Placement

The heart of the solver is a loop with two halves: propagate finds every placement that the constraints uniquely determine, and pick places one element when nothing is forced. They alternate until both are quiescent.

The loop

A pick can unlock more propagation: placing one vertex by a canonical fixer may give another vertex a second known neighbour, which then resolves exactly via propagate. The loop is guaranteed to terminate — every iteration either places an element or exits.

Propagate — forced placements

A placement is forced when the constraints uniquely determine the element. No canonical choice, no representative pick. Every forced rule fires in priority order until nothing more can be placed.

Every constraint on a vertex defines a locus — the set of positions where that vertex could legally sit.

Constraint	Locus
Known distance from a placed point	Circle
On a named line	Line
On a segment (both endpoints placed)	Line segment

A vertex with two or more loci is placed at their intersection (finite solutions, dof = 0). Lines work the same way — a line with a placed point and a known coefficient pair is exactly determined.

1. Line ∩ Line

Condition: vertex is constrained to two or more named lines.

Two lines in general position intersect at exactly one point — no placed neighbours are needed. This fires before the circle variants.

Given lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0, the intersection is found via Cramer's rule:

x = (b₁c₂ − b₂c₁) / det
y = (a₂c₁ − a₁c₂) / det
where det = a₁b₂ − a₂b₁

If det ≈ 0 the lines are parallel and the solver throws a constraint error.

When three or more lines are specified, the solver intersects the first two to get a candidate point, then verifies that the candidate satisfies every remaining line equation. If any line does not pass through that point, the constraints are inconsistent and the solver throws.

2. Circle ∩ Circle

Condition: two or more placed neighbours each connected by a known distance.

Each known distance draws a circle around its placed neighbour. The vertex lies at the intersection of those circles.

      a ·····
       ·     ·····
        ·          ·
   r₁   ·      s₁ ×  ← solution 1 (left of a→b)
         ·    ×
          ·  s₂     ← solution 2 (right of a→b)
           ·
            b ·····

Given placed points a and b at distance d, with radii r₁ and r₂:

1. Distance along a→b to the radical axis: A = (r₁² − r₂² + d²) / 2d
2. Perpendicular offset: h = √(r₁² − A²)
3. Solutions: m ± h·n̂ where m is the point along ab at distance A from a, and n̂ is the unit normal to ab

Solution 1 is left of a→b (counter-clockwise). Solution 2 is right (clockwise).

3. Circle ∩ Line

Condition: vertex is on a named line, and has at least one placed neighbour with a known distance.

The named line and the distance circle intersect at up to two points.

Given placed neighbour p at distance r, and line ax + by + c = 0:

1. Project p onto the line → foot f
2. Distance from p to line: d = |ap_x + bp_y + c| / √(a² + b²)
3. Offset along the line: h = √(r² − d²)
4. Solutions: f ± h·t̂ where t̂ is the unit tangent of the line

Solution 1 has the higher y-coordinate (or larger x if y values are equal).

Both circle rules: if the two loci are tangent, there is exactly one solution. If they do not reach each other, the solver throws a constraint error.

4. Line completion

Condition: a named line has placed points on it.

If two or more distinct placed points are on the line, both direction and position are determined. If one placed point plus a known coefficient pair are on the line, the missing coefficient is solved exactly.

5. Direction propagation

Condition: a line is declared parallel or perpendicular to another line whose direction is already known.

The direction of the partner copies over. If parallel at d is specified and the partner is fully resolved, position is determined too (two solutions, one on each side).

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Pick — one element at a time

When propagate stops firing but the model still has unplaced elements, pick takes over. Each pick call places one element, then propagate runs again to chase the consequences.

A pick has two sub-cases:

1. Canonical — a global degree of freedom (T, R, or S) is still free. Consuming it places the element at a canonical position (dof = 0). This is the anchor machinery.
2. Representative — no gauge is available. The element is placed at a representative position along its free locus, marked as underconstrained (dof > 0).

Representative — locus

1 placed neighbour with a known distance. The vertex lies somewhere on the circle around that neighbour, but the angle is unconstrained. The solver uses a rotating heading: starts along +x, then rotates 90° counter-clockwise after each such placement (+x → +y → −x → −y → repeat).

The rotation prevents degenerate layouts where everything collapses onto a line. A rhombus with only side lengths would place all four vertices in a row without it; the rotation produces a square-like layout instead.

On a named line, no distance neighbours yet. The vertex lies somewhere on the line. The solver places it at the foot of the perpendicular from the origin to the line — the closest point on the line to (0, 0).

On a segment, both endpoints already placed. The vertex lies somewhere along the segment. If multiple vertices share the same segment, they are distributed evenly:

a ──── p₁ ──── p₂ ──── p₃ ──── b
       t=¼     t=½     t=¾

For n unplaced vertices, vertex i is placed at t = (i+1) / (n+1).

Representative — fallback

When even a locus is unavailable, the solver still has to place the element somewhere visible.

Segment neighbour. The vertex shares a segment with a placed neighbour but no length is known. Placed 3 units along +x from that neighbour as a placeholder.

Isolated. The vertex has no connection to any placed vertex at all — it belongs to a disconnected component of the constraint graph. Isolated vertices are seeded one at a time, stacked vertically below the main figure so they do not overlap. Each seeded vertex then allows its neighbours to be reached in the next loop iteration.

Line default. A bare line with no constraints is canonicalized to y = x. A line with a single placed point gets the canonical slope and solves c from the point. Partial lines (one coefficient unknown) fill the missing coefficient with the canonical default.

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DOF propagation

The dof value controls rendering: dof = 0 gives a crisp dot and solid lines; dof > 0 gives a wavy circle and squiggly edges.

dof propagates through the constraint graph: when a vertex is placed by intersecting loci, it inherits dof from those loci. If a placed neighbour was underconstrained, the new vertex is underconstrained too — its position is only determined relative to something that was already moving.

Elements placed by representative rules are always underconstrained (dof > 0). Elements placed by a canonical fixer (a consumed gauge) are determined (dof = 0) — the canonical position represents the entire equivalence class under that gauge.

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Multiple solutions and pick

When two loci intersect at two distinct points and no pick statement has been declared, both positions are stored and shown on the canvas, numbered 1 and 2. Edges connected to an ambiguous vertex are drawn for every combination.

pick v 1   ← use solution 1
pick v 2   ← use solution 2

A picked vertex is treated as determined from that point forward.