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3331. Principles of the Three-Point Fix 



The theory of the three-point problem is well known. It depends on the following 

 three principles: 



(a) The circumference of a circle can be described through any three given points. 



(b) If two of the points are fixed in position, the angle between them measured at a third point 

 will be the same for all points on that part of the circumference of the circle on the same side of a 

 line joining the two fixed points. 



(c) If, in addition to the first angle, a second angle is measured from the same unknown point 

 to two points, one of which always, and both of which occasionally, differ from the first two, the posi- 

 tion of the unknown point will also be defined by a second circle. Since the unknown point lies on the 

 circumferences of two circles, its position will be defined by the intersection of these. 



3332. Strength of Three-Point Fix 



Theoretically, the strength of a position determination by a three-point fix depends 

 directly on the angle of intersection of the two circumferences defined by the two 

 angles and the three known points. The more nearly this intersection approaches 90°, 

 the stronger is the fix. Conversely, the nearer the circles approach tangency, the weaker 

 the position becomes until it is indeterminate when the two circles coincide. An inde- 

 terminate fix is called a revolver or swinger because, when an attempt is made to plot 

 it with a protractor, the protractor will swing along the arc of the coincident circles, 

 since any point on them will satisfy the conditions. The strength of a three-point 

 fix, therefore, depends directly on the relative positions of the three fixed points and 

 the unknown position. 



An experienced hydrographer can visualize almost automatically the circle passing 

 through two fixed points and his position. When he realizes that the two circles 

 involved will intersect at an acute angle, he knows that the objects responsible for this 

 condition must be avoided, if another choice is available. The three-point problem 

 is analyzed in various treatises on surveying and this need not be repeated here, but 

 beginners who may have some difficulty in selecting the most suitable objects should 

 find the following general rules useful: 



(1) The strongest fix is when the observer is inside the triangle formed by the three objects. And 

 in such case, the fix is strongest when the three objects form an equilateral triangle, the observer is at 

 the center, and the objects are close to the observer. 



(2) The fix is strong when the sum of the two angles is equal to or greater than 180° and neither 

 angle is less than 30°. The nearer the .angles equal each other the stronger will be the fix. 



(3) The fix is strong when the three objects are in a straight line, or the center object lies between 

 the observer and a line joining the other two and the center object is nearest to the observer. 



(4) The sum of the two angles should not be less than about 50°, better results being obtained 

 when neither of the angles is less than 30°. 



(5) The fix is strong when two of the objects a considerable distance apart are in range or nearly 

 so and the angle to the third is not less than 45°. 



(6) A fix is strong when at least one of the angles changes rapidly as the survey vessel moves 

 from one location to another. 



3333. Selection of Objects 



The theoretical strength of the tlu-ee-point fix is based on the supposition that 

 there will be no more than a certain angular error in any measured angle regardless of 

 the size of the angle or the distance of the objects. There are other practical limitations 

 which must be considered. 



Small angles should generally be avoided as they result in weak fixes in most cases 

 and are usually difficult to plot. A strong fix will be obtained, however, with one small 

 angle when the vessel is a little off a range and the nearer of the two objects in range is 

 the center object. In this case the small angle must be observed very accurately, and 



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