T A UC-NRLF 753 COORDINATES OF ELEMENTARY SURVEYING J. C. L. FISH ' LIBRARY OF THE UNIVERSITY OF CALIFORNIA. Class COORDINATES OF ELEMENTARY SURVEYING By J. C. L. FISH Linear Drawing and Lettering For Beginners, pp. v-f-65, 4 folding plates; limp cloth, 7 x 10% New York : Engineering News Publishing Co. $1.00 Blank-Book for Lettering With 30 sheets ruled for lettering, and plate of standard alphabets; paper, 7x10^ New York : Engineering News Publishing Co. .25 Lettering of Working Drawings With 13 full-page plates; boards, 9x12 New York: D. Van Nostrand Co i.oo Descriptive Geometry With instructions and exercises for the drafting- room ; paper, 6x9 Stanford University, Cal. : Stanford Bookstore i.oo Mathematics of Paper Location of a Railroad Giving the steps taken to write out the field notes from the location drawn on the map ; paper, 4^x7^ Chicago : Myron C. Clark Publishing Co 25 Coordinates of Elementary Surveying With examples and algorithms; limp cloth, 6x9 Stanford University, Cal.: Stanford Bookstore i.oo COORDINATES OF ELEMENTARY SURVEYING BY J. C. L. lSH, M. AM. soc. C. E. Member American Railway Engineering and Maintenance of Way Association; some time Division Engineer Lake Shore and Michigan Southern Railway; Professor of Railroad Engineering in Leland Stanford Jr. University STANFORD UNIVERSITY, CALIFORNIA THE AUTHOR 1909 COPYRIGHT, 1909, BY JOHN CHARLES LOUNSBURY FISH STANFORD UNIVERSITY PRESS UNIVERSITY Of PREFACE In my first eight years of teaching" surveying in the custom- ary way, stress was laid on those fundamental principles of the subject which are independent of the circumstances of surveying, namely personnel, equipment, nature of the survey and of the ground. Nevertheless it was to be observed that on subsequently taking up the study of railroad surveying too many of the stu- dents were deficient in grasp of fundamentals. Six years ago the experiment was tried of devoting a part of the lectures of the course in surveying to the fundamentals exclusively, and the following year the students were supplied with a mimeograph text covering this ground and were required to work many prob- lems in connection therewith. The result of the experiment was gratifying. During the following three and a half years I had occasion to observe critically the work of several surveying parties em- ployed on heavy railroad construction and was struck by the fact that nearly all of the cases of defective surveying were due to lack of grasp of the coordinates of surveying rather than to want of skill in the use of implements. The lack of familiarity with fundamentals was most often exhibited in failure to devise a scheme of check measurements which should be independent of the original measurements. It has been observed not infrequently that the young sur- veyor looks upon the different branches of surveying, e. g., land, city, mine and hydrographic surveying, as involving different fundamental principles of surveying, and this is because in his training the varying circumstances of surveying have been per- mitted to overshadow the unvarying principles. The foregoing facts supply the reason for making much of and formally presenting the subject of this book. 191650 VI PREFACE The general absence of explanations is due to design : to make the book brief, advantage has been taken of the fact that in this institution the study of coordinate geometry precedes the study of surveying. It is assumed that the teacher who uses this drill book will enliven it by illustrations taken as far as possible from the experiences- of his students, and require from them accurate and neatly arranged solutions of numerous prob- lems. Ten to fourteen lectures may be profitably devoted to illustrating the text, and the student may well give upwards of twenty hours to the solution of original problems. The Tables relating to interchange of bearings, azimuth and deflections are intended to be used only in checking computation. Lest the forbidding mathematical aspect of some of the pages should discourage the reader on first opening the book, it ought to be said here that the subscript is used to gain brevity without loss of precision of statement, and does not indicate the presence of "higher mathematics." CONTENTS PART I. INTRODUCTION CHAPTER I. GENERAL STATEMENTS ARTICLE PAGE 2. Definitions . . . . . . ... . . ... 3 3. Survey of Straight Line . . <; . . . . . . 4 4. Circle. . . . . .4 5. Irregular Curves . . . . ; . . . 4 6. of a Plane . . . ; . . . ;. . . 4 7. Irregular Surfaces . 4 CHAPTER II. -SYSTEMS OF COORDINATES 9. Systems of Coordinates in Horizontal Surveying . . 5 10. Vertical . . 6 11. Space . . 6 CHAPTER III. NOTATION 12. Symbols , . # 13. Greek Alphabet . * ; . . 7 PART II. HORIZONTAL SURVEYING CHAPTER IV. -RECTANGULAR COORDINATES 14. Rectangular Coordinates _. . . . . . . . . 8 15. General and Local Rectangular Coordinates . . . 9 16. .r- Difference and ^'-Difference . . . / . . . 10 17. Equation of a Straight Line . . . . . . . . . 10 18. Intersection of Two Straight Lines 11 19. Area of Closed Figure 12 20. From General to Local Coordinates 14 viii CONTENTS ARTICLE PAGK 21. Local to General Coordinates: Single Point . 14 Series of Points 14 23. To Make Coordinates All Positive . . . . . . 15 24. From Local Rectangular to Local Polar Coordinates 15 25. General Polar Coordinates . . 17 CHAPTER V. POLAR COORDINATES 26. Polar Coordinates . . . . . 18 27. Direction and Distance . 18 28. Azimuth 18 29. Bearing ....... . .19 30. Deflection . . ' . .20 31. Angle Between Two Lines: Azimuths . -, . . . 21 32. " Bearings 21 33. Distance Between Two Points . . ... . . 22 34. Area of Polygon 23 35. To Change Zero Direction : Azimuth 24 36. " Bearings 25 37. From Azimuth to Bearing 27 38. " " Deflection ....... 27 39. " Bearing to Azimuth 28 40. " " Deflection 28 41. " Deflection to Azimuth 29 42. " ." Bearing 30 43. To Change Pole Without Changing Zero Direction : Azimuths _ * . .32 44. From Local Polar to Local Rectangular Coordinates : Azimuths 32 45. From Local Polar to Local Rectangular Coordinates : Bearings 33 46. From Local Polar to Local Rectangular Coordinates : Deflections . . . . . . ... . . . 35 47. From Polar Coordinates to Biangular Coordinates . 35 CHAPTER VI. BIANGULAR COORDINATES 48. Biangular Coordinates . . , . . . . . -. 36 49. Distance Between Two Points . . . , . . . 36 50. Area of the Triangle . . . ,. . . ... 38 CONTENTS IX ARTICLE PAGE 51. From Biangular. to Polar Coordinates: Azimuth and Distance .; , . . . . . . . . . ..;. . 38 52. From Biangular to Polar Coordinates : Bearing and Distance , .. . . . . . .... . . 39 53. From Biangular to Biradial Coordinates : Single Point 40 54. "' Series of Points 40 55. From Biangular to Rectangular and Other Coordinates 43 CHAPTER VII. BIRADIAL COORDINATES 56. Biradial Coordinates 44 57. Area of the Triangle 44 58. From Biradial to Biangular Coordinates .... 44 59. " Polar .... 45 60. " Rectangular and Other Coordinates 46 CHAPTER VIII. BIPOLAR COORDINATES 61. Bipolar Coordinates 47 62. From Bipolar to Polar Coordinates 47 63. Rectangular and Other Coordinates 47 CHAPTER IX. TRIPOLAR COORDINATES 64. Tripolar Coordinates .49 65. From Tripolar to Polar Coordinates 49 66. " Rectangular and Other Coordinates 51 PART III. VERTICAL SURVEYING CHAPTER X. RECTANGULAR COORDINATES 67. Rectangular Coordinates ^ . . . . ... . 52 68. .From General to Local Rectangular Coordinates . . 53 69. Local to General ^ . . 53 70. '* " Rectangular to Polar . . 54 71. " General " " . . 54 72. Local Rectangular Coordinates 54 X CONTENTS CHAPTER XI. POLAR COORDINATES ARTICLE 73. Polar Coordinates "... 56 74. From Polar to Local Rectangular Coordinates . . 56 75. " Polar-Rectangular . . 56 CHAPTER XII. POLAR-RECTANGULAR COORDINATES 76. Polar-Rectangular Coordinates . . . . . . . 58 77. From Polar-Rectangular to Local Rectangular Coordinates 58 78. From Polar Rectangular to Polar Coordinates . . 