correlation of the two is demonstrated by the diagonal line, along which the P/A ratio is equal to the error.

ments are the personal experiences of the author, and greater care may give different results which, it is hoped, will be more gen

AP

PPROXIMATELY 20,000 air photos, covering a 38,000 square mile area in Central British Guiana, will be delivered to the Colonial Surveys Office, London, at the conclusion of a three-month photographic mission started in August by Spartan Air Services, Ottawa. Two PV-1 Venturas were scheduled to fly at

15,000 feet; the chief problem is the weather, because clear cloudless days are infrequent in the Tropics. The same company has recently completed a large-scale Shoran-controlled photo mission north of the Arctic Circle in the Northwest Territories.

CENTERLINE

and property-survey calculations frequently require a solution for the value of some unknown which cannot be directly obtained by trigonometric methods because some additional condition has been imposed, such as a fixed length, a fixed angle, or the tangency of a curve to a fixed line.

While a solution can usually be obtained through the use of higher mathematics, it is likely to be laborious because of the size and number of decimal places of the coefficients. A correct value of the unknown can be obtained by the method of double position, employing only calculations with which the surveyor is familiar. The steps involved are:

tion in a trigonometric problem is illustrated by the following example which refers to figure 1. The distance AB between the projected centers of the fixed 80.00-foot and 140.00-foot radius curves is to be held at 219.52 feet. The perpendicular distances from the centers to the line AB are also fixed The reverse curve must be tangent to these two curves and to the straight line AB. Platting shows this radius to be approxi mately 20 feet. What is the exact radius to two decimal places?

1. Assume trial values of the unknown and solve by regular trigonometric methods so as to obtain results on each side (hence double position), one greater and one smaller than the result required by the imposed condition.

2. Apply the rule of double position to determine a small correction to the trial value of the unknown.

CURVE TANGENT TO A LINE AND TWO REVERSE CURVES

The two essential parts of the method of double position are (1) the rule and (2) the method of solution.

The rule

Difference of assumed numbers
Difference of computed values

Correction to first assumed number.
First error

The first error is the amount by which the first computed value differs from the fixed value. The fixed value is the condition to be met, which is that AB must be 219.52 feet.

The method of solution

r's 19.0

which differ from each other by one or two units. These trial values are taken here as 19.0 and 21.0 feet, since the required radius scaled 20 feet.

The first approximation of r through the use of the method of double position is carried only to tenths of a foot.

In trigonometric problems the curve of error may have a rapidly changing curvature, so to be certain of the correct answer a second approximation should be made. The assumed numbers for this second calculation are the first approximate value of ↑ and another assumed value 0.1 or 0.2 greater. The result will be correct to two decimal places.

=

Final value of r = €19.800 +0.0007 19.8007. This small change from the first approxi

mation shows this curve of errors to be almost a straight line within the limits taken. The second approximation was necessary to determine the value of r to the second decimal place.

CURVE TANGENT TO AN ARC AND TWO REVERSE CURVES

A more complicated problem, which might occur in the layout of a subdivision or in connection with the off-and-on ramps of a freeway, is illustrated in figure 3. The coordinates of the centers of curvature, A, C, and D and the lengths of the respective radii from these points are fixed. It is required to find the length of the radius r of a curve to be tangent to the arc with a radius

of 325 feet and to join properly the incoming curves at the P.R.C. An assumed value for r, determined from a careful plat, plus the known radii gives a preliminary value to AB and BC. Computing the distance and bearing AC from the coordinates gives tion of this triangle permits the preliminary a third side to the triangle ABC. The solu coordinates of B to be computed and the distance BD to be found. BD-r gives the first computed value. The difference be tween this first computed value and the fixed radius value of 325.00 will be the first error. A second value for r is assumed and a second error found. The method of dou ble position is then followed, as previously outlined, until the value of r is found to two decimal places.

AREA PROBLEMS

The method of double position will als furnish a solution to area problems such as may occur when a straight line running in a fixed direction must be so located as t

cut off a fixed area from a property bounded by curves.

This is an old mathematical device rarely seen in modern mathematical texts, being one of the simplest of the group connected with the theories of finite differences, to

which engineering owes so many useful formulae.

This method has been used very successfully by the Los Angeles Department of Public Works on certain complicated surveying problems.

R. THOMSON'S article is particularly interesting because of the practical slant that comes from actual experience. Many Many problems that may present difficulties if a direct solution were attempted can well be solved by a series of approximations such as the author has used. Thus the problem represents a type with which the surveyor should be familiar. There are interesting ramifications and opportunities for philosophical contemplation, all of which may be good discipline for the mind. Readers can do well to seek more in such examples than at first meets the eye.

Let us imagine first that the transition curve (fig. 1) might be required to be tangent to the line AB but to only one of the curves of given radius. In this case the radius could have any value and yet meet these two conditions, but the location of the center would be constrained to a definite path or locus. What is this locus and how may it be defined? It will be readily appreciated that by varying the radius, or distance from the center to the base line, equal amounts must be added to or subtracted from the distance between this center and the center of the fixed-radius curve to which it is tangent. It should be recalled that it is just in this same manner that the generation of a parabola was depicted in the study of conic sections, by maintaining equal distances from any point on the curve to both the focus and a line known as the directrix. Under the condition that r may be 0, it follows that the directrices of the

be difficult to solve for the coordinates of the intersection by using a pair of simultaneous equations.

Now simultaneous quadratics are not always the nicest things to worry one's head about, but if one should happen to like that kind of recreation just for the sake of dusting the cobwebs out of the brain, there are a few convenient tricks. One of these is to select as the origin of the coordinates the vertex of one of the parabolas, so that for one equation, at least, there is a simple relation y = mx2, which is handy for substituting in the equation for the other parabola.

When this is done the mess looks rather formidable, but have courage, for by grouping everything together under x2, x, and a constant term, the aggregate of which is equal to zero, it can all be expressed in the classic form

If you can do this without looking it up in your algebra book, rigor mortis of the brain has not yet begun to set in. By substitution of the coefficients in their proper places, x may be found, and again by substitution in the simpler of the two equations for the parabolas, y is likewise

from the base line.

To satisfy the condition that the transition curve must be tangent to both fixed-radius curves means that the only possible location for the center is where the two parabolas cross each other. Each of these parabolas can be represented by a quadratic equation, and, consequently, if one is not too rusty, it should not

be computed. Subtracting the fixed radii from these quantities should, in both cases, give identical results, being the radius of the transition curve sought.

Try this out. It will be a good exercise and it will show that the author has in two approximations derived the correct answer to the nearest ten thousandth of a foot.

Now let the imagination run riot still a little