Case Study: Canvas-013

Module 1: Metadata Table

Metadata Item	Specification Details
Work ID	Canvas-013
Perceptual Image	Uphill and Downhill Slopes
Primary Structure	Layered Slope Paths
Thread Spec	DMC / No. 25 thread, 4 strands folded
Color Palette	Green, Off-white
Stitching Direction	Divided into two paths Phase ① / Phase ②, extending from top-left center to bottom-right
Syntax Type	Relief-accentuating Syntax

Module 2: Starting & Vector Operations

The structural arrangement consists of three major regional modules (Stitches 1–24 serve as the base unit, while Stitches 25–48 and 49–72 replicate the pattern shifted diagonally downward to the right).

Phase ①: Red Short Diagonal Module (1×1 Grid)

1. First Diagonal Sequence (Stitches 1–8):
   • 4 parallel short diagonal stitches (Stitches 1–2, 3–4, 5–6, 7–8), embroidered from top-right to bottom-left.
   • Single stitch vector: $(\Delta x, \Delta y) = (-1, +1)$, with a length of $\sqrt{1^2 + 1^2} = \sqrt{2}$ grid units.
   • Node coordinates (e.g., Stitches 1–2): Start $(x, y) \rightarrow$ End $(x-1, y+1)$, inclination angle $-45^\circ$.
2. Second Vertical Extension (Stitches 9–12):
   • 2 short diagonal stitches (Stitches 9–10, 11–12), connecting to the right side of the first sequence and extending downward along the Y-axis.

Phase ②: Blue Long Diagonal Module (2×2 Grid)

1. First Parallel Long Diagonal Sequence (Stitches 13–20):
   • 4 parallel long diagonal stitches (Stitches 13–14, 15–16, 17–18, 19–20). The starting points connect directly to the end points of the red stitches (e.g., Stitch 13 starts at the endpoint of Stitch 2 at 140, 100), spanning extensively toward the bottom-left.
   • Single stitch vector: $(\Delta x, \Delta y) = (-2, +2)$, with a length of $\sqrt{2^2 + 2^2} = 2\sqrt{2}$ grid units (twice the length of Phase ①).
2. Second Vertical Extension Long Sequence (Stitches 21–24):
   • 2 long diagonal stitches (Stitches 21–22, 23–24), stepped downward longitudinally to the right.

Block Replication Mechanism

• Unit 1 (Stitches 1–24): Establishes the base dual-color, dual-layer framework in the top-left area.
• Unit 2 (Stitches 25–48): Replicates the exact sequence of Stitches 1–24, offset by translation vector $(\Delta X, \Delta Y) = (+3, +3)$ grid units.
• Unit 3 (Stitches 49–72): Further replicates Stitches 1–24, offset by translation vector $(\Delta X, \Delta Y) = (+6, +6)$ grid units, creating cascading slope paths.

Module 3: Structural Mechanics & Topology

1. Dual-Layer Interlocking Topology:
   • Short diagonal stitches (Red) and long diagonal stitches (Blue) alternate at a $1:2$ span ratio. The short stitches anchor the fabric base densely, while the long stitches float on the surface as extended diagonal guides.
2. Stepped Backbone & Tension Balance:
   • Modules 25–48 and 49–72 overlap with a 3-grid displacement toward the bottom-right. Shared grid coordinates at transition points absorb and distribute thread tension evenly, preventing fabric hole distortion.
3. Relief Dynamics:
   • The high-relief texture of blue stitches (2-unit span) combined with the flat anchoring of red stitches (1-unit span) generates pronounced height variations, creating the visual image of “Uphill and Downhill Slopes.”

Module 4: Vector API & Dynamic Display Data

SVG Geometric & Vector Parameters

• ViewBox Dimensions: 0 0 720 600 (effective rendering area $640 \times 520\text{px}$).
• Grid System: $40 \times 40\text{px}$ per unit cell.
• Path ① (Red – Phase ① Short Diagonal Stitch):
  • Start-to-End Span: $\Delta X = -40\text{px}, \Delta Y = +40\text{px}$ (hypotenuse length $56.57\text{px}$).
  • Color Coding: #ef4444 (represents green main body in diagram).
• Path ② (Blue – Phase ② Long Diagonal Stitch):
  • Start-to-End Span: $\Delta X = -80\text{px}, \Delta Y = +80\text{px}$ (hypotenuse length $113.14\text{px}$).
  • Color Coding: #0284c7 (represents off-white layers in diagram).

Module 5: Visual Gallery

Front Face	Back Face	Draft Design
Front view of physical embroidery showcasing green and off-white relief texture.	Back structural view revealing clean stitching paths of Phase ① and Phase ②.	Original design draft detailing grid coordinate transitions between $3 \times 3$ and $2 \times 2$.

Module 6: Algorithm Extensions & Variant Syntax

Coverage & Metrics

• Physical Thread Length per Cycle:
  • Phase ①: $4 \times \sqrt{2} + 2 \times \sqrt{2} = 6\sqrt{2} \approx 8.495$ grid units.
  • Phase ②: $4 \times 2\sqrt{2} + 2 \times 2\sqrt{2} = 12\sqrt{2} \approx 16.971$ grid units.
• Physical Coverage Ratio: Reaches approx. $78.5\%$ density under dual-track interleaving, creating excellent depth variations and structural shadows.