FlatPoint

Builds the flat point (px, py, pz), unitized and zero-sanitized.

The attitude, degenerate for a point — pure weight, no direction.

Whether this point lies at infinity — a free vector rather than a position.

The Euclidean magnitude √(px² + py²) — raw.

The e41 component — the Euclidean x-coordinate when finite.

The e42 component — the Euclidean y-coordinate when finite.

The e43 weight — 1 when finite; rounds to 0 at infinity (kept raw).

This free vector as a Direction — the refinement carrying the at-infinity guarantee.

This position as a RoundPoint — the conformal lift.

FlatPoint contract RoundPoint => RoundPoint
Returns the flat point's position as a round point sized by minus the squared distance to the given point.

Returns this object rebuilt around the replacement geometry flatPoint — the hook the shape-preserving operations return through.

FlatPoint distanceTo FlatPoint => Double
Returns the raw Euclidean distance between the two points.

FlatPoint distanceTo Line => Double
Returns the raw perpendicular distance from the point to the line.

FlatPoint distanceTo RoundPoint => Double
Returns the raw Euclidean distance between the flat point and the round point's position.

FlatPoint dot Circle => Double
Returns the raw point-against-circle probe: positive inside, zero on the circle, negative outside, for a finite point and a classic circle.

FlatPoint dot FlatPoint => Double
Returns the raw inner product of two flat points — zero when they coincide, −½·d² for finite points d apart.

FlatPoint dot Line => Double
Returns the line's equation evaluated at the point: zero when the point lies on the line, the raw signed distance to it otherwise.

Rounded identity: equal when every component matches at the SCALE quantum.

FlatPoint expand Circle => Line
Returns the line through the point and the circle's center.

FlatPoint expand Line => Line
Returns the line through the point perpendicular to the given line.

Returns the hash of the rounded components, consistent with equals.

FlatPoint join FlatPoint => Line
Returns the line through both points, oriented so its normal points to the left of the travel from this point to other.

FlatPoint join RoundPoint => Line
Returns the line through the flat point and the round point.

FlatPoint - FlatPoint => Direction
Returns the free vector from other to this point: the displacement that carries other onto this point.

FlatPoint on Polygon => Boolean
Returns whether the point lies on the polygon's boundary.

FlatPoint on Segment => Boolean
Returns whether the point lies on the segment itself, endpoints included.

FlatPoint on Circle => Boolean
Returns whether the point lies on the circle.

FlatPoint on Line => Boolean
Returns whether the point lies on the line.

FlatPoint project Segment => FlatPoint
Returns the point of the segment nearest to this point: the perpendicular foot when it falls within the segment, else the nearer endpoint.

FlatPoint project Circle => Dipole
Returns the dipole where the circle crosses the radial line through the point.

FlatPoint project Line => FlatPoint
Returns the point's perpendicular foot on the line.

Returns the raw components — full stored precision, no rounding.

Returns the components rounded at the SCALE quantum — the debug-output convention of every primitive.