Constraint Model

Before the solver loop can run, every statement in the program gets walked once to build the constraint model — a graph of vertices and the relationships between them. This is the data structure the placement loop works from.

What gets built

The model holds:

Structure	What it stores
points	Every declared vertex, initially unplaced (dof = 2, coordinates null)
segments	The set of all declared edges, as canonical pairs
lengths	A length value for each segment that has one, or null
lines	Named lines, stored as (a, b, c) where any coefficient may be null for partial/bare declarations
onLine	Which vertex is constrained to which named line(s)
onSegment	Which vertex is constrained to lie on which segment
solutionPicks	Which solution index was chosen for each ambiguous vertex

How shapes become vertices and edges

A shape declaration registers its vertices and the edges between them. No positions are assigned yet — that's the placement loop's job.

let triangle abc

This registers vertices a, b, c and three edges: a–b, b–c, c–a.

let segment ab

Registers vertices a, b and one edge: a–b.

Inline length constraints attach length values to the edges at the same time:

triangle abc with ab = 3 and bc = 4 and ca = 5

Registers the triangle and sets |ab| = 3, |bc| = 4, |ca| = 5.

How constraints layer on top

Constraints declared separately (via with blocks or standalone statements) add information to what's already registered. A length constraint sets a length on an existing or new edge. An on-line or on-segment constraint records that a vertex must lie on a specific line or segment.

Pick statements

pick v 2 tells the solver that when vertex v has two possible positions, it should use position 2. These are recorded into the constraint model and consulted during placement.

What the model captures

The model does assign coordinates to explicitly placed vertices — point a = (3, 4) or a = (3, 4) inside a with block resolves immediately to dof = 0. Every other vertex enters the loop as an unplaced working point: both coordinates null, dof = 2. The constraint model is purely relational for those: it says what exists and what constraints apply, not where anything is.