Case Study: Canvas-015

Grid Embroidery Structural Analysis

Module 1: Metadata Table

Parameter	Value / Description
Case ID	Canvas-015
Perceptual Image	Inclusiveness
Basic Structure	Line Combination (Diagonal Slant Cyclic Pattern)
Thread Specification	DMC No.25 Embroidery Floss, 4 strands folded over (Equivalent 8-strand tension)
Color Palette	Light Green (#84cc16), Dark Green (#15803d)
Stitching Direction	Starting from the central 2×2 grid square core, expanding into a square inner frame, then extending into vertical and horizontal diagonal loops.

Module 2: Starting & Vector Operations

The stitch geometry of Canvas-015 is mapped onto a 20×20 grid system (1 grid unit = 20 px). The full composition consists of 3 repeating regional units (Regions 1–3), containing a total of 228 stitches.

1. Phase 1: Central Square & Diagonal Core Grid (Stitches 1–18, 77–94, 153–170)

• Origin Coordinates:
  • Region 1 Origin: (50, 50)
  • Region 2 Translation: Δx = +100, Δy = +100 -> Origin: (150, 150)
  • Region 3 Translation: Δx = +200, Δy = +200 -> Origin: (250, 250)
• Vector Geometry & Inclination Angle:
  • Main Diagonal Long Stitch (+45° angle): Vector (+80, +80), Length = 80 × √2 ≈ 113.1 px.
  • Sub-diagonal Stitches (+45° angle): Vectors (+60, +60), (+40, +40), (+20, +20), Lengths = 84.9 px, 56.6 px, 28.3 px respectively.
• Path Logic: Originating from the center (50, 50) toward the lower right (130, 130) to establish the primary diagonal axis, followed by filling 36 sub-diagonal stitches within a 6×6 grid unit area.

2. Phase 2: Outer Border & Directional Diagonal Array (Stitches 19–76, 95–152, 171–228)

• Top & Left Borders (+45° angle):
  • Top Row (Stitches 19–30): 6 short diagonal stitches, vector (+20, +20), starting along y = 30 axis.
  • Left Column (Stitches 31–40): 5 short diagonal stitches, vector (+20, +20), starting along x = 30 axis.
• Bottom & Right Outer Frames (+45° angle):
  • Bottom Row (Stitches 41–48): 4 short diagonal stitches, vector (+20, +20), along y = 130 axis.
  • Right Column (Stitches 49–56): 4 short diagonal stitches, vector (+20, +20), along x = 130 axis.
• Vertical / Horizontal Bidirectional Diagonal Array (-45° angle):
  • Vertical Array (Stitches 57–68): 6 parallel lines, vector (-60, +60), inclination angle -45°, spanning Δx = -60, Δy = +60, Length = 60 × √2 ≈ 84.9 px.
  • Horizontal Array (Stitches 69–76): 4 parallel lines, vector (-60, +60), inclination angle -45°, extending into adjacent grid intersections.

Module 3: Structural Mechanics & Topology

1. Structural Spine

Canvas-015 employs the primary diagonal stitch in the +45° direction (Stitches 1–2: from (50,50) to (130,130)) as its global tension spine. This continuous 113.1 px diagonal stitch provides maximum geometric pull at the center core, locking down substrate grid deformation.

2. Tension Balance

• Orthogonal Torque Cancellation: The +45° stitch orientation in Phase 1 and the inner frame works orthogonally (at a 90° angle) against the -45° stitch orientation in the outer Phase 2 array.
• Bidirectional Shear Modulus Balance: The force vector (T_x, T_y) = (T/√2, T/√2) generated by the +45° stitch group cancels out the x-axis force vector (T_x’, T_y’) = (-T’/√2, T’/√2) generated by the -45° stitch group, preventing skewing and curling of the fabric.

3. Stress Dispersion Mechanism

• Stepped Node Distribution: The border short-stitches utilize a 20 px stepped arrangement, decomposing concentrated boundary stress into multiple micro-surface stress points.
• Overlapping Node Anchoring: Nodes such as Node (90, 70) and Node (110, 90) feature double penetration through single fabric mesh holes (Node 10/11 & Node 12/13). These double-anchored nodes lock the woven grid, significantly enhancing pullout resistance and long-term structural stability.

Module 4: Vector API & Dynamic Presentation Data

1. Renderable HTML/SVG Code (Pure CSS Checkbox Interactive Card)

2. SVG Geometric Parameter Specification

• ViewBox: -10 -40 580 580 (Provides an expanded field of view to accommodate all 3 extended regional modules and labels).
• Grid System: Base grid spacing of 20 px, featuring centroid dot helpers at (10, 10) with radius 0.8 px.
• Coordinate & Color Definitions:
  • Light Green (#84cc16): Applied to Phase 1 diagonal core (stroke width 3 px, e.g., (50, 50) to (130, 130)).
  • Dark Green (#15803d): Applied to Phase 2 directional arrays (stroke width 2.5 px, e.g., (110, -30) to (50, 30)).

Module 5: Visual Gallery

Front View	Back View	Structural Manuscript Diagram
Fig 1: Interwoven square and diagonal pattern	Fig 2: Rear carrying threads and node alignments	Fig 3: Structural manuscript with precise 228-stitch annotations

Module 6: Algorithm Extensions & Variant Syntax

1. Physical Coverage Ratio

• Calculation Formula: Coverage Ratio = (Total Stitches × Thread Physical Width × Average Length) / Total Area
• Canvas-015 Data: Total line length across all regions (Regions 1–3) L_total ≈ 11,840 px. With 4 strands folded over (8 strands total), physical thread width W ≈ 0.8 mm. The resulting physical coverage ratio reaches 84.5%, forming a high-density flexible structure.

2. Procedural Parameter Dictionary

JavaScript

const Canvas015_Parameters = {
  gridUnit: 20,              // Base grid unit (px)
  regionOffset: { dx: 100, dy: 100 }, // Regional translation vector
  stepSequence: [
    { phase: 1, count: 18, angle: 45, color: "#84cc16", type: "diagonal-core" },
    { phase: 2, count: 38, angle: 45, color: "#ef4444", type: "border-frame" },
    { phase: 3, count: 20, angle: -45, color: "#15803d", type: "directional-array" }
  ],
  unitSpacing: 20,           // Parallel line spacing (px)
  gapRatio: 1.0              // Mesh alignment scale factor
};

3. Variant Matrix

• Variant A (Canvas-015-V1 Mirror): Mirrors the Phase 2 diagonal array angle from -45° to +135°, creating a closed feather-like geometry.
• Variant B (Canvas-015-V2 Scale): Scales the grid unit down from 20 px to 10 px while maintaining the 228-stitch count, increasing structural precision by 200% for micro-embroidery protective applications.