58 PART IV. SPACE SURVEYING CHAPTER XIII. RECTANGULAR COORDINATES 79. Rectangular Coordinates 60 80. From General to Local Rectangular Coordinates . 61 81. " Local to General Single Point 61 82. From Local to General Rectangular Coordinates: Series of Points 62 83. From General Rectangular to Polar Coordinates . . 63 84. " -Rectangular Coordinates 64 CHAPTER XIV. POLAR COORDINATES 85. Polar Coordinates 65 86. From Polar to Polar-Rectangular Coordinates . . 65 87. Local Rectangular . . 65 88. " General Rectangular . . 66 CHAPTER XV. POLAR-RECTANGULAR COORDINATES 89. Polar-Rectangular Coordinates ........ 67 90. From Polar-Rectangular to Local Rectangular Coordinates . , ., . . , . . . ... 67 91. From Polar-Rectangular to General Rectangular Coordinates , . ,, . . . . . . .... 67 92. From Polar-Rectangular to Polar Coordinates . . 67 PART I. INTRODUCTION. CHAPTER I. GENERAL STATEMENTS. 1. For the purpose of study the subject of Elementary Survey- ing may be divided into the following parts : ( i ) Coordinates of surveying, which are the "magnitudes which serve to fix the position of a point, in space, on a surface, or on a line." (2) Surveying instruments, by means of which the magnitudes are measured. (3) Errors of measurement. (4) Methods of surveying, which deal with the practical application of systems of coordinates, either singly or in combination, through the use of various surveying instruments. (5) Computing and mapping, by which the measurements are combined and the results presented for convenient use. Coordinates of surveying, unlike the other four topics, have to do only with mathematical laws. A thorough grasp of the coordi- nates of surveying is absolutely necessary to the intelligent planning of a survey, whether it involves few or many points. This book deals only with the systems of coordinates used in surveying, and the following definitions are accordingly restricted. 2. Definitions : Surveying. Surveying consists in making measurements of angles or of distances or of both, with the object (i) of finding the mathematical relations which exist between given points of the earth's surface ; or (2) of marking on the earth's sur- face points which bear given mathematical relations to given points of the earth's surface. A survey made with the second object, just stated, is called a location survey or simply a "location"; and the marked points are said to be "located." In Horizontal Surveying, all the points considered lie in the same horizontal plane, and are either actual points or the horizontal projections of actual points. In Vertical Surveying, all the points considered at one time lie in the same vertical plane. 4 COORDINATES OF ELEMENTARY SURVEYING In Space Surveying, the points are considered in their actual positions in space. 3. Survey of Straight Line. A limited straight line is deter- mined by its extremities. Therefore, to survey a system of straight lines it is sufficient to determine the position of the two extremities of each line. When two lines have a common extremity, the two lines are determined by the common point and the two other extremities. In general, a broken line is determined by a number of points one greater than the number of segments of the broken line. In the case of a polygon, as the boundary of a field, two adjacent lines have a common extremity and the n lines of the figure may be surveyed by determining the positions of the n double extremities. 4. Survey of a Circle. To survey a circle it is sufficient to determine the radius and the position of the center of the circle, or to determine any three points of the circumference. 5. Survey of Irregular Curves. When it is required to survey an irregular curve (whether it lies in a horizontal or a vertical plane, or is a non-planar curve) the surveyor determines the positions of a number of points on the line. The distribution and number of points determined in any survey depend upon the object of the survey. The surveyor chooses the points so that for the purpose for which the survey is required the resulting segments of the curve may be considered as straight lines, or the smooth curve drawn through the points may be considered as the true curve. Theoretically the accuracy of the survey of the curve depends upon the number of points determined, other things being equal ; but the labor of surveying increases with the number of points determined. 6. Survey of a Plane. To survey a plane it is sufficient to determine the positions of three points (which do not lie on the same straight line) of the plane. (We are not here considering the survey of a limited plane area. ) The ground surface is not plane, but limited areas sometimes are (or are made to be) approximately plane, and are often con- sidered to be plane. 7. Survey of Irregular Surfaces. To survey an irregular sur- face the surveyor determines the positions of a number of points of the surface. The points are selected with regard to number and distribution, so that, for the purposes of the survey, the broken plane surface determined by the points or the smooth surface drawn through the points, may be taken to represent the actual surface. SYSTEMS OF COORDINATES CHAPTER II. SYSTEMS OF COORDINATES. 8. The distances or angles, or distances and angles, which are measured for the purpose of determining the position of any point with respect to any given points or lines, are called coordinates of the point. In Elementary Coordinate Geometry are discussed two systems of coordinates, Rectangular and Polar. These and several other systems which in this book are given arbitrary names, are used in surveying. These arbitrary names are not used in the practice of surveying, and are introduced here simply for convenience in discussing systems of coordinates. 9. Systems of Coordinates in Horizontal Surveying : Y 0. FIG. i Rectangular coordinates FIG. 2 Polar coordinates a) P. Pa FIG. 3 Biangular coordinates * P, Pa FIG. 4 Biradial coordinates f FIG. 5 Bipolar coordinates \ Pa FIG. 6 Tripolar coordinates * Referred to by Pence and Ketchum in Surveying Manual as "angular intersection." t " Focal coordinates or tie lines " of Surveying Manual. { " Modified polar coordinates " of Surveying Manual. \ " Resection " of Surveying Manual. 6 COORDINATES OF ELEMENTARY SURVEYING 10. Systems of Coordinates in Vertical Surveying : Datum FIG. 7 Rectangular coordinates FIG. 9 Polar-rectangular coordinates 1 1 . Systems of Coordinates in Space Surveying -. P t Iz. -^ X FIG. 10 Rectangular coordinates (JTand Y are horizontal) FIG. 8 Polar coordinates FIG. ii Polar coordinates X and Y are horizontal ) Frc. 12 Polar-rectangular coordinates NOTATION CHAPTER III. NOTATION. 12. Symbols. The following symbols are typical of the uniform notation used throughout this book. P lt P,, . . . P k y points. ;F, abscissa of the point /\ referred to a general origin. of the point P. 2 referred to the origin P l . j/! ordinate Zi elevation jtr,. 2 abscissa y r . 2 ordinate z rt elevation A x . , X- AjY a }'~ difference of the points P l and P 2 = A^.JJ s- a r2 azimuth /a,. 2 true azimuth ;a r 2 magnetic azimuth j3 r 2 bearing *- 2 *\ i r 2 7i; s*., #, . true bearing of the line magnetic bearing deflection angle to the of the line P 1 P,. right ; left. d r ^ (horizontal) distance sd r . 2 slope-distance between the two points P l and "r-2 D &i slope-angle of the line P l P. 2 . datum-line, or plane, through Y, general origin ; origin P l . rectangular axes through general origin ; origin the (horizontal) angle P l P^ P 3 . 13. Greek Lower-Case Letters. a alpha f zeta /3 beta 77 eta 7 gamma 6 theta & delta (lower case) L iota A delta (capital) K kappa e epsilon X lambda a mil v nu r tau f xi v upsilon o omicron phi TT pi % chi /o rho i/rpsi a sigma oj omega PART II. HORIZONTAL SURVEYING. CHAPTER IV. RECTANGULAR COORDINATES. 