on
FlatPoint on Line => Boolean
Returns whether the point lies on the line.
True when this dot line — the signed Euclidean distance for finite operands — rounds to zero at the SCALE quantum. The gate is all-linear, robust at the quantum. A direction (at infinity) counts as on the line when it runs along it.
Receiver
The point to test.
Return
true when the rounded signed distance is zero.
Parameters
The line tested against.
FlatPoint on Circle => Boolean
Returns whether the point lies on the circle.
True when this dot circle — the raw probe ½·(r² − d²) against a classic circle — rounds to zero at the SCALE quantum. Orientation-blind: an inside-out circle negates the probe, which leaves the zero test unchanged. Against an imaginary circle the raw probe is negative everywhere. The probe is quadratic, reached through the conformal lift, so its residual grows with the radius — a point placed within the quantum of a large circle's boundary may still fail the test; callers wanting a "near" test need their own tolerance.
Receiver
The point to test.
Return
true when the rounded probe is zero.
Parameters
The circle tested against.
FlatPoint on Segment => Boolean
Returns whether the point lies on the segment itself, endpoints included.
Two gates, both all-linear and rounded at the SCALE quantum: the point must lie on the segment's Segment.carrier, and its rounded distance to the Segment.cocarrier — the perpendicular bisector of the endpoints — must not exceed the rounded half-length Segment.radius.
Receiver
The point to test.
Return
true when both rounded gates pass.
Parameters
The segment tested against.
FlatPoint on Polygon => Boolean
Returns whether the point lies on the polygon's boundary.
A disjunction of FlatPoint on Segment probes over the polygon's Polygon.edges, each all-linear and rounded at the SCALE quantum.
Receiver
The point to test.
Return
true when the point is on at least one edge.
Parameters
The polygon whose boundary is tested against.
RoundPoint on Line => Boolean
Returns whether the point lies on the line.
True when this dot line rounds to zero at the SCALE quantum — the same all-linear gate as FlatPoint on Line; the dot never reads the round bulk RoundPoint.aw, so the flat and round forms agree exactly.
Receiver
The point to test.
Return
true when the rounded signed distance is zero.
Parameters
The line tested against.
RoundPoint on Circle => Boolean
Returns whether the point lies on the circle.
True when this dot circle rounds to zero at the SCALE quantum — the same orientation-blind quadratic gate as FlatPoint on Circle, whose residual grows with the radius. The pairing reads the stored round bulk RoundPoint.aw, so the boundary reading holds for null-lifted points; a semantic aw (a center, an attitude) feeds that data into the probe instead.
Receiver
The point to test.
Return
true when the rounded probe is zero.
Parameters
The circle tested against.
RoundPoint on Segment => Boolean
Returns whether the point lies on the segment itself, endpoints included.
The same two all-linear gates as FlatPoint on Segment, rounded at the SCALE quantum; neither dot reads the round bulk RoundPoint.aw, so the flat and round forms agree exactly.
Receiver
The point to test.
Return
true when both rounded gates pass.
Parameters
The segment tested against.
RoundPoint on Polygon => Boolean
Returns whether the point lies on the polygon's boundary.
A disjunction of RoundPoint on Segment probes over the polygon's Polygon.edges, each all-linear and rounded at the SCALE quantum.
Receiver
The point to test.
Return
true when the point is on at least one edge.
Parameters
The polygon whose boundary is tested against.