Direct single intersection: Calculation by Jung formulas

To define coordinates of point P (Xp, Yp) if coordinates of points A (Хa, Ya) and B (Xb, Yb) are known and angles a and b are measured. (Example of data input)

The task is solved by Yung’s formulas which are looking like:

  Xp = (Xa*Ctg(b)+Xb*Ctg(a)+Ya-Yb)/(Ctg(a)+Ctg(b)),

  Yp = (Ya*Ctg(b)+Yb*Ctg(a)+Xb-Xa)/(Ctg(a)+Ctg(b))

Designations in these formulas correspond to a relative location of defined point P and to starting points A and B shown on fig.1. If from starting side AB to look at defined point P then the right point and the angle measured on it are designated accordingly A and a, and the left  point and the angle measured on it - B and b.

With a view of the control and to exclude calculations in the second hand the design formulas are transformed. Below the derivation of control formulas of the solution of this task is shown.

1. A direction angle of a line (AB):

tg(AB)=(Yb-Ya)/(Xb-Xa).

2. Length of a line (distance between initial marks):

d = (Xb-Xa)*sec(AB)= (Yb-Ya)*cosec(AB).

3. Length of line AC:

AC= d/sin(a+b)*sin(b)= d*cosec(a+b)*sin(b).

4. Length of line BC:

BC= d/sin(a+b)*sin(a)= d*cosec(a+b)*sin(a).

5. A direction angle (AC):

(AC) = (AB)+ (a).

6. A direction angle (BC):

(BC) = (BA)- (b).

7. Coordinates of point P:

Xp = Xa + AC*cos(AC) = Xb+BC*cos(BC).

Yp = Ya + AC*sin(AC) = Yb+BC*sin(BC).