Constraints

Constraints express geometric relationships between elements. They can appear as standalone statements or inline after with ... and ... on a shape declaration.

triangle abc with ab = 3 and bc = 4 and ca = 5   # inline
ab = 3                                             # standalone

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Length

Sets the distance between two vertices. Units are optional — omitting them uses abstract units.

ab = 5
ab = 5cm
ab = 5mm
ab = 2.5in

Available units: cm, mm, m, in, inches.

click to pan & zoom

triangle abc with ab = 3 and bc = 4 and ca = 5 pick c 1

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Coincidence

Declares that two vertices occupy the same position.

a = b

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On and through — the matrix

Two operators express positional constraints between elements:

• X on Y — X (a point) lies on Y (a locus: line, segment, circle).
• X through Y — X (a line or circle) passes through Y (a point).

Both are forms of the same underlying relation. They desugar to the same internal constraint at parse time, so the choice is purely about readability: point p on line l and line l through point p say the same thing.

Here's what each pair of LHS/RHS element types means:

LHS / RHS	point	line	circle	segment
point on …	—	point on line	point on circle	point on segment
line on …	(error: line isn't a point)	—	—	—
circle on …	(error)	circle's centre is on line	(error — not supported)	circle's centre is on segment
line through …	line passes through point	—	line passes through circle's centre	—
circle through …	circle passes through point	—	(error — not supported)	—

The non-empty cells are the constraints you can write. Everything else is rejected at elaboration.

Circle conventions. When a circle name appears in a point position (the LHS of on, or the RHS of through), it stands in for its centre. So:

c on l         # c's centre is on l
l through c    # same constraint, read from l's side
c on ab        # c's centre is on segment ab
l through c, d # both c's and d's centres are on l

Two circles can't relate this way: c1 on c2 and c1 through c2 are rejected because there's no obvious one-point reading. (To express "the centre of c1 lies on c2" you'd write c1.center on c2 — though there's no such syntax today; you can express it by giving the circle a named centre point and constraining that point directly.)

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On-line

Constrains a point to lie on a named line. The point's position along the line is underconstrained unless a distance to a known neighbor pins it down.

p on l
p on line l
point p on l
point p on line l

The same constraint can be expressed from the line's side using through:

line l through p          # inline on the line declaration
line l through p, q       # multiple points inline
l through p               # standalone statement

All forms are equivalent — through is desugared to on at parse time.

click to pan & zoom

line l = (1, -1, 0) point a = (0, 0) b on l ab = 3 pick b 1

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On-circle

Constrains a point to lie on a circle. With one circle, the point's position along the circle is underconstrained; combine with a second locus (line or another circle) to pin it down.

p on c
point p on c

The same constraint can be expressed from the circle's side using through:

circle c through p             # inline on the circle declaration
circle c = (o, 1) through p, q # multiple points inline
c through p                    # standalone statement

When three or more points are known to lie on the same circle, the circle's centre and radius are uniquely determined and the solver places them automatically.

A circle written in a point position — c on l or l through c — stands in for its centre. See On and through — the matrix above.

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On-segment

Constrains a point to lie somewhere on a segment. Its position along the segment is always underconstrained — free to be anywhere on it. Multiple underconstrained points on the same segment are distributed evenly.

p on ab
p on segment ab
point p on ab

click to pan & zoom

segment ab = 6 p on ab q on ab r on ab

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Equality

Coming soon

Two segments share the same length without specifying a value.

ab = cd

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Angle

Coming soon

Sets the interior angle at a vertex.

angle abc = 90
angle abc = 1.571rad

Available units: deg, rad. Required for squares, rectangles, and right-angle constructions.

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Parallel

Constrains two lines to have the same direction.

l parallel m
line l parallel m         # inline on the line declaration
line l parallel line m    # optional 'line' hints

Add at to specify the distance between the lines. This produces two symmetric solutions — one on each side.

line l parallel m at 3    # l is 3 units from m (two solutions)

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Perpendicular

Constrains two lines to meet at a right angle.

l perpendicular m
line l perpendicular m    # inline on the line declaration

Add at to name the intersection point — it is placed exactly at the crossing of the two lines.

line l perpendicular m at p   # l ⊥ m, p is their intersection