14. Rectangular Coordinates. In Fig. 13 is shown a point P n referred to the rectangular coordinate axes X k and Y k which in- tersect at the origin P k , by the rectangular coordinates x k . n and y k . n . x k . n is the abscissa, and y k . n is the ordinate, of P n . Quad.ET fX Quad. I Xoeg ypos. X neg. yneg. Xpos. | ypos. j, Xpos. Quad.IH _ FIG. 13 Quad.H F? -X FIG. 14 We number the quadrants, I, II, III, IV, as shown in Fig. 13, beginning with that quadrant between the two arrow-heads, and counting in clockwise direction. It is customary to consider the coordinates positive or negative according to the following table : SIGN OF X AND OF y FOR EACH QUADRANT. Quad. I Quad. II ' Quad. Ill Quad. IV X y + -f _j_ + RECTANGULAR COORDINATES 9 Example. Fig. 14 represents a piece of "coordinate paper" upon which have been marked the two axes JT and Y with origin P , and four points /\ , P 2 , P 3 , P t . One side of a small square is taken as unity in measuring the coordinates of the points. Using symbols we express the coordinates of the four points thus: Required to rewrite these expressions substituting for each x and each y its numerical value with proper sign. The result is : COORDINATES OF POINTS. ORIGIN AT P . ^1(3, 5), P*( i, 3), P 2 ( 4 , -4), />,( 3, 2), in which those quantities which are not preceded by the sign ( ) are understood to be positive. For convenience we may express the result as shown in this table. 15. General and Local Rectangular Coordinates. If all the points of a survey are referred to the same origin and axes Point x y P 3 5 P. 4 4 P-i, i 3 P 3 2 The origin axes coordinates will be called general origin ; axes ; coordinates. E. g. , the general coordinates of a point P n are x n and y n , and when xoryis written with a single subscript we understand that it refers to the general origin and axes. If the points of a survey are divided into groups and each group is referred to a separate origin, then for each group The origin axes coordinates will be called local origin ; axes; coordinates. If the local origin is P tt the local axes are X k and Y k , and the local coordinates of a point P n (referred to P^) are x k . n , y k . n ; and when x or y is written with a double subscript we understand that the first part of the subscript refers to the local origin, and the sec- ond part refers to the point determined. 10 COORDINATES OF ELEMENTARY SURVEYING 16. jr-Difference and y-Difference. If two points P k and P H are referred to the same origin so that we have then of the points P k and P u the ^"difference is {*?*"" Zf" ~ **' JT~ ^y k-n J n y k > in which the sign of the result is to be regarded. The ^"difference of a limited straight line is the ab ^ lssa o fthe y- ordinate terminal point diminished by the Q r jj^^ f tne initial point of the line ; e. g., %**'* II ** ~ ** II ^'difference of line P k P n , the sign of the re- sult to be regarded ; while &*.* -= x k --x n = ^-difference of the line P H P ky the sign of the *y'jt y * ** y~ result to be regarded. Terms equivalent to ^-difference and ^-difference, to be found in books on surveying, are given in the following table: EQUIVALENT TERMS. x- Difference ^/-Difference Longitude difference Longitude Departure J Easting ( Westing Latitude difference Latitude Latitude | Northing ( Southing 17. Equation of a Straight Line. The equation of a straight line which passes through the two points P k (x k , y^ and P n (x tlt y,^) is commonly written y^ax-^b, in which a = y n y k jx H x k and b = y A ax k . Example. Write the equation of the line which passes through points P,(2, 5) and P. 2 (6, 3). We use the following algorithm in solving. RECTANGULAR COORDINATES 11 ALGORITHM : EQUATION OF STRAIGHT LINE. I Line i"2 2 X k 2. 3 x* 6. 4 y* 5- 5 }'n 3- 6 y* y* 2. 7 *n *k 4- 8 log (v, t y*) o . 30 i n 9 log (x n X A ) 0.602 10 log a 9 . 69911 ii a 0.5 12 log 4-,, 9.301 13 log ax k o.ooon 14 ax k I .0 15 b (=>***) 6. 16 Eq. : y ax-\-b y = o . *x + 6 A B 18. Intersection of Two Straight Lines. Given the equation of each of two straight lines, /and /', to find the coordinates x t , j z of their point of intersection, P t . The equations are : for line /, y = ax -f b . (0 and for line /' y = a'x -\~ b' ... (2) Putting (ax -\- b} for y in (2) we obtain ax -\- b = a 'x -\- b' or, (a a')x= (b' -- b). We can, therefore, say : the abscissa of the point of intersection is *,= (b' - b} I (a '), and the ordinate of the point of intersection is y. = aXi -r b, and the check equation is y t = a'Xi + b'. Example. Find the coordinates of the point of intersection P & of the two lines P l P^ and P 3 P i whose equations are respectively y = 0.5*+ 4 y = zx -f- 7- 12 COORDINATES OF ELEMENTARY SURVEYING ALGORITHM : INTERSECTION OF Two LINES. I />. = (P.P.,, P 3 PJ 2 Eq. y = ax -f b y = 0.5*4-4 3 Eq. y = a'x-\- b' ^/ = 2*4- 7 4 a 0.5 5 a' 2. 6 b 4- 7 b' 7- 8 b' b 3- 9 a a 1.5 10 log. (b'b) 0.477 ii log. (a a') o. I76n* 12 log. Xi = diff. log. 0.30 in 13 log. a 9.699 14 log. a 0.301 15 log. axi o. ooon 16 log. a'Xi o. 6o2n 17 axi i.o 18 a'Xi 4.0 19 Xi 2.0 20 axi -\- b yi 3- 21 a'xi-\- b' = yi 3. (check) A B 19. Area of Closed Figure. If P l P. 2 , P 2 P 3 , . . . P n _iP n , P n P l are the sides, taken in order, of a closed plane figure, the area of the figure can be expressed in two ways, one of which is conven- ient for slide-rule and the other for logarithmic computations. For slide-rule computation we use this equation (Coord. Geom.): Area = ^[(^ x, + j/ a x, + / 3 x, + . . . -f y n x,) - (ji *n -I- j, ^ 4- y* x, + - . -f- }' n *_i)] in which x , y v are the general coordinates (Art. 1 5) of point P l ; x^ y. z , of P 2 ; etc. Or, in words : Multiply the y of each point by the x of the and add the products, obtaining the sum 3 ' The "' Area = ?=!' * This " n " shows that the log. is of a negative number, t PI is the point following P n . Pn is the point preceding /\. RECTANGULAR COORDINATES 13 For logarithmic computation we use the following equation (Coord. Geom.) : Area = ^[j^fe #) -f- yx* x^ -f- y.,(x, x. t ) -f . . . + ^fe-^-0] in which ;r, , j^ are the general coordinates of P l ; .r, , y tt of P. 2 ; etc. ; or, in words : 1. Subtract the x of each point from the x of the second point following,* obtaining an .^-difference, A;r. 2. Multiply each A,r by the y of the intermediate point. 3. Then, Area = ^(algebraic sum of products). Example. P l , P 2 , P. A , P i , P & are the corners of a field, taken in order around the perimeter. Given P l (2, i), P 2 (3, 2), P 9 (4, 7), PI (3, 2), P, (5, 6). What is the area of the field ? I 2 Point x J *-diff. = log log^r-diff. Partial area 3 ^.-i) sum log (+) ( ) 4 p. - 5 5 P\ 2 I 0.000 6 8 0.903 7 0.903 8 8 r> 3 2 0.301 9 2 0.301 10 0.602 4 ii r> 4 -7 o.845n 12 6 0.77811 13 1.623 42 14 P. 3 2 o.3om 15 9 o.954n 16 1-255 18 17 P* -5 6 0.778 18 5 0.699 19 1-477 30 20 P l 2 21 Sum -(-i 02 -o Therefore area = ^(102 o) = +5i.f *The " second point following /-.-, fThe sign of the area has no significance, as it depends only on whether we take the points in clockwise or counter-clockwise order. 14 COORDINATES OF ELEMENTARY SURVEYING 20. From General to Local Coordinates. We are given the gen- eral coordinates x k , y k of the point P k , and x nt y of the point P n . We wish to know the local coordinates x k . n , y k . n of P n referred to the local axes X k and Y k taken through the local origin, P k . X k is || to X and Y t is || to Y. Or, briefly, Given Pk(x*> yd and P n (x Ht y n ) , To find Pn(x k ., t , y k .,d- ** * ^ = A*X.. ^difference of line P h P n . y =y y k =^ fy# =7- 21 . From Local to General Coordinates ; Single Point. We are given the local coordinates x k . n , y k . n of the point P n , and the gen- eral coordinates x k , y k of the point P k . We wish to know the gen- eral coordinates x n , y n of P n . X is || to X k and F is || to Y k . Or, briefly, Given P k (x k , y^ and P n (x k . n , y*. n ), To find /^(^ M , - 22. From Local to General Coordinates ; Series of Points. We are given a series of points, P l , P. 2 , P 3 , . . . . We are given the general coordinates x^ , y l , of the first point, /\ , of the series. For each point after the first we have the local coordinates referred to the preceding point as origin. All the ^r-axes are || and all the jj/-axes are || . We wish to find the general coordinates of all the points after the first. Or, briefly, Given ^i (X , yd To find P., O, yd ^ 3 and zero direction = -+- Y k (axis), Required P H (a k . n , d k .