A 2D coordinate system is present, but the filled region is the wrong shape (e.g., a full circle, a square, or left unfilled) or the wrong color; at least one black boundary line is missing, mispositioned, or disconnected from the others; no vertex is labeled; red contour lines are absent or cannot be visually distinguished from the black boundary lines.
level_3: A 2D coordinate system is shown; a light-blue filled region appears roughly confined to the first quadrant; black boundary lines approximating the x-axis segment, the y-axis segment, and the arc x²+y²=4 are all present; at least two of the three vertices (0,0), (2,0), (0,2) are labeled; red contour lines are present and visually distinct from the boundary lines, though the region's edges or the vertex labeling may be imprecise.
level_4: All elements from 3-score are present with correct placement: the light-blue fill stays within x ≥ 0, y ≥ 0, and x² + y² ≤ 4 without spilling into another quadrant or past the arc; all three boundary edges (x=0 segment, y=0 segment, and the arc) are solid black and meet at shared endpoints; all three vertices (0,0), (2,0), (0,2) are labeled, though a label may sit on the wrong side of its point or use inconsistent styling; red contour lines are present, colored consistently, and clearly show more than one position or size across the video.
level_5: All elements from 4-score are present AND: the light-blue region's outline visually matches the quarter-disk with no visible gaps or overshoot at the three vertices; each vertex label reads clearly as (0,0), (2,0), or (0,2) and sits next to its point (exact font, kerning, or decimal formatting is not required); the red contour lines keep a consistent color and line weight and remain clearly distinguishable from both the black boundary lines and the light-blue fill throughout the video.

Scientific and Causal Correctness rubric:
level_1: The red contour lines bear no visible relation to z = x² + 2y (e.g., they are circles or ellipses centered away from the axes, or every line shown looks identical); or the light-blue region plainly violates the constraints, extending into x<0, y<0, or beyond x²+y²=4.
level_2: The red contour lines are recognizably curved consistent with the family x² + 2y = z (each dips from higher up down toward the x-axis as x increases), but the sequence of lines shown does not follow a single consistent size order (a later line is sometimes not clearly smaller or larger than an earlier one), or a line visibly closes into a loop instead of remaining an open arc, or the lines visibly shrink toward a point other than (0,0), such as (2,0) or (0,2).
level_3: Each red contour line visually matches the expected shape for x² + 2y = z inside the quarter-disk (an open arc, not a closed loop), and the sequence of lines shown is ordered by size; but one issue remains: either the arcs curve in the wrong orientation (e.g., dipping toward the y-axis instead of toward the x-axis), or the lines shrink toward a location other than (0,0).
level_4: All elements of 3-score are correct: contour lines are open arcs; smaller lines are visibly closer to (0,0) than larger ones, and the shrinking sequence visibly targets (0,0) specifically, not (2,0) or (0,2), but one small mismatch remains, such as a single line briefly extending past x<0 or y<0 before correcting itself, or two adjacent lines whose relative size is momentarily hard to tell apart.
level_5: All elements from 4-score are correct with no remaining mismatch: every red contour line at every moment is an open arc consistent with x² + 2y = z, staying fully inside the quarter-disk and touching its edge only where the arc meets a boundary; across the video the lines shrink continuously and exclusively toward (0,0), reflecting that z = x² + 2y reaches its lowest value only at the origin within this region; no line at any point suggests a different target point or closes into a loop.

Spatiotemporal Consistency rubric:
level_1: The coordinate system, the light-blue region, the boundary lines, or a vertex label appears, disappears, or changes position/size abruptly between shots; the viewpoint itself pans, rotates, or zooms; red contour lines pop in or out with no visible motion connecting one state to the next.
level_2: The coordinate system and vertex labels stay in place and the viewpoint is fixed, but the light-blue fill or a boundary line visibly flickers while appearing; red contour lines jump abruptly from one size or position to a clearly different one instead of gliding continuously; at least one vertex label separates from its point at some point in the video.
level_3: The coordinate system, boundary lines, and vertex labels stay fixed and stable throughout; the light-blue fill appears once and keeps a constant shape and color; red contour lines move along a continuous, visually smooth path, but the motion includes a stretch where a line stops moving for a noticeable pause, or its speed visibly lurches (suddenly faster or slower) instead of progressing steadily, or a line appears to poke outside the region before settling back inside it.
level_4: All elements from 3-score are satisfied, and in addition the red contour lines move at a steady, even pace with no pauses or sudden speed changes; every vertex label stays right next to its point for the whole video; the light-blue fill and black boundary lines show no flicker or shape distortion at any point, but one small blemish is visible somewhere, such as a boundary line's edge looking momentarily uneven or a contour line's leading edge looking slightly rough for an instant.
level_5: All elements from 4-score are satisfied with no visible blemish anywhere: the red contour lines glide continuously and evenly from their first appearance to their final position at (0,0); the light-blue region, black boundary lines, and vertex labels stay crisp, correctly placed, and free of flicker or momentary distortion for the entire video; nothing in the scene ever appears to jump, flash, or detach from where it belongs.