dot
FlatPoint dot FlatPoint => Double
Returns the raw inner product of two flat points — zero when they coincide, −½·d² for finite points d apart.
The negated half squared Euclidean distance, at full Double precision. If either operand is at infinity the distance reading is gone — the value degenerates to the plain dot of the two carried vectors.
Receiver
The left flat point.
Return
The raw inner product; −½·d² when both points are finite.
Parameters
The right flat point.
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.
The raw value px·gx + py·gy + pz·gz — the classical n·p + d form, derived from the antiwedge F ∨ L. For a finite point and a finite line the magnitude is the Euclidean distance and the sign is positive on the side the line's normal (gx, gy) points toward. A direction (at infinity) dots with the normal instead: zero when it runs along the line.
Receiver
The point to evaluate at.
Return
The raw signed value; the signed Euclidean distance for finite operands.
Parameters
The line whose equation is evaluated.
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.
The value ½·(r² − d²) — half the negated power of the point with respect to the circle, d its distance to the center. The identity takes r² signed, so an imaginary classic circle probes negative everywhere — no point is inside. An inside-out circle negates every component and so negates the value. Evaluated on the conformal lift; a point at infinity has no lift and the value degenerates to a directional pairing with no distance reading.
Receiver
The point to probe with.
Return
The raw probe value; ½·(r² − d²) for a finite point and a classic circle.
Parameters
The circle probed against.
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.
The strict inner product under the conformal metric; the cross terms carry the null pairing e3·e4 = −1. The −½·d² reading holds for two null lifts; an operand carrying a semantic round bulk aw (a center, an attitude) feeds that data into the same bilinear form instead.
Receiver
The left round point.
Return
The raw inner product; −½·d² for two finite null lifts.
Parameters
The right round point.
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.
The raw value ax·gx + ay·gy + az·gz — the flat-point form with RoundPoint.az in the weight role; the round bulk aw plays no part (a line has no e321 component). Sign and at-infinity behavior match FlatPoint dot Line: positive on the normal's side, and a direction dots with the normal.
Receiver
The point to evaluate at.
Return
The raw signed value; the signed Euclidean distance for finite operands.
Parameters
The line whose equation is evaluated.
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.
The value ½·(r² − d²) — half the negated power of the point with respect to the circle. The identity takes r² signed, so an imaginary classic circle probes negative everywhere — no point is inside. Derived from the antiwedge R ∨ C: the full bilinear pairing cx·ax + cy·ay + cz·az + cw·aw, so an operand carrying a semantic round bulk aw feeds that data into the pairing. An inside-out circle negates the value.
Receiver
The point to probe with.
Return
The raw probe value; ½·(r² − d²) for a finite point and a classic circle.
Parameters
The circle probed against.
Circle dot Circle => Double
Returns the raw inner product of two circles — zero exactly when they cross at right angles.
The strict inner product: ½·(d² − r₁² − r₂²) for two classic circles with centers d apart, which vanishes precisely on orthogonal circles and gives −r² for a circle dotted with itself. Both r² are taken signed, so the identity extends to imaginary operands. Flipping one operand inside-out negates the value.
Receiver
The left circle.
Return
The raw inner product; ½·(d² − r₁² − r₂²) for two classic circles.
Parameters
The right circle.