^ ; or />(&., 4*. M ). From Fig. 15 the tangent of the direction angle is tan a k . n = tan &. = x k . H \ y k . n . The quadrant in which the azimuth FIG. 15 terminates (/'. e. , the bearing quadrant in which P n lies) is determined by the following : I a terminates in II quadrant III IV If* is and y is an d/3i is NE; SE; SW; NW. The distance is d k . n = jr^/sin a k . n = jtr^/sin &.. The check equation is <^V x2 , the distance (radius vector). The expression "Given P M (B^ a R t */*.)" means angle ^>^ n between P k P n and the preceding line produced (the angle fc . n being on the right side of the latter), and the horizontal dis- tance dL ." FIG. 21 Given the POLAR COORDINATES 21 31. Angle Between Two Lines: Azimuths. Given the azi- muths (referred to one zero direction) of two lines, Pr P, the angle between the two lines (following in clockwise direction the first line), is /j ?= <*T.S 0,&n> If the azimuth of the first line is less than that of the second line add 360 to the azimuth of the first line before subtracting. 32. Angle Between Two Lines: Bearings. The following table is constructed on the supposition that the angle sought follows (in clockwise direction) the line whose quadrant is given at the top of the table. ANGLE BETWEEN Two LINES OF KNOWN BEARINGS. NE SE s w N W NE diff. 1 80 -(- sum i8o + NE-SW sum * SE 1 80 sum diff. 360 - sum i8o + NW-SE SW i8o-fSW-NE sum diff. 1 80 -f- sum NW 360 sum i8o+SE-NW 1 80 sum diff. Example. What is the angle (measured in clockwise direction from first named line) between a line of NVV SE SE NW bearing and a line of bearing ? We enter column SE NW and follow down to line NW, SE, where we find 1 80 + SE NW, 1 80 4- NE SW, which means that for the sought angle we are to add 180 to the NW subtract the SW SE NE bearing and from the sum bearing. * " Sum" means "sum of given bearings," and " diff." means " differ- ence of given bearings." 22 COORDINATES OF ELEMENTARY SURVEYING 33. Distance Between Two Points. We are given the azimuth a k and the distance d k of the point P k ; and the azimuth a n and the distance d n of the point P n . Both points are referred to the same pole. Required the distance d k . H between P k and P n . Or, briefly, Given P k (a k , ^), P M (a H , d n \ Required d k . H . The required distance is d k . n = \d -f d* 2d* d n cos Aa,.,,]* where Aa x .. w = azimuth difference == >t If a n < a, , add 360 to a n before sub- tracting a k . If Aa^. M > 90 and < 270, its cosine is negative. The following table may be of use where the computation for d^. n is made with a slide-rule, or with a table of logarithmic cosines in which the angles are written for the first quadrant only. EQUIVALENTS OF Cos Aa. When Aa lies in Quad. I Quad. II Quad. Ill Quad. IV cos Aa *sin (90 Aa) sin (Aa-9o) cos (Aa-i8o) *-sin (27O-Aa) sin (Aa-270) Example. Two points, P. z and P. A , are referred to the same pole. Given P. z (40, 20), P 3 (350, 40), the directions being azimuths. What is the distance between P 2 and P z ? * The starred values of cos Aa are given for slide-rule use, since that instrument is not figured for cosines. POLAR COORDINATES 23 I d k 20. 2 d n 40. 3 a k 40 4 a n 350 o 5 Aa = a n a k 6 log cos Aa 9.808 7 log^ 1.301 8 log<4 i .602 9 log 2 0.301 10 sum log 3.012 ii log d? 2.602 12 log d* 3-204 13 dk 400. 14 dn 1600. 15 df + d* 2000. 16 2d k d n cos Aa 1030. 17 d k -n 970. 18 log d; 2.988 19 log d k . n 1.494 20 d*. n 31.2 A B 34. Area of Polygon. Given the azimuths* and distances (re- ferred to one pole) of a series of points P lt P 2 , P 3 . . . P n) taken in order. Required the area Q of the polygon bounded by the lines />!/>, P,P S9 P 3 P<, . . . Pn-iPn, P n P^ Or, using symbols, Given P l (a iy */,), Required the area Q of the polygon P 1 P.,P^ . . . P n P l . In Coordinate Geometry it is shown that the area is : Q = Y-2. (d^d., sin Aa 1>2 + d^ sin Aa,. s -+ . . . 4- d n _i d, t s\n Aa w _ rw + d n d^ sin Aa M>1 ) where Aa,. 2 = a. 2 a l , Aa 2 . 3 = a 3 a^ , etc. Add 360 to an azimuth when it is less than the preceding azi- muth, before making the subtraction for azimuth-difference. E. g. , * If bearings or deflections instead of azimuths are given, compute the corresponding azimuths (Arts. 39 and 41) before beginning the work of find- ing the area. 24 COORDINATES OF ELEMENTARY SURVEYING if a 2 90 (M80-M+4) SW IfmJKA NW 26 COORDINATES OF ELEMENTARY SURVEYING Table B. FROM MAGNETIC TO TRUE BEARING, For use where True Bearing of Magnetic North is NA W, /'. e. , where Declination of Needle is A West. NE SE sw NW NE SE SW NW To use either table follow down the column headed with the quad- rant of the given magnetic bearing m/3 till the "If" . . . statement is satisfied by the given bearing, and use the equation for t$ there found and that quadrant which stands on the same horizontal line. It will be evident that the tables serve for changing from true to magnetic bearing by interchanging the words and symbols for "true" and "magnetic" in the tables and interchanging the tables. Example. The magnetic bearing of a line is S 82 E and the declination of the needle is 25 East. What is the true bearing of the line? The declination of the needle being East, we use Table A. We run down column ' * SE ' ' and find that the given bearing and declination satisfy the condition "If mfl > A" (/. e., 82 > 25). We there- fore use the equation there found : t$ m@ A (z. e.> t$ = 82 25 = 57), and quadrant SE which is named on the same hori- zontal line. Thus we find the required bearing is S 57 E. Example 2. The true bearing of a line is N 80 W, and the de- clination of the needle is 15 W. What is the magnetic bearing of the line ? If the problem were to change from magnetic to true bearing we should use Table B, but for the problem as stated we interchange I UNIVERSITY V OF / X^^UFosSifc^ POLAR COORDINATES 27 the tables, 2. e. , use Table A ; and in this table read //3 for m/3 and m(B for //3. The true bearing of the given line being N 80 W we enter Table A and find in ~-*4*- ___ column "NW" that the ^^ ^To^^^n data satisfy the condition &/*" O "If m/3 > A" (it being 1 80, 360 B 1 . M = B . H L. Example. The azimuth of P^P 2 is 210, of P 1 P 3 is 10. If P 1 P 2 is taken as the zero direction for deflection, what is the deflec- tion of PJ 28 COORDINATES OF ELEMENTARY SURVEYING -^ 0\.3 = 10 -f 360 210 360 ^Q y I6o < I8 o f> I CX^P \ 39 From Bearing to N \ Azimuth. The diagram, T terms of bearing, for each \ 5 ' quadrant. The diagram as- \ (X = \80 e 4G ' a = l80 q -f3 / sumes that the bearing of Q ^ zero azimuth is N. ^o Jy Example. The mag- ty o O J& 160 O^ netic bearings (Art 29) of "-- ,,L - "" certain lines are N 35 E, FlG . 