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Vector Network: Core Concepts and Terminology

This document defines the key mathematical and structural concepts used in the VectorNetwork model for representing vector graphics.

Overview​

  • A Vector Network is a graph-based structure for 2D vector shapes.
  • It generalizes SVG <path> data with explicit support for multiple subpaths, compound shapes, holes, and both linear and Bézier segments.
  • The core components are: vertices, segments, loops, regions, and fill rules.

Vertices​

  • Vertex: A point in 2D space, defined as (x, y) ∈ ℝ².
  • Vertices are indexed by position in the vertices list.
  • All segments reference vertices by their indices.

Segments​

  • Segment: An edge connecting two vertices, possibly with Bézier control tangents.
    • Defined by:
      • a: start vertex index
      • b: end vertex index
      • ta: (optional) tangent vector at start vertex (for Bézier control)
      • tb: (optional) tangent vector at end vertex (for Bézier control)
  • Types:
    • Linear: if both tangents are zero or omitted.
    • Cubic Bézier: if either tangent is nonzero.
  • Segments are directed (from a to b).

Loops​

  • Loop: A closed, ordered sequence of segment indices forming a contiguous path (contour) that starts and ends at the same vertex.
  • Each loop must be topologically closed (forms a ring).
  • Analogous to a "subpath" or a single closed shape in SVG.

Regions​

  • Region: A filled area defined by one or more loops.
    • The first loop: defines the outer boundary ("positive" area).
    • Subsequent loops: define holes ("negative" areas) inside the region.
  • A region may represent:
    • A simple shape (single loop, no holes)
    • A compound shape with holes (multiple loops)
    • Disjoint islands (multiple regions)

Fill Rules​

  • Fill Rule: Determines how overlapping loops and holes combine to define filled areas.
    • EvenOdd: A point is inside the region if it crosses an odd number of loops (parity rule).
    • NonZero: A point is inside if the sum of winding numbers (directional crossings) ≠ 0.
  • Fill rules affect how compound shapes and holes are rendered.

Planar vs Nonplanar Networks​

  • Planar Network: All segments and vertices lie in a single 2D plane, and no segments cross except at shared vertices.
  • Nonplanar Network: The graph may have crossing segments (not embeddable in the plane without intersections), which can lead to ambiguous or undefined regions.
  • The VectorNetwork model is primarily intended for planar networks, but does not enforce planarity.

Implicit Regions​

Vector Network Implicit Region

  • If no explicit regions are defined:
    • The network is interpreted as a set of open or closed paths (for stroking or outline rendering).
    • No fill is applied; only the connectivity of segments is used.
  • If regions are defined:
    • Only areas described by the regions and their fill rules are considered filled.