RESECTION

THE RESECTION Determine Coordinates of P from observations to known points A, B and C. Note: Since P is un-known, the raw observations at P need to be oriented. Use the Q-Point Method to orient the observations at P and then intersect P with the oriented forward directions from A and B. Compute α and β from observations a = 180° - α , b = 180° - β Compute AzAB from XA, YA, XB,YB AzAQ = AzAB – b, AzBQ = AzBA + a Compute XQ and YQ with AzAQ and AzBQ from A and B (forward intersection) Compute AzQC (= AzPC) from XQ, YQ, XC,YC AzAP = AzPC + α ± 180°, AzBP = AzPC - β ± 180° 8. Compute XP and YP with AzAQ and AzBQ Q A b a α P b a Note: The solution is ambiguous when A,B,C and P all lie on a circle (the danger circle) β B C For best orientation results use the furthest point for Orientation.

THE INACCESSIBLE POINT AP = AB sin (B) = AB sin (B) sin(180-(A+B) sin (A+B) BP = AB sin (A) sin (A+B) ∆hA = AP tan (vA) ∆hB = BP tan (vB) Elev.PA = Elev.A + ∆hA + hiA Elev.PB = Elev.B + ∆hB + hiB Elev P = Elev.PA + Elev.B 2 Can be applied in reverse to determine height of A (or B) from known height of P. Provided sufficient outside orientation is given, this model is also used to close traverses on inaccessible points with known coords. (e.g. church spire) ∆hA ∆hB vA B p hiA Outside Orientation A Caution: Steep sights – observe on both faces and level instrument carefully.

THE ‘DOG’S LEG’ CHECK When fixing or placing a point from only one known station by means of a ‘single polar’, the ‘dog’s leg’ is often applied to obtain an independent check. O U From known point B observe two orientation points to ensure correct orientation. Measure direction and distance to unknown point U Place auxiliary point A in such a way that at least one known point in addition to B is visible and that the angle at U is greater than 30°. Measure direction and distance to A Move instrument to A and observe outside orientation point O and B and measure direction and distance to U. Compute Coordinates of A and U to obtain independent coordinates for U A B To strengthen the check A is sometimes placed on line from B to an orientation point.

THE 2D CONFORMAL COORDINATE TRANSFORMATION Given A(E,N) and B(E,N) and A(X,Y) and B(X,Y) compute C(E,N) from C(X,Y) 1. Compute the Swing Az(XY)AB = atan (∆XAB/∆YAB) Az(EN)AB = acot (∆NAB/∆EAB) ‘Swing’ θ = Az(XY)AB - Az(EN)AB Az(EN)AB Az(XY)AB B C ∆NAB θ X D 2. Compute the Scale  ∆EAB2 + ∆NAB2 Scale s =  ∆XAB2 + ∆YAB2 Y A XAB YAB ∆XAB ∆YAB ∆EAB E

THE 2D CONFORMAL COORDINATE TRANSFORMATION Given A(E,N) and B(E,N) and A(X,Y) and B(X,Y) compute C(E,N) from C(X,Y) 1. Compute the Swing Az(XY)AB = atan (∆XAB/∆YAB) Az(EN)AB = acot (∆NAB/∆EAB) ‘Swing’ θ = Az(XY)AB - Az(EN)AB 2. Compute the Scale  ∆EAB2 + ∆NAB2 Scale s =  ∆XAB2 + ∆YAB2 3. Compute the Translations TX and TY a = sYAsin θ, b=sXAcos θ Hence X’A = s(XAcos θ - YAsin θ) c = sXAsin θ, d=sYAcos θ Hence Y’A = s(XAsin θ + YAcos θ) EA = X’A + TX hence TX = EA – X’A and NA = Y’A + TY hence TY = NA – Y’A C NC Y’ B D Y A X A YA d θ Y’A XA 4. Apply the transformation parameters EC = s(XCcos θ - YCsin θ) + TX NC = s(XCsin θ + YCcos θ) + TY c θ θ TY X’ X’A a b E Tx EA EC

Ø ✔ Ø ✔ ✔ Ø SOME ‘GOLDEN RULES’ Unless traversing, always observe two orientation rays to control correct orientation of instrument. Long observation rays give better orientation than short rays. Orientation rays should always be longer than fixing rays. The formulae derived for trilateration, intersection and resection use the minimum number of observations to determine one unchecked set of coordinates for unknown points. Additional independent observations should be made to provide sufficient redundancy and control of the quality of coordinates. Pay attention to the strength in the geometry – keep intersection angles larger than 30°. Be aware of situations that yield multiple solutions. (Intersection/Trilateration). Add value to your survey by connecting it to the most widely used coordinate system in the area. If possible, avoid local coordinate systems. Interpolate – do NOT extrapolate! Connect to nearest available known points Orientation B A Mirror image due to orientation mistake of 180° Geometry Ø ✔ Interpolation Ø ✔ Neighborhood Principle ✔ Ø