Case Study: Canvas-012

Embroidery Algorithm & Topological Analysis

Module 1: Metadata Table

Item	Specification Details
Work ID	Canvas-012
Perceptual Image	Stepped 2.3.2 grid arrangement cutting into diagonal slopes like airplane wings, creating a sleek geometric rhythm
Basic Structure	2.3.2 grid line combination (wing structure) alternating with four 2-grid x 2 diagonal stitches
Thread Specification	DMC No. 25, 4 strands folded
Color Scheme	Sky Blue, Green, Ash White
Embroidery Direction	Executed in phases: green sections from bottom-left to top-right first, followed by central sky blue and ash white fills
Syntax Type	Wing & Diagonal Stitch Combination Type

Module 2: Starting Logic & Vector Operations

1. Fundamental Geometric Unit & Scalar Parameters

• Grid System: Canvas grid system set at 10 x 10 pixels per unit square.
• Single Stitch Vector & Inclination Angle:
  • Main Diagonal Stitch (-45° Axis): Vector span Δx = -40, Δy = +40 (length L = 40 × √2 ≈ 56.57 pixels) or Δx = -20, Δy = +20 (length L = 20 × √2 ≈ 28.28 pixels), inclination angle θ = -45°.
  • Wing Span Structure: Composed of 2-grid, 3-grid, and 2-grid segments, with a 1-grid gap (Gap = 10 pixels) maintained between adjacent 2.3.2 groups along the 45° slope.

2. Execution Phases & Sequence Logic

• Phase 1 (Corner Diagonals & Wing Skeleton):
  • Starting Point: Point 1 (50, 10).
  • Stitch Sequence: Execute segments 1-2, 3-4, 5-6 (Blue 2.3.2 wing group), then segments 7-8 through 13-14 (Yellow/Green group) to set the -45° wing orientation framework.
• Phase 2 (Corner Green Fills & Center Color Placement):
  • Fill bottom-left and top-right green areas along the 45° diagonal plane, then complete the central interlocking area using sky blue and ash white threads.

3. Array Copy & Transformation Operators

• Base Group (Segments 1-14): Contains segments 1 through 14 (7 segments total).
• Translation Group 1 (Segments 15-28):
  • Translation Vector: T1 = (+40, 0) (Shift right by 4 grid units).
  • Coordinates & Color: Segment 15 starts at (130, 10); Segment 28 ends at (150, 50). Segments 15-20 color changed to Red.
• Translation Group 2 (Segments 29-42):
  • Translation Vector: T2 = (+80, 0) (Shift right by 8 grid units).
  • Coordinates & Color: Segment 29 starts at (210, 10); Segment 42 ends at (230, 50). Color unchanged.
• Translation Group 3 (Segments 43-64):
  • Translation Vector: T3 = (-60, +20) (Shift left by 6 grid units, down by 2 grid units).
  • Coordinates & Color: Cloned from segments 21-42. Segment 43 starts at (30, 50); Segment 64 ends at (110, 90).
• Translation Group 4 (Segments 65-84):
  • Translation Vector: T4 = (+20, +20) (Shift right by 2 grid units, down by 2 grid units).
  • Coordinates & Color: Cloned from segments 15-34. Segment 65 starts at (170, 50); Segment 84 ends at (230, 90).
• Global Translation Group (Segments 85-168):
  • Translation Vector: T5 = (0, +40) (Shift down by 4 grid units).
  • Coordinates & Color: Cloned from segments 1-84. Segment 85 starts at (50, 90); Segment 168 ends at (230, 170). Specific segments reassigned colors (85-90, 113-118, 135-140, 153-168 changed to Red; 99-104, 149-154 changed to Blue).

Module 3: Structural Mechanics & Topology

1. Structural Backbone & Geometric Guidance

Canvas-012 pairs “2.3.2 grid wing line combinations” with “four 2-grid x 2 diagonal stitches” to create a strongly directional diagonal geometry. All stitches align along the -45° axis, establishing an aerodynamic wing-like topological network through alternating 2-grid and 3-grid segment lengths separated by 1-grid gaps.

2. Tension Balance & Stress Dispersion

• Axial Tension & Shear Stress: All diagonal stitches produce a pull vector F_shear towards the bottom-left (135°). Longer 3-grid stitches build higher tension, whereas shorter 2-grid stitches provide tighter node anchoring.
• Gap-Interleaved Stress Relief: The 1-grid blank space (Gap = 10 pixels) between 2.3.2 grid groups acts as a physical Stress Buffer Zone. This break interrupts continuous tension lines, preventing fabric puckering during long-distance continuous stitching.
• Dual-Diagonal Grid Anchoring: Through the geometric interleaving of corner green fills and central blue/white zones during Phase 2, tensile forces reach mechanical equilibrium across fabric boundaries, keeping the overall canvas exceptionally flat.

Module 4: Vector API & Dynamic Render Data

1. Interactive SVG Dynamic Card

2. SVG Vector Geometry Parameters

• ViewBox Definition: 0 0 250 180.
• Grid System: Base 10 x 10 pixel grid for precise coordinate alignment.
• Color Palette Codes:
  • Sky Blue: #1e88e5
  • Green: #43a047
  • Yellow: #fdd835
  • Red: #e53935

Module 5: Visual Gallery

Front View	Back View	Draft Chart
File ID: CP_012_4.jpg	File ID: CV_012B_4.jpg	File ID: CV_012H_4.jpg

Module 6: Algorithm Extensions & Variant Syntax

1. Physical Coverage Calculation

• Stitch Density: Geometric full-fill coverage rate reaches 92%.
• Total Thread Consumption: Approximately 520 mm per unit structure (calculated based on DMC No. 25 thread, 4 strands folded).

2. Procedural Parameter Definitions

• stepSequence = [2, 3, 2] (Wing group stepped span sequence)
• gapRatio = 0.20 (Diagonal wing connection gap ratio, fixed to 1 grid unit)
• unitSpacing = 4 units (Horizontal translation factor)
• globalShift = (0, +4) (Global vertical translation operator applied to 42 lines)