antiproject
Line antiproject FlatPoint => Line
Returns the parallel to the line through the point.
The antiprojection flatPoint ∧ (g ∨ flatPoint^☆), normalized by the flat point's self-pairing flatPoint ∘ flatPoint so the parallel keeps this line's weight and orientation — its normal points the same way. A point on the line antiprojects the line onto itself, and a line at infinity antiprojects to the all-zero line, which the Line constructor accepts. A point at infinity is not rejected: its self-pairing vanishes and the division floods the components.
Receiver
The line whose direction is carried over.
Return
The raw Line through the point, parallel to this line.
Parameters
The point the parallel passes through.
Line antiproject RoundPoint => Line
Returns the parallel to the line through the point.
Line antiproject FlatPoint through RoundPoint.toFlat — the antiprojection reads only the target's location, so the round form delegates to the flat one; a conformal component carrying a size plays no part. A point at infinity is not rejected: the delegated division floods the components.
Receiver
The line whose direction is carried over.
Return
The raw Line through the point, parallel to this line.
Parameters
The point the parallel passes through.
Line antiproject Dipole => Circle
Returns the circle through the dipole's endpoints incident with the line's crossing of their perpendicular bisector.
The antiprojection dipole ∧ (g ∨ dipole^☆), normalized by the dipole's self-pairing dipole ∘ dipole — its raw signed Dipole.radius2, so an imaginary dipole negates every component. The circle passes through the dipole's endpoints, and its raw pairing with the sized crossing point g contract dipole vanishes. A line perpendicular to the carrier bends into the circle with the endpoints as a diameter. A dipole with coincident endpoints is not rejected: its self-pairing vanishes and the division floods the components.
Receiver
The line bent through the dipole.
Return
The raw Circle through the dipole's endpoints.
Parameters
The dipole the circle passes through.
Throws
When the line parallels the dipole's carrier at the SCALE quantum — the bent circle straightens into the carrier, carried as the payload.
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.
The antiprojection flatPoint ∧ (c ∨ flatPoint^☆), normalized by the flat point's self-pairing flatPoint ∘ flatPoint. The result is a line, not a circle: every flat-point blade carries a factor of e4, so the outer wedge can never rebuild a round weight — the flat limit of carrying the circle's attitude over to the point. Its normal points from the point toward the circle's center, so a point on the circle antiprojects to the tangent there, oriented inward. A point at the circle's center antiprojects to the all-zero line, which the Line constructor accepts; a point at infinity is not rejected: its self-pairing vanishes and the division floods the components.
Receiver
The circle whose bearing is carried over.
Return
The raw Line through the point, perpendicular to its radial; all-zero at the circle's center.
Parameters
The point the line passes through.
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 antiproject FlatPoint through RoundPoint.toFlat — the antiprojection reads only the target's location, so the round form delegates to the flat one; a conformal component carrying a size plays no part. A point at infinity is not rejected: the delegated division floods the components.
Receiver
The circle whose bearing is carried over.
Return
The raw Line through the point, perpendicular to its radial.
Parameters
The point the line passes through.
Circle antiproject Dipole => Circle
Returns the circle through the dipole's endpoints incident with this circle's pairing point on their perpendicular bisector.
The antiprojection dipole ∧ (c ∨ dipole^☆), normalized by the dipole's self-pairing dipole ∘ dipole — its raw signed Dipole.radius2, so an imaginary dipole negates every component. The result passes through the dipole's endpoints and its raw pairing with the sized point c contract dipole vanishes; a circle already through the endpoints antiprojects to itself. A dipole with coincident endpoints is not rejected: its self-pairing vanishes and the division floods the components.
Receiver
The circle bent through the dipole.
Return
The raw Circle through the dipole's endpoints.
Parameters
The dipole the circle passes through.
Throws
When the round weight collapses at the SCALE quantum — the bent circle straightens into a line, carried as the payload.