RoundPoint

Builds the round point (ax, ay, az, aw), unitized and zero-sanitized.

Builds the Euclidean point (ax, ay) with its conformal lift.

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

The e4 conformal component — (ax² + ay²)/2 for a plain Euclidean point.

The e1 component — the Euclidean x-coordinate when finite.

The e2 component — the Euclidean y-coordinate when finite.

The e3 round weight — 1 when finite; rounds to 0 at infinity (kept raw).

The flat point at this position — the carrier car(a) = e4 ∧ a.

The co-carrier, degenerate for a point — just its weight az.

The smallest circle containing this point — zero radius for a plain Euclidean point.

Whether this point lies at infinity — a direction with magnitude rather than a position.

Whether this point is imaginary — a negative squared radius, no real locus.

The Euclidean magnitude √(ax² + ay²) — raw.

The radius magnitude √(|radius2|) — raw, non-negative.

The squared radius — raw and signed; negative for an imaginary point.

This position as a FlatPoint — drops the conformal component.

This direction as a RoundDirection — the refinement carrying the at-infinity guarantee.

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

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

RoundPoint distanceTo Line => Double
Returns the raw perpendicular distance from the point's position to the line.

RoundPoint distanceTo RoundPoint => Double
Returns the raw Euclidean distance between the two points' positions.

RoundPoint 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.

RoundPoint 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.

RoundPoint dot RoundPoint => Double
Returns the raw inner product of two round points — zero when two finite positions coincide, −½·d² when they sit d apart.

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

RoundPoint expand Circle => Dipole
Returns the dipole holding the point and its inversion through the circle.

RoundPoint expand Dipole => Circle
Returns the circle through the point whose inversion swaps the dipole's endpoints.

RoundPoint expand FlatPoint => Circle
Returns the circle through the round point centered at the flat point.

RoundPoint expand Line => Dipole
Returns the dipole holding the point and its mirror image across the line.

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

RoundPoint join Segment => Circle
Returns the circle through the segment's endpoints and this point.

RoundPoint join Dipole => Circle
Returns the circle through the point and the dipole's endpoints.

RoundPoint join FlatPoint => Line
Returns the line through both points.

RoundPoint join RoundPoint => Dipole
Returns the dipole holding both points — this point as Dipole.start, other as Dipole.end.

RoundPoint - RoundPoint => Direction
Returns the free vector from other's position to this point's position: the displacement that carries other onto this point.

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

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

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

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

RoundPoint project Circle => RoundPoint
Returns the round point that pairs the point into the circle: on the radial line through the point, incident with the circle.

RoundPoint project Dipole => RoundPoint
Returns the point of the dipole's carrier dividing the endpoints in the ratio of their squared distances from the given point.

RoundPoint project Line => RoundPoint
Returns the point's perpendicular foot on the line, sized by the squared distance to it.

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

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