distanceTo

FlatPoint distanceTo FlatPoint => Double
Returns the raw Euclidean distance between the two points.

The measurement ‖att(a ∧ b)‖● of the round representations: the join of the operands is the dipole through both points, its attitude the separation as a direction at infinity, and that direction's round-bulk magnitude the distance — componentwise, the coordinate differences, which the body evaluates directly. The result is non-negative and raw at full Double precision. A point at infinity — zero weight — carries a direction rather than a position, and the result is no distance; gate with FlatPoint.isAtInfinity.

Parameters

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

The round point contributes its position via RoundPoint.toFlat — the flat weight copies the round weight, so a point at infinity stays at infinity — and the measurement is FlatPoint distanceTo FlatPoint. A sized or imaginary round point measures from its position all the same: the conformal component takes no part.

Parameters

FlatPoint distanceTo Line => Double
Returns the raw perpendicular distance from the point to the line.

The magnitude of the meet of the round representations — for a finite point and a finite line, the line's equation evaluated at the point, |px·gx + py·gy + pz·gz|. The sign discarded here is the point's sidedness; the signed reading is FlatPoint dot Line. A point at infinity pairs its direction against the line's normal instead, and a line at infinity — zero normal — pairs only weights: neither result is a distance; gate with FlatPoint.isAtInfinity and Line.isAtInfinity.

Parameters

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

The measurement ‖att(a ∧ b)‖●: the join of the operands is the dipole through both points, its attitude the separation as a direction at infinity, and that direction's round-bulk magnitude the distance — componentwise, the coordinate differences, which the body evaluates directly. The result is non-negative and raw at full Double precision. A sized or imaginary round point measures from its position: the conformal component takes no part. A point at infinity — zero weight — carries a direction rather than a position, and the result is no distance; gate with RoundPoint.isAtInfinity.

Parameters

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

This point contributes its position via RoundPoint.toFlat — the flat weight copies the round weight, so a point at infinity stays at infinity — and the measurement is FlatPoint distanceTo FlatPoint. A sized or imaginary round point measures from its position all the same: the conformal component takes no part.

Parameters

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

The magnitude of the meet of the operands — for a finite point and a finite line, the line's equation evaluated at the point, |ax·gx + ay·gy + az·gz|; the conformal component takes no part, so a sized or imaginary round point measures from its position. The sign discarded here is the point's sidedness; the signed reading is RoundPoint dot Line. A point at infinity pairs its direction against the line's normal instead, and a line at infinity — zero normal — pairs only weights: neither result is a distance; gate with RoundPoint.isAtInfinity and Line.isAtInfinity.

Parameters