Case Study: Canvas-001

Computational Embroidery Syntax Card

Property	Specifications & Details
Archive ID	Canvas-001
Perceptual Image	Grid-counting geometric rhythm, smooth wheat-ear and diagonal woven texture, regular dense filling
Basic Structure	Surface filling, diagonal stitch ($2 \times 2$ grid half-cross fixed)
Thread Spec	DMC Stranded Cotton / No. 25, 4 strands folded
Color Palette	Gradient Green Shades (Variegated Green tones: Magic 3407/16, 3407/18, 3407/8, etc.)
Composition Vector	Left to Right, then Right to Left (S-shaped serpentine bi-directional filling)

Starting Rules:
- The embroiderer must strictly follow the grid coordinate system.
- Front Side: $2 \times 2$ grid diagonal half-cross stitch (Exit: $(x, y+2)$ $\rightarrow$ Enter: $(x+2, y)$).
- Back Side: 2-grid vertical stitch (Enter: $(x+2, y)$ $\rightarrow$ Exit: $(x+2, y+2)$).
- Execution Order: Stitch a single $2 \times 2$ diagonal unit $\rightarrow$ Repeat $2 \times 2$ diagonal units horizontally across the row $\rightarrow$ Upon reaching the row end, reverse direction and fill the next row from right to left.
Key Visual Property: Single $2 \times 2$ diagonal unit. To maintain the hand-drawn draft aesthetic, the diagonal line ends do not join directly at grid intersections, leaving a gap offset of approximately $18\%$.
Polar Coordinate Vectors & Angle Operations (Syntax Type: Grid Counting / 45° Diagonal Pure Shear Array):
- Standard Diagonal Half-Stitch Vector A: Angle $45^\circ$, physical length $6.1\text{ mm}$ (Diagonal of a $2 \times 2$ square unit: $\sqrt{2^2 + 2^2} \times 2.15\text{ mm} \approx 6.08\text{ mm}$).
- Back-Side Vertical Anchor Vector B: Angle $270^\circ$ (straight downwards), physical length $4.3\text{ mm}$ ($2\text{-grid}$ span).
- Terminal Gap Ratio Offset: Both ends of the diagonal vector shrink by $18\%$ to create hand-drawn independent stitch gaps.

Core Topological Node Network:
- Formed by square array nodes $P_{(i,j)} = (2i, 2j)$ and $Q_{(i,j)} = (2i+2, 2j+2)$, constructing an $N \times M$ parallel diagonal tension network.
- Alternating front diagonal stitches $\slash$ ($45^\circ$) and back vertical stitches $\vert$ ($270^\circ$) form a closed, single-direction inclined tension zigzag chain.
Structural Mechanics Analysis:
- Front $45^\circ$ Diagonal Shear Tension: Distributes diagonal shear stress evenly across the fabric surface, creating an extraordinarily flat and uniform filling texture.
- Back $90^\circ$ Vertical Structural Support: Vertical line segments on the back absorb lateral tension, preventing fabric curling/warping and maintaining self-balancing tension across the textile.

準備運針中…

ViewBox: 0 0 400 400
Grid Cell Standard: 2×2 grid units=40×40 px (with 18% Gap Ratio Offset)
Stitch Path 1 (Row 1, Unit 1): Start (37.2, 62.8) -> End (62.8, 37.2) | Angle: 45∘ | Color: #3c6e47
Stitch Path 2 (Row 1, Unit 2): Start (77.2, 62.8) -> End (102.8, 37.2) | Angle: 45∘ | Color: #3c6e47
Stitch Path 3 (Row 1, Unit 3): Start (117.2, 62.8) -> End (142.8, 37.2) | Angle: 45∘ | Color: #3c6e47
Stitch Path 4 (Row 2, Unit 1 [Reverse Direction]): Start (117.2, 102.8) -> End (142.8, 77.2) | Angle: 45∘ | Color: #2e5b37

Thread Physics & Fabric Coverage:
- Mesh Specification: Optimized for 18-count Mono Canvas (mesh grid spacing $\approx 1.41\text{ mm}$).
- Thread Bulk & Coverage Rate: DMC No. 25 thread (4 strands folded, equivalent to 8 strands effective width) achieves a fabric coverage rate of $95\% – 98\%$ over a $2 \times 2$ grid, providing an exceptionally dense and solid fill.
Stitch Variant Matrix & Syntax Switching:
- Full Cross Stitch Transition:
  - Base Layer (Under-stitch): Execute Canvas-001 $2 \times 2$ diagonal half-cross stitch ($45^\circ$, vector $(2, -2)$).
  - Top Layer (Over-stitch): Superimpose a $2 \times 2$ reverse diagonal stitch ($135^\circ$, vector $(-2, -2)$).
- Gobelin Stitch Extension (Elongated Diagonal):
  - Vertical Expansion: Alter the $2 \times 2$ matrix to $2 \times 3$ or $2 \times 4$ aspect ratios. Vector angles shift from $45^\circ$ to $56.3^\circ$ or $63.4^\circ$, optimized for varying directional texture fills.
Procedural Parameters for Hand-Drawn Aesthetics:
- gapRatio (Terminal Gap Offset): Default set to $0.18$ (Adjustable range: $0.10 – 0.25$). Higher values emphasize the independent, hand-drawn draft aesthetic; a value of $0$ reverts to a continuous industrial line render.