25 S 53 E, S 5 oW, N 7 oW. What are the correspond- ing magnetic azimuths? By use of Fig. 25 we make out the table : 40. From Bearings to Deflection. Given /3 Z = fi (Given) N 35 E S 5 3E S 50 W N 70 W a (Required) 35 127 230 bearing of zero direction for deflection ; /3 n = bearing of any other line. Required = corresponding deflection of other line. R indicates deflection right, and L left. The required deflection is found by means of Table C, page 29. To use the table : In the column headed with the quadrant of the given bearing of the zero deflection line, and on the line begun with the quadrant of the given bearing of the other line, note which con- dition there written is satisfied by the given bearings, and use the equation for 8 written immediately below. Example. The bearing of a line a is N 40 W, and that of a line b is S 50 E. What is the deflection of the line b referred to the line a ? In Table C, in column "NW" and on line " SE " we find two conditions : ' ' If ft > &/ 'and " If &<&" The data satisfy the second condition (i. e., 40 < 50), and we therefore use the second equation : S = (180 -f- fi z &,) R = (180 -f 40 50) R. The required deflection is, therefore, B = 170 R. POLAR COORDINATES 29 Table C. BEARING TO DEFLECTION. QUADRANT or j8 z NE SE sw NW NE 5-&-JUL 5=180-1 4 SE H80+|5 r jyR sw S=(l80+j3 n -j3 z )R NW lfJ5z>j3n &=(l80+j3 n -jyL 41. From Deflection to Azimuth. The deflection 8 n of a line is given, and the azimuth a z of the zero direction for deflection is also given. Required the azimuth a n of the line. a n = a z ^_ g" L. Or in words : If the deflection is muth of the zero deflection line, the given deflection to obtain the required azimuth. If a z < 8 n L add 360 to a z before making the subtraction. If (a z -f- & U R) > 360, decrease the sum by 360. R L rig! left, in it, dicates deflectior add to subtract from i 1 ri^ht. left. ;he azi- 30 COORDINATES OF ELEMENTARY SURVEYING Example. The deflection of line b, referred to line a, is 153 L. The azimuth of line a is 327. What is the azimuth of line bl Az. of b = 327 - 153 = 174. 42. From Deflection to Bearing. The deflection of a line is 5, and the bearing of the zero deflection line is & . Required the bear- ing /3 corresponding to 8. For the solution of this problem the following two tables are given, one for use when the given deflection is right, and the other when it is left. Table D. DEFLECTION RIGHT TO BEARING. 1 QUADRANT OF fi n QUADRANT OF J3 Z NE SE SW NW NE IfS<90-j3 z frfrrf If 8>!80-J3 Z j3=]3 2 +6-l80 IfS>J3 z <90+J3 Z J3-S-J3 Z If=90-J3 z J3=90 IfS=l80-J3 z ^=0 SE IfS>90-J3 z < I80-J3 Z JH80-(S Z +S) IfS90+0z J3H80+&-6 If<5=90+)3 2 J3-90 SW If S> I80-& J3=J3 Z +H80 If>j3 z <90+j3 z -$-& If6<90-|3 z ^5 ffW8(HBz J3=0 If6=90-j3 2 J3=90 NW If6>90+J3 z J3=jS z +l80-S If6>90-J3 z 90-j3 z j3 2 <90+J3, j3=S-^ 2 FS>I80-J3 Z JH+J3 Z H80 If=90-J3 2 J3-90 IfS-l80-j3 2 j8-0 sw IfS>90-4J 2 J3=J3 Z -H80-S If690-J3 z l80-J3 z =A+S-I80 If8<90-j3 2 ^=^z + 8 If8-I80-J3 z J3-0 F6-90-J3 2 ^-90 fied by the data, and immediately beneath this find the required equation for /3, and on the same line on the left find the quadrant for/3. Example. The deflection of a line a is 160 R, and the bearing of the zero deflection line is S 40 E. What is the bearing of line a ? In Table D in column SE we find that the condition : "If S > 90 -f &" is satisfied by the data (/. e., 160 > 90 + 40). We therefore use the equation fi = $ z -f 180 B there found, and use the quadrant NW found on the same line. The bearing of line a is therefore N (40 -f 180 160) W, or N 60 W. 3-. COORDINATES OF ELEMENTARY SURVEYING 43. To Change Pole Without Changing Zero Direction: Azi- muths. Given the azimuth a . k and distance d . k of the point P k referred to the origin P ; and the azimuth a k . n and distance d k . n of the point P n referred to the pole P k . Required to find the azi- muth a . n and distance d . n of P H referred to the pole P . d . n and d k . n form two sides of the triangle P P k P n , and the angle 6 . A . n included between them is known in terms of a . k and a k . n . d . n is the third side of the triangle, and the problem re- duces to finding the distance between two points, for which see Art. 33. 44. From Local Polar to Local Rectangular Coordinates : Azi- muths. Given the azimuth a k . n and the distance d k . n of the point P n referred to the local pole P k . Take P k for local origin for rectangu- lar coordinates, making the axis + Y k identical with the zero direc- tion for azimuth. Required the rectangular coordinates x k . n , y k . n of P n referred to P k . Or, briefly, Given Required Pn (* , /U***i = d k . = d, sn a k . n cos a fc . n FIG. 26 the point P H lies, this table. The signs of sin a k n and cos a k n depend on the quadrant in which The resulting signs of x fc . H and y k . n are given in Quad. I Quad. II Quad. Ill Quad. IV Sign of x Sign of y .+ + + + Logarithmic sines and cosines are preferably taken from a table which is figured at top and bottom of each page for the four quad- rants, as exemplified in Hussey's Mathematical Tables.* When a table figured for only the first quadrant must be used, functions of first-quadrant azimuths which are equivalent to the azimuths of the various quadrants are found by means of the following table. *Allynand Bacon, Boston. POLAR COORDINATES 33 Table F. EQUIVALENT FIRST-QUADRANT FUNCTIONS. When oc Terminates in Quad. i. 5//7 a= Cos a- Tan OC= Cot a= FIRST QUADRANT EQU/I/ALENTS sin OL cos a. [sin (90- a)] tan a cot a [Tan (90-00] i. cos(cc-9o) [sin(iao-CCJ] -5in(GC-90) -cot(tt-90) [-Tan(l80-ttj -tan((I-90) ft -sin (OH 80) -cos(a-90) [-sin(70-d| tan(a-ISO) cot(OH80) [tan(70-a)] K -cos(a-c70) ;-sin(360-al sin(d-270) -cot(CC-70) [-tan(360-dj -tan(tt-c70] The. bracketed equivalents are to be used when computations are made with the slide-rule, since that instrument has only sine and tangent scales. Example. The pole is P l . The polar coordinates of P. L are 1 5, 16 ; of P. A , 200, 32. What are the rectangular coordinates of P 2 and P z referred to origin P l if the azimuth of -f- F (axis) is zero ? (See solution In tabular form on following page.) Note. If log sin and log cos are taken from a table having only first-quadrant headings, insert a column between cols. C and D, in which to write the equivalent first-quadrant angles as found in .table. 45. From Local Pclar to Local _^( Rectangular Coordinates : Bearings. Given the bearing /3 /i ,. M (with the letters of its quadrant) and the dis- tance d k of the point P n , referred to the local pole P k . We take P k for local origin for rectangular coord i- V nates making -|- Y (axis) identical FlG - 2 7 with north. Required the rectangular coordinates x k . H , y k . t referred to P k . Or, briefly, of P.. 34 COORDINATES OF ELEMENTARY SURVEYING Point Polar Coords. log.*- 1 log sin a I log <* 1 log cos a j log j/ Rect. Coords. I 2 3 4 5 Az. a Dist. d X y + + 6 7 8 9 10 P, 15 16 0.617 9-4I3 1 .204 9.984 1.188 4-1 15-4 ii 12 13 14 15 P, 200 32 i .O39n 9-534^ 1-505 9-973 n i -478n 10.9 30.1 A B C D E F G H Given Required sn COS The signs of sin /9 fe . w and cos fi k . n depend on the quadrant in which the point P n lies, i. e,, on the quadrant letters of the bearing. The resulting signs of x k . n and y k . n are given in the following table. NE SE sw NW Sign of x Sign of y + -f 4- For slide-rule computation sin (90 /3) should be substituted for cos /3 because the slide-rule carries no scale of cosines. Example. The bearings and distances of two points, referred to P l as a pole, are P, (N 15 E, 16), P 3 (S 20 W, 32). What are the rectangular coordinates of P 2 and P 3 , referred to origin P l , if the bearing of -f Y (axis) is north ? POLAR COORDINATES 35 Polar Coords. Rect. Coords. I log^ 2 f log sin /3 X y 3 Point Bearing Dist. { log d \ 4 ft d log cos /3 j 5 logy + + 6 0.617 4.1 7 9-4I3 8 P a N 15 E 16 i .204 9 9.984 10 1.188 1,5.4 1 1 1.039 10.9 12 9-534 13 P 3 S20W 32 1-505 14 9-973 15 1.478 30.1 A B C D E F G H 46. From Local Polar to Local Rectangular and Other Coordi- nates : Deflections. Either change from deflections to azimuths (Art. 41 ), and proceed as in Art. 44 ; or, change from deflections to bearings (Art. 42), and proceed as in Art. 45. 47. From Polar Coordinates to Biangular and Biradial Co- ordinates. See Arts. 51, 52 under Biangular, and Art. 59 under Bipolar Coordinates. CHAPTER VI. BIANGULAR COORDINATES. 48. Biangular Coordinates Fig. 28. In the triangle P t P 2 P, , we are given the distance d r . z , and the angles # rl . 3 , rr3 . The position of .P 3 with respect to P l and P. 2 is determined by the data. # rl . 3 , O rv * are the biangular coordinates of P 3 . d r2 is the base, and P l , P 2 are the poles of this system of coordinates. For the purpose of computation the trigonometric notation for a triangle will be substituted for the foregoing, and in such a way that the base will always be represented by c. FlG . 28 49. Distance Between Two Points. Fig. 29. Given the distance PjP.2, the biangular coordinates rl . 