Distance-Distance (Swing-Tie) Intersection

The tape swing-tie, as-built, and batter-board method. Geometrically it is where two circles cross: with the control points d apart, the chord foot is a = (r0^2 - r1^2 + d^2)/(2d) from point 1 and the crossings sit h = sqrt(r0^2 - a^2) to each side, so there are TWO mirror-image solutions across the line between the points (the field sketch or a third tie picks the side). Flags the impossible tape readings: distances too short to meet (d > r0 + r1) or one swing circle inside the other (d < |r0 - r1|), plus the tangent (single-solution) case. Plane northing/easting grid geometry; grid scale factor and elevation are separate. First-principles Euclidean geometry (trilateration; Ghilani & Wolf). A computational aid; the project control governs.

d = |P1-P0|; a = (r0^2 - r1^2 + d^2)/(2d); h = sqrt(r0^2 - a^2); Pm = P0 + a*(P1-P0)/d; solutions = Pm +/- h*(perpendicular unit). Real iff |r0-r1| <= d <= r0+r1.

First-principles Euclidean geometry (two-circle / trilateration intersection), the standard swing-tie locate as compiled in the route-surveying references (Ghilani & Wolf, Elementary Surveying), by name.

The two-circle intersection is public first-principles geometry; the control coordinates and the two measured distances are the user's field data.

Estimate. AHJ and licensed professional govern.

Field names used by the API: n0_ft, e0_ft, dist0_ft, n1_ft, e1_ft, dist1_ft, d_ft, sol1_n_ft, sol2_n_ft

• Two-circle intersection a = (r0^2 - r1^2 + d^2)/(2d), h = sqrt(r0^2 - a^2); solutions = midpoint +/- h*perpendicularEuclidean geometry / trilateration
• Two solutions mirror images across the control line; the field sketch or a third tie resolves the sidesurveying practice
• Plane geometry plane grid northing/easting; grid scale factor and elevation are separateplane-survey assumption