Circle

Builds the circle (cx, cy, cz, cw), unitized and zero-sanitized.

Builds a classic circle with the given center and radius.

The purely directional part of this circle — a Dipole at infinity.

The canonical representative of this circle — useful for unoriented equivalence.

The carrier, saturated for a circle — the whole plane, as the antiscalar weight cw.

The center of the circle, carrying its size — a RoundPoint.

The co-carrier — the circle's center as a FlatPoint, dropping the size.

The smallest circle containing this circle — itself, orientation-normalized to inside-out.

The e321 round weight — −1 for a classic circle whose interior is the bounded disk, +1 for an inside-out circle whose interior is everything outside it.

The e423 component — the center's x-coordinate for a classic circle; negated when inside-out.

The e431 component — the center's y-coordinate for a classic circle; negated when inside-out.

The e412 component — minus half the origin's power with respect to the circle, (radius² − ‖center‖²)/2, for a classic circle; negated when inside-out.

Whether this circle is imaginary — a negative squared radius, no real points.

Whether this circle is inside-out — its interior is everything outside the bounded disk.

The center's distance from the origin √(cx² + cy²) — raw.

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

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

Circle antiproject Dipole => Circle
Returns the circle through the dipole's endpoints incident with this circle's pairing point on their perpendicular bisector.

Circle antiproject FlatPoint => Line
Returns the line through the point running the circle's way at the point's bearing — perpendicular to the radial through it.

Circle antiproject RoundPoint => Line
Returns the line through the point running the circle's way at the point's bearing — perpendicular to the radial through it.

Circle contains Circle => Boolean
Returns whether the other circle's region lies within the region this circle bounds, boundaries included — internal tangency and identical circles count.

Circle contains FlatPoint => Boolean
Returns whether the point lies in the circle's interior or on the circle.

Circle contains RoundPoint => Boolean
Returns whether the point lies in the circle's interior or on the circle.

Circle contract Dipole => RoundPoint
Returns the round point pairing the circle against the dipole, on the endpoints' perpendicular bisector.

Circle contract FlatPoint => RoundPoint
Returns the circle's tangent direction at the point's bearing, as a point at infinity.

Circle contract RoundPoint => Dipole
Returns the dipole where the circle crosses the radical axis between it and the point.

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

Circle dot Circle => Double
Returns the raw inner product of two circles — zero exactly when they cross at right angles.

Rounded, orientation-sensitive identity: equal when every component matches at the SCALE quantum.

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

Circle meet Circle => Dipole
Returns the dipole of the two points where the circles cross — imaginary when they don't touch.

Circle meet Line => Dipole
Returns the dipole of the two points where the circle crosses the line — imaginary when they miss.

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

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