3 , #^.3 of the point P 3 , and the biangular coordinates 2<1 . 4 , rri of the point P 4 . Required the distance P a P 4 . Or, briefly, Given d r . z length of base, Required d yi . The figure is a quad- rilateral with diagonals. The lines of the figure form four triangles which are here num- bered I, II, III IV, I and II being those for which the base P l P 2 is a common side; III and IV being those for which P 3 P 4 is a common side. We use the ABCabc triangle notation, employing a BIANGULAR COORDINATES 37 Solution of example in Article 49, page 38. I Triangle I Triangle II 2 A 100 110 3 B 35 45 4 A + B 155 155 5 C 45 25 6 c 20. 20. 7 logr 1.301 I.30I 8 log sin C 9.850 9.626 9 log c 1 sin C 1.675 10 log sin A 9-993 9-973 ii log sin B 9-759 9.850 12 log a 1.444 1.648 13 log i .210 1-525 14 a 27.8 44-5 15 b 16.2 33-5 16 AI 100 ^n 110 17 #ii 45 i 35 18 Am 55 ^iv 75 19 Triangle III Triangle IV 20 ^ 55 75 21 16.2 33-5 22 c 44-5 27.8 23 log 2 0.301 0.301 24 log $ I . 2IO 1-525 25 log c I .648 1.444 26 log cos ^4 9-759 9-4^3 27 log 2.bc cos ^ 2.918 2.683 28 log P 2.420 3-050 29 log r 2 3.296 2.888 30 $ 2 263. 1 1 20. 31 <: 2 1980. 773- 32 b 1 -j- <: 2 2243. 1893- 33 2&T COS /4 828. 482. 34 tf 2 1415. 1411 . 35 logrt 2 3.151 3-150 36 log tf I-576 1-575 37 a 37-7 37-6 A B C 38 COORDINATES OF ELEMENTARY SURVEYING subscript to designate the triangle to which each part belongs, and distribute the letters as shown in the figure. Notice that Ci = c llt a ll = cm t ^ = c lv , b l = b lUt b u =-' b lv . We employ the following equations : C = 1 80 (A -{- B} . . (i) I (a and b are biradial coordi- a = (c I sin 7) sin A . . (2) J> nates of the point opposite the b = (c I sin C} sin B . . (3) j base c. See Art. 56.) The three equations above apply to triangles I and II ; the follow- ing applies to triangles III and IV. a == (b 1 -f c* 2bc cos A)% . . . (4) It should be remembered that an a of (i), (2), (3) is a c in (4). Example. Given the following data, referring to Fig. 29 : = no, BI == 35 U = 45 = 100 = 20 ; find the distance P 3 P 4 . (Solution is given on page 37.) 50. Area of the Triangle. From Trigonometry we have : Area = C L sin A sin B / 2 sin C : in which c base ; A, D biangular coordinates ; C = 180 - (A+B). Example. The biangu- lar coordinates of P 5 , refer- red to base P 7 P 2 , are 0.,. 7 . 5 = 30, O r 2 . 5 = 70, 4.,= 12. What is the area of the triangle P 7 P 5 P,? 51. From Biangular to Polar Coordinates (Azimuth and Distance). Fig. 30. Given the base PjP 2 = ^. 2 , and the biangular coordinates 2 -i-3 > 01-2-s f tne P omt ^a referred to this base, and the azimuth a r . 2 of the line P l P 2 . Required the azimuth a,. 3 and the distance d rz of P 3 referred to the pole P l . Or, briefly : Given P z (0ri-s 0r-s) and a i- 2> Required P 3 (a r3 , ^. 3 ). I Base P P *1*2 2 Point opp. base P, 3 P 7 P 2 = ^ 12. 4 # 2 . 7 . 5 :=r ^4 30 5 # 7 . 2 . 5 ^^ .5 7 6 ^ +^ 100 7 C 80 8 log^ 1.079 9 log^ 2 2.158 10 log sin A 9.699 ii log sin B 9-973 12 sum log 2.830 13 log 2 0.301 14 log sin (7 9-993 15 sum log 0.294 16 log area 2-536 17 area 344- A B BIANGULAR COORDINATES 39 Using the ABCabc triangle notation, we put c base, A = O ri . 3t B = rv9 . We know that C = 180 (A + B). The required distance is b *-*"J.JJ IS W4.J.X J^f I ^AJJ. \^ j * 1 and the required azimuth is rz = c sin B / sin C ; in which is positive when reck- oned in clockwise, and negative when reckoned in counter-clock- wise direction from the base. Example. The line P^ is 40 ft. long, and its azimuth is 240. A point P 3 is referred to P : P 2 as a base, by the biangular coordi- nates 2 . r3 = 50, 1<2 . 3 = 80. 2<1 . 8 is reckoned in counter-clock- wise direction from the base. What is the azimuth and length of the lineP 1 P 3 ? I BaseEEE^P, O E> 2 Point = P 8 2 ~p 3 base c 40. 4 5 biang. f A coords. 1 B 50 80 6 A+B I30 7 C 50 8 log^r 1.602 9 log sin B 9 993 10 sum log 1-595 ii log sin C 9.884 12 log b i. 711 13 b = d r3 51-4 H a ri 240 15 vrz = A 50 neg. 16 ,, 190 A B 52. From Biangular to Polar Coordinates (Bearing and Dis- tance). Fig. 31. Given the base P 1 P 2 = d v% and the biangular coordinates rl . t , r . 2 - 9 of the point P 3 referred to this base, and the bearing /3 r2 of the line PjP^ Required the bearing /3 r3 and distance d r s of P 3 referred to pole P t . Or, briefly, 40 COORDINATES OF ELEMENTARY SURVEYING Given P 3 (0 rr3 , 1>2 . 3 ) and ft.,, Required P, (ft.,, ^.,). Using the ABCabc triangle nota- tion, we put c = base, A = # rl . 3 , B = e m . Now C= 1 80 04 -f ), FIG. 31 and ^ = ^. 3 = c sin .Z? / sin 7. ft. 3 is found by entering Table D (or E), Art. 42, with fi z = ft. a and $ = ,.!.,. 53. From Biangular to Bira- dial Coordinates: Single Point (i. e. , given one side and two an- gles of a triangle, to find the two other sides). Fig. 32. Given the length d rz of the base P,P 2 , and the biangular coordinates $i- 2 -3 > ^2-i-3 f a point P 3 referred to this base. Required the bi- radial coordinates d d. rz of P 3 referred to the same base. Or, briefly, Given P 3 (0 r2 ,, 0,^), and base d^ , Required P, (^.,, d^. To enable us to use the ABCabc triangle notation we ^ Now C = 1 80 (A + ), # = and the check-equation is cos C = (a* -\- b z ^ 2 ) / 2ab. For an example see the following Article. 54. From Biangular to Biradial Coordinates : Series of Points (i. e.) to solve a chain of triangles). In Fig. 33 we have a series of points P l , P 2 , P 3 , . . . . Given the biangular coordinates and base for each point of the series, required the biradial coordinates, as follows : BIANGULAR COORDINATES 41 Point Base (Given) Biangular Coordinates (Required) Biradial Coordinates P, P t P. P. P,P, = d^ P. i P, = d. i . z p.p<^4< P.P. aU 2 -i-3 > 01-2-3 #3-2-4 > #2-3-4 #4-3-5, #3-4-5 P-4-6> #4-5-6 ^1-3 > ^2'3 ^2-4 y d y d^y ^4-5 di.~ % d^~ In other words, Given two angles of each triangle of a chain, and one side of one of the triangles, required to compute all other triangle sides of the chain. 1. Plot the points of the series, using the data. Fig. 34. (This plot need be nothing more than a mere sketch or diagram upon which to write the notation, but if made with some care serves as a rough check on the computations.) 2. Mark the data* upon the plot as shown. 3. Number the triangles I, II, III, . . . , as shown. 4. Letter the sides and angles of each triangle with the ABCabc notation, using red ink, according to this scheme : (a) Place the letters within the triangle as shown, (b) Call the given base c, (c} Call the base of each succeeding tri- angle c, (/sin O 1-363 1.464 10 log sin ^4 9-985 9.699 ii log sin .# 9.850 9-993 12 log^ 1-348 i . 163 13 log b 1.213 1-457 H log 2 0.301 0.301 15 # 22.3 14-5 16 b 16.3 28.6 17 logtf 2 2.696 2.326 18 log ^ 2.426 2.914 19 log c 1 2 .6O2 2 .696 20 c? 497- 212. 21 & 267. 820. 22 a 1 -f a 764. 1032. 23 r' 2 400. 497- 24 ^2 + ^ 3 _ <* 364- 535- 25 log( 2 + ^ 2 ^-) 2.561 2.728 26 log 2^ 2.862 2.921 27 log (a'+fr <*)l*ab 9.699 9.807 28 log cos C 9.699 9.808 * c of triangle II is identical with a of triangle I. BIANGULAR COORDINATES 43 55. From Biangular to Rectangular and Other Coordinates We first make the tranformation from biangular to polar coordinates ; and second, convert these polar coordinates into the required co- ordinates as directed under Polar Coordinates, Chap. V. CHAPTER VII. BIRADIAL COORDINATES. 56. Biradial Coordinates. Fig. 36. In the triangle P^P,, we are given the distance d r . 2 and the two distances d rz , d rz . The position of P 3 with respect to P t and P 2 is determined by the data. d v ^ d TZ are the biradial coordinates ofP 3 . d V i is the base, and P 1 and P 2 are the poles of this system of coordinates. 57. Area of the Triangle Trigonome- try gives us Q = area = [s (s a) (s b} (s c)]^ in which s = }4(a H- b -f- r). The check equation is (s a) -f- (s b} -f ( s c) = s. Example. The three sides of a triangle are 12, 8, 6. What is the area of the triangle ? 