Once rectangular components are obtained in either the math/science or engineering/surveying system, standard coordinate geometry (often referred to as COGO) operations are the same. Admittedly, it becomes a challenge to keep track of various conventions employing x/y coordinates and eastings/northings in the same rectangular system, but the COGO procedures are the same in each case. John (1984) gives a derivation for the following 2-D COGO operations listed as follows:
1. Forward (traverse to new point)
2. Inverse (find direction and distance from standpoint to forepoint)
3. Line-line (also called bearing-bearing) intersection
4. Line-circle (also called bearing-distance) intersection
5. Circle-circle (also called distance-distance) intersection
6. Perpendicular offset distance from a line to a point
(Use signs of Δβ and Δη to determine the proper quadrant.)
When performing design or other COGO computations, it is often necessary to determine where lines intersect (see Figure 4.3). Three methods are as follows:
1. Line-line: in this case, points 1 and 2 are given. The directions from point 1 and the direction to point 2 are also given. The problem is to compute the coordinates of the intersection point. Line-line is also called a bearing-bearing intersection.
2. Line-circle: in this case, points 1 and 2 are given. The direction from point 1 and the distance from the intersection point to point 2 are given. The problem is to compute the coordinates of the intersection point. Line-circle is also called a bearing-distance intersection.
FIGURE 4.3 Geometry of Intersections
3. Circle-circle: in this case, points 1 and 2 are given. The distance from point 1 to the intersection point and the distance from the intersection point to point 2 are given. The problem is to compute the coordinates for the intersection point. Circle-circle is also called a distance-distance intersection.
In each case, the computation starts at the standpoint and ends at the forepoint while establishing the location of an intermediate intersection point. The rules to be used in establishing the computed intersection point vary according to the type of intersection, but they fall into one of the three categories listed. Symbols and conventions for the computations are follows:
In each case, the solution starts by computingAlso note that no solution exists if, in the first case, the lines are parallel; in the second case, the line does not intersect the circle; and, last, the circles do not intersect. A different mathematical impossibility is encountered in each case.
Line-line: if the lines are parallel, azimuths a and β are the same and the denominator of equation 4.14 goes to zero. As a result, the distance d1 is undefined. There is no intersection. See Figure 4.4a.
Line-circle: if the line does not intersect the circle, the perpendicular offset distance from the line to the circle is greater than the radius of the circle.
If that happens, the quantity under the radical in equation 4.15 is negative. Since it is not possible to take the square root of a negative number, d1 is undefined and there is no intersection. See Figure 4.4b.
FIGURE 4.4 Examples of Intersections That Fail
Circle-circle: if the two circles do not intersect, it is not possible to find the angle γ using equation 4.16 because, in one case, the value of cos γ is greater than 1.0 and, in the other case (one circle entirely within the other), the value of cos γ is less than -1.0. See Figure 4.4c.
Other than the nonintersection cases described, the following formulas provide very efficient procedures for computing intersections in a two-dimensional plane.
Line-Line: One Solution or No Solution if Lines Are Parallel
(d1 may be either a positive or negative value.)
Check: inverse intersection point to forepoint, and compare computed direction β with given direction β. If they are not the same, a mistake was made.
Line-Circle: May Have Two Solutions, One Solution, or No Solution
(d1 normally has two values, one for each solution.)
Check: inverse intersection point to forepoint, and compare computed distance d2 with given distance d2. They should be the same. Also make sure the solution obtained is the one desired. Depending upon where the line intersects the circle, the values of d1 could both be positive, could be one positive and one negative, or could both be negative.
Circle-Circle: May Have Two Solutions, One Solution, or No Solution
Check: inverse intersection point to forepoint, and compare computed distance d2 with given distance d2. They should be the same. Also make sure the solution obtained is the one desired. Several solutions may exist. Also be aware that equation 4.16 has no solution if the two circles do not intersect.
In many situations, it is desirable to find the perpendicular distance from a line to a point. Given one is on a standpoint P(e1,n1) and looking in the direction of a line, α, the problem (as illustrated in Figure 4.5) is to find the perpendicular offset distance right (positive) or left (negative) from the line to the point specified as the forepoint P(e2,n2). Note that the azimuth, α, may be any azimuth from 0° to 360°. Using the conventions given earlier, the perpendicular offset distance is the distance d2 computed as
FIGURE 4.5 Perpendicular Offset to a Line
Area by Coordinates
Area is the product of length times width and is generally presumed to be computed on a flat surface. Computing area on a spherical or ellipsoidal surface is more of a challenge and is addressed.The method generally used for computing the area of an irregular tract is area-by-coordinates.Area by double-meridian-distance (DMD), useful if working with latitudes and departures, is described in many surveying texts and is not presented here. Development of the area (equation 3.1) involves adding and subtracting trapezoids, as shown in Figure 4.6.
Area = Trapezoid I + Trapezoid II – Trapezoid III – Trapezoid IV
Combining the four trapezoids into one equation and multiplying by two gives
Considerable algebraic manipulation and combination of terms are needed to get
FIGURE 4.6 Area by Coordinates
Comments about equation 4.20 are as follows:
• The coordinates are often arranged in tabular form with the beginning values listed again at the end. The computation is then illustrated by showing accumulation of cross products, and area is computed as half the difference of the two accumulated sums.
• If points around a figure are chosen in a clockwise sequence, as was done in the derivation, a positive area is computed. If a counterclockwise sequence is used, a negative area will be the result. Although each user is free to “traverse” a parcel in either direction, the convention is to use positive area.
• Care must be exercised to assure that area is being computed for a legitimate figure. If one line of the figure crosses another (e.g., due to erroneous point sequence input), the equation will provide an answer that reflects removal of the crossover area.
• Significant figures may become an issue if equation 4.20 is used with large (i.e., state plane) coordinates.Equation 3.2 is recommended as the preferred area-by-coordinates formula.