58. From Biradial to Bian- gular Coordinates. Fig. 37. Given the biradial coordinates d rz , d. 2 . 3 of the point P 3 referred to the base P l P 2 whose length d r . 2 is also given. Required the biangular coordinates # 2 .j. 3 , ^.g.g of P 3 referred to the same base. Or, briefly, Given the base d r ^ and P 3 G^.g, d r3 ), Required P 3 (# 2 .,. 3 , O vrs ). We reletter the triangle to conform to the ABCabc triangle notation, calling the base c as usual. The distribution of the remaining letters is shown in the figure. I a 12 2 b 8 3 c 6 4 2S 26 5 S 13 6 s a i 7 s b 5 8 s c 7 9 (check) s 13 10 log s i . 114 ii log (s a) 0.000 12 log (s b} 0.699 13 log (s c} 0.845 log Q> 2.658 J 5 log<2 1.329 16 area = Q 21.3 A B BIRADIAL COORDINATES 45 The three sides of the triangle being a, b, c, i. Compute s == y^(a -\- d -{- c) . (i) I 2. Computer a, s b, s c and solve the check equation i? s =(s a) + (s b) + (s c) (2) > 3. Compute k = [>/(j -a)(s b) (s eft* (3) 4. Find the angles A, B, C, from FIG. 37 and solve the check equation cot y 2 B - cot y 2 c cot which is derived thus : multiply together (4) , (5) and (6), ob- taining cot y 2 A cot y 2 B - cot y 2 c = fr(s a) (s b ) c) = k* [k(s a) (s fi)(s cft Replace ft? by its value in (3), obtaining (7). The required biangular coordi- nates are O ri .* = A, O m = B. Example. Given the bira- dial coordinates d^ = 12, d,^ = 20 of the point P 3 referred to the base P 1 P. 2 whose length is 1 6. Required the biangular coordinates rl . 3 , rrB of P 3 referred to the same base. 59. From Biradial to Polar Coordinates. We (i) convert the biradial into biangular co- ordinates (Art. 58); and (2) con- vert the biangular coordinates into either azimuth or bearing, cot%A=k(s d) . (4) cot y 2 B =k (s b) . (5) i a 20. 2 b 12. 3 c 16. 4 2S 48. 5 S 24. 6 s a 4- 7 s-b 12 . 8 s c 8. 9 (check) s 24. 10 log (s a) 0.602 ii log (s b} 1.079 12 log (s e) 0.903 13 sum log 2-584 H log s 1.380 15 log k'- 8.796 16 log k 9- 398 17 log cot y 2 A o.ooo 18 log cot ^ 0.477 19 log cot y 2 c 0.301 20 sum log 0.778 21 (check) log ks 0.778 22 y?A 45 23 %B i8.4 24 y-zC 26. 6 25 A 90 26 B 36. 8 27 C 53. 2 28 check :A+B-\-C i8o.o A B 46 COORDINATES OF ELEMENTARY SURVEYING as desired. We then have from the given biradial coordinates and the derived azimuths (or bearings) two sets of polar coordinates for the point P 3 , viz. : PS ( a i'3> ^1-3) referred to the pole P 1} P 3 (a 2 . 3 , Example. PjP 2 is 20 ft. long. The angle 6. 2 . r3 = 40, P 2 P 3 is 25 ft. What are the polar coordinates r2 . 3 , <^ 2 . 3 of P s referred to the base P, P 2 ? We put PiP 2 = c, d. r s = a, O rr $ = A, and so on. (Solution in tabular form given on next page.) 63. From Bipolar to Rectangular and Other Coordinates We convert the bipolar into polar coordinates (Art. 62). The derived polar coordinates are converted into rectangular or other coordinates as shown in Chap. V. 48 COORDINATES OF ELEMENTARY SURVEYING Solution of Example in Article 62, page 47 : I A 40 2 c 20 3 a 25 4 log^ I .301 5 log sin A 9.808 6 log c sin A I . lOQ 7 log a 1.398 8 log sin C 9.7II 9 C 30. 9 10 A + C 70. 9 ii B 109. i A B CHAPTER IX. TRIPOLAR COORDINATES. 64. Tripolar Coordinates. Fig. 40. Given the distances P : P 2 = c , and P. 2 P 3 = c' 9 angle P^P^P* = 6. Given, also, the two angles P l P, P 2 , P 2 P 4 P 3 . These data are sufficient to determine the position of the point P 4 with respect to P lt P. 2 , P 3 (except when P 4 falls on the circumference drawn through P, , P 2 , P 3 ). C, C' are the tripolar coordinates of P 4 referred to the P P P P 65. From Tripolar to Polar Coordi- nates. Fig. 41. Given the two segments P l P 2 = c, P.,P 3 = c', and the included angle 6, of the double base P l P t P 2 P 3 . FIG. 41 Given also the tripolar coordinates C, C' of the point P 4 referred to this double base. Re- quired the polar coordinates (de- flection S,. 4 = B and distance d r4: = a), of the point P 4 referred to zero direction P 2 Pj and pole P 2 . We use the ABCabc triangle notation as shown in Fig. 41 where the letters of the right-hand tri- angle are primed, a and a' are identical. a = a ' = c sin A / sin C = c' sin A' /sin C' (i) A + A' = 360 - (C + C + 0) - From (2) we have and sin A' = sin (7 A} = sin 7 cos A cos 7 sin A . (2') 50 COORDINATES OF ELEMENTARY SURVEYING Substituting this value of sin A' in (i), we obtain c sin A I sin C -- (c r sin 7 cos A c' cos 7 sin A) / sin C' or, cot A = (c sin C' /sin C + ^' cos 7) / r' sin 7 . (3) where M csm "/sin N = c' cos 7 O = c' sin 7 (2') (4) (4) I c 20 2 c' 40 3 C 15 4 C' 28 5 6 140 6 c 'H- c* -\- o 183 7 7 177 8 log c 1.301 9 log sin C' 9.672 10 sum log 0-973 ii log sin C 9-4I3 12 log M i .560 13 log N i .6oin 14 log cos 7 9-999* 15 log c[ i .602 16 log sin 7 8.719 17 log (9 0.321 18 Jf 36.3 19 N 39 9 20 M + N 3-6 21 log M+ N o . E \56n 22 log cot ^4 0.23 5* 23 ^4 149. 8 * 24 7 A = A ' 2 7 . 2 25 A \ /~* JT ~p tx 164. 8 26 B 15. 2 27 A' + C' 55 -2 28 B' 124. 8 29 log sin A 9.702 3 log sin ^4 1.003 3 1 log ,4 1-590 S 2 log sin A ' 9.660 33 log c' sin ^4 ' i .262 34 check : log a ' 1-59 35 a' 38.9 # == I8o (A + C) # = =18004'+ C") The polar coordinates (de- flection and distance) are, there- fore, respectively, *n = B . . (5) d>2-i = a = c s i n A /sin C (6) the zero direction being P. i P l and the pole P 2 . The check equation is d ri = a' = c'smA f /sin C' (6') There are two possible po- sitions of the point P 4 , one on the concave side of the double base, and the other on the con- vex side. Example. The point P 4 is referred by tripolar coordinates to points P t P 2 P 3 . Given C = 15, r = 28-, p,p>p 3 = e = I 4 0, P 1 P,= *=20, P,P. = ' = 40. What is the angle p i p i p i==j g =: 5 r4i a nd the distance ^ 2 . 4 = a ? Plot the points P x , P a , P 3 on one piece of paper. Draw on a piece of tracing-paper three lines from any point P so as to form two adjacent angles equal respectively to the given angles C and C '. Move the tracing-paper about on the plot of P lt P a , P 3 until the lines TRIPOLAR COORDINATES 51 radiating from P respectively pass through P v P 2 , P 3 simulta- neously. Prick through the tracing-paper at P on to the other paper. The pricked point is P v Measure B and a to check the computations. 66. From Tripolar to Rectangular and Other Coordinates. The tripolar coordinates are first converted into polar coordinates after which any other desired coordinates are derived according to the directions given under Polar Coordinates, Chap. V. PART III. VERTICAL SURVEYING. Any one of the systems of coordinates which have been presented under ' ' Horizontal Surveying " (Part II) can be used in Vertical Surveying, but for practical reasons the only systems commonly em- ployed are rectangular, polar and polar-rectangular coordinates. It will be well to bear in mind the fact that while only one hori- zontal plane can be drawn through a given point, an infinite number of vertical planes can be passed through the point. Nearly all vertical surveying is for the purpose of determing eleva- tions, in which case it is called leveling. CHAPTER X. RECTANGULAR COORDINATES. 67. Rectangular Coordinates. In Fig. 42 we have a point P n referred to the two rectangular coordinate axes Z k and D k (which intersect at the origin P^) by the rectangular coordinates d k . n , z k . n . >Pn Z Datum Line K-n *+o* (Horizontal) FIG. 42 FIG. 43 The two axes lie in the vertical plane determined by P k and P n . The 2 below the datum line and horizontally distant d rz from P! . P 3 is z rz above the datum and horizontally distant d rz from P! , the origin. All these relations are expressed briefly by writing 68. From General to Local Rectangular Coordinates. Fig. 44. Given the general coordinates d k , z k of the point P k , and d n , z n of the point P M . Required the local coordinates d,,.,,, #*. of 7 jJ7 _l_ / ~ /. y^ the point P n referred to the -L __A CI K r\ ^1 local axes /? A and Z k drawn through the local origin P k . Or, briefly, Given P k (d k , ^,), and P n (d n , ^), Required P w (^. w , ^. w ). Evidently d k . w = d n d fc . = elevation difference General Datum FIG. 44 where the signs of all quantities are to be considered when making numerical substitutions. 69. From Local to General Rectangular Coordinates ___ Fig. 44. Given the coordinates d k . H , z fc . n of the point P n referred to the local origin P k , and the coordinates d k , z k of P k referred to the genarel origin. Required the general coordinates d n , z n ofP n . Or, briefly, Given P(^., ^-) and P, (^, z k \ Required P n (d n , ^). Evidently d n = d k + d A . n , Z H = z k -f ^ , in which the signs of all quantities are to be considered when making numerical substitutions. 54 COORDINATES OF ELEMENTARY SURVEYING 70. From Local Rectangular to Polar Coordinates. Fig. 45. Given the rectangular coordinates d k . n , z k . n of the point P n referred to the pole P k . Required the polar coordinates Required P n (tr k . H , sd k . t ^). The tangent of the slope is tan >, J^..,,). The tangent of the slope angle 72. From Lccal Rectangular to p Polar -Rectangular Coordinates.- Fig. 47. Given the rectangular co- FIG. 47 RECTANGULAR COORDINATES 55 ordinates d A . Ht z k . n of the point P n referred to the poleP*. Re- quired the polar-rectangular coordinates d k . n of P M referred to the datum line D k and the pole-origin P k . Or, briefly, Required P n (cr k . H , ^. M ). The slope angle is 0V =s= o- A . M . The horizontal distance is d k . n = j^. M cos o- A . H . CHAPTER XII. POLAR-RECTANGULAR COORDINATES. 76. Polar-Rectangular Coordinates In Fig. 51 is shown a point P n referred to the datum line D k and pole-origin P k by the polar- rectangular coordinates a k . n , d k the (horizontal) distance. is the slope angle; d k . n is 77. From Polar-Rectangular to Local Rectangular Coordinates. Fig. 52. Given the polar-rectangular coordinates dfrn f the point P n referred to the datum line D k and POLAR-RECTANGULAR COORDINATES 59 the pole-origin P \ . Required the polar coordinates cr k . H , sd k . n of P n referred to the datum line D k and the pole P k . Or, briefly, Given P n (a k . n , */*.), Required P H (0*. M , J^. M ). The slope angle is y& %k f the point Pj, and^T M , y n , z n of the point P n referred to the same origin. Re- quired the rectan- gular coordinates P M referred to the origin P k , the lo- cal axes X k , Y k , Z k being taken parallel respect- ively to the gen- eral axes X, Y, Z. Or, briefly, Given Required General Datum FIG. 55 P k (x k , y k , P n \ An inspection of Fig. 55 shows that and The signs of the coordinates must be considered when substitut- ing numerical values. 81. From Local to General Rectangular Coordinates. Single Point. Fig. 55. Given the coordinates x ky y ky z k of the point P k referred to a general origin ; and the coordinates x k . n , y k . n , z k . n of the point P n referred to the local origin P k , the general axes being parallel respectively to the corresponding local axes. Required the corrdinates x n briefly, y n , z n of P n referred to the general origin. Or, 62 COORDINATES OF ELEMENTARY SURVEYING *fo. y*n> Zk-n Given P k (x k , y** **) and P H (x Required /*(#, y, t , z, t ). An inspection of Fig. 55 shows that X n r== %k ~T %k'n) and z = The signs of the coordinates must be considered when making numerical substitutions in these equations. It is evident that the x's and jy's can be computed as in Art. 21, and the z's computed by themselves ; or the three coordinates can be computed in one algorithm. 82. From Local to General Rectangular Coordinates. Series of Points. Given a series of points, P lt P^, . . . . Given the gen- eral coordinates x^ , y l , z l , of the first point P l of the series. For each point after the first we are given the local coordinates referred to the preceding point as origin. All the ^r-axes are parallel ; all the j-axes are parallel ; and all the ^-axes are parallel. Required the general coordinates of any point of the series. Or, briefly, Given P l (X , j, , ^), Pi(*r,> -Ti-2> *i-.), * 3 V^-3 J^2'3 ^2-S >> Required P n (Xn* ?, ^). The required coordinates of P H are x n = x^ -f x v i + XY* + . . + x n _ rn , y* =}'i -\-yr-* +7 2 -3 -f . . -f- j*w, ^ = 2, + ^., + * a . 8 + . . . -f r w _ 1>w . Or, in words, abscissa The general ordinate elevation of any point of the series is equal to the algebraic sum of the local abscissas ordinates elevations of all the points after the first point up to and including the given point, and the general of the first point. abscissa ordinate elevation RECTANGULAR COORDINATES If we are considering the lines P l coordinates may be expressed thus : 63 . the required y tt = Or, in words : The general . 2 -f a . s -f . . . abscissa ordinate elevation of any given point of the series is equal to the algebraic sum of the general I x- abscissa ordinate elevation of the first point and the differences of all the lines between the first point and the given point. Them's andjy's can be considered by themselves (as in Art. 22) and the ^'s separately considered (as in Art. 69); or the three co- ordinates can be computed in one algorithm. 83. From General Rectangular to Polar Coordinates. Fig. 56. Given the rectangular coordinates x k > y k , z k of the point P k , and x n , y n , z n of the point P n referred to the same origin. Required the polar coordinates a A . M , Evidently, a k . n = a k . n and 4 >w = ^.^ . cos 87. From Polar to Local Rectangu- FlG> 58 lar Coordinates. Given the polar coordinates a fc . n , (r k . ni sd k . n of the point P n referred to the pole P k . Required the rectangular coordinates x k . n , y k . n , z k . n of P n referred to the origin P k ; the axes being so taken that Z k and Y k lie in the zero-azimuth plane of polar coordinates. Or, briefly, 66 COORDINATES OF ELEMENTARY SURVEYING Given P n ( a k-n> o- k . nt sd k . n ^ Required P n (x k . n , y k . n , z k . n ). In Fig. 57 d k . n = sd k . n cos V*.*. = d }t . n cos a l k . n k . n , The check equation is x\. n + y\. n -f z\. n = (sd k . n )\ Example. sd r2 =- 20, sd r3 = 40, a V2 = 340, a r3 = 110, o- r2 = 12, o- 1>3 = 8. Required ^. 2 , 7^, ^. 2 , and ^ r3 , ^.3, ^1-3 ; + Y (axis) being taken as zero azimuth. I /\/> P.P. 2 a 34 o 110 3 (T 12 8 4 sd 20. 40. 5 log sd I.30I i .602 6 log cos cr 9.990 9.996 7 log^ I.29I 1.598 8 log sin a 9-534" 9-973 9 log cos a 9-973 9-534 n 10 log sin cr 9.318 9-144" ii log x 0.82 ^n 1 . 571 12 log/ 1.264 i.i32n 13 log* 0.619 o.746n 14 X -6.7 37-2 15 y 18.4 -13-6 16 4.2 -5-6 17 log;r 2 1.650 3-I42 18 log y j 2.528 2 . 264 19 log T 2 1.238 1.492 20 log sd 2 2 .6O2 3.204 21 X* 44-7 1387. 22 j/ 2 337- 184. 23 ^ 2 17 3 31- 24 *+/.+ 2 2 399- 1602. 25 check : (j<^) 2 400. 1600. 88. From Polar to General Rectangular Coordinates. Change (i) from polar to local rectangular (Art. 87) and (2) from local rectan- gular to general rectangular coordinates (Art. 81 or Art. 82). CHAPTER XV. POLAR-RECTANGULAR COORDINATES. 89. Polar-Rectangular Coordinates. The polar coordinates be- come the polar-rectangular coordinates of a point if we substitute horizontal distance for slope distance. In Fig. 59 the point P n is referred to the zero-azimuth plane Z^ Y k , the datum plane D k (i. e., the horizontal plane J k Y k ") t and the pole-origin P k , by the polar-rectangular coordinates a k . n , called azimuth, n (*., d k . n ), Required P n (x k . n) y k . n , z k . n ). An inspection of Fig. 59 will show that x^n = dk>n sin a k . n yk-n = 4-n cos a k-n z k-n = dk-n tan o- k . n . FIG. 59 91. From Polar-Rectangular to General Rectangular Coordi- nates. After finding the local coordinates, convert these into gen- eral coordinates by Art. 81 or Art. 82. 92. From Polar-Rectangular to Polar Coordinates. Given the polar-rectangular coordinates a k . n , OVn ^4-n), Required Pn( a k-n> 0V n> -"4-n)- Evidently, a k . n = a k . n , and = d k . n / cos a k . n . THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE FOURTH DAY AND TO $1.OO ON THE SEVENTH DAY OVERDUE. SEP 15 1933 86l 9S d3S AU6 LD 21-50m-l,'3: YC 13548