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Math for Surveyors

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Math for Surveyors Math For Surveyors James A. Coan Sr. PLS Topics Covered 1) The Right Triangle 2) Oblique Triangles 3) Azimuths, Angles, & Bearings 4) Coordinate geometry (COGO) 5) Law of Sines 6) Bearing, Bearing Intersections 7) Bearing, Distance Intersections Topics Covered 8) Law of Cosines 9) Distance, Distance Intersections 10) Interpolation 11) The Compass Rule 12) Horizontal Curves 13) Grades and Slopes 14) The Intersection of two grades 15) Vertical Curves The Right Triangle B Side Opposite (a) A C Side Adjacent (b) a b a SineA = CosA = TanA = c c b c c b CscA = SecA = CotA = a b a The Right Triangle The above trigonometric formulas Can be manipulated using Algebra To find any other unknowns The Right Triangle Example: a a SinA = SinA· c = a = c c SinA b b CosA = CosA· c = b = c c CosA a a TanA = TanA·b = a = b b TanA Oblique Triangles An oblique triangle is one that does not contain a right angle Oblique Triangles This type of triangle can be solved using two additional formulas Oblique Triangles The Law of Sines a b c = = Sin A Sin B Sin C C b a A c B Oblique Triangles The law of Cosines a2 = b2 + c2 - 2bc Cos A C b a A c B Oblique Triangles When solving this kind of triangle we can sometimes get two solutions, one solution, or no solution. Oblique Triangles When angle A is obtuse (more than 90°) and side a is shorter than or equal to side c, there is no solution. C b a B A c Oblique Triangles When angle A is obtuse and side a is greater than side c then side a can only intersect side b in one place and there is only one solution. C a b B A c Oblique Triangles When angle A is acute (less than 90°) and side a is longer than side c, then there is one solution. C b a A c B Oblique Triangles When angle A is acute, and the height is given by the formula h = c Cos A, and side a is less than h, but side c is greater than h, there is no solution. h b a A B c Oblique Triangles When angle A is acute and side a = h, and h is less than side c there can be only one solution C b a = h A c B Oblique Triangles When angle A is an acute angle and h is less than side a as well as side c, there are two solutions. C b h C’ a’ a A c B Azimuth Angles Bearings Azimuth, Angles, & Bearings Azimuth: An Azimuth is measured clockwise from North. The Azimuth ranges from 0° to 360° Azimuth, Angles, & Bearings Azimuth: N 360° 0° 270° 90° 180° Azimuth, Angles, & Bearings Azimuth to Bearings In the Northeast quadrant the Azimuth and Bearing is the same. N E AZ = 50°30’20” = N 50°30’20”E Azimuth, Angles, & Bearings Azimuth to Bearings In the Southeast quadrant, subtract the Azimuth from 180° to get the Bearing. 180° - 143°23’35” = S 36°36’25”E Azimuth, Angles, & Bearings Azimuth to Bearings In the Southwest quadrant, subtract 180° from the Azimuth to get the Bearing 205°45’52” – 180° = S 25°45’52”W Azimuth, Angles, & Bearings Azimuth to Bearings In the Northwest quadrant, subtract the Azimuth from 360° to get the Bearing. 360° - 341°25’40” = N 18°34’20”W Azimuth, Angles, & Bearings Bearings to Azimuths In the Northern hemisphere Bearings are measured from North towards East or West N 47°30’46”E N 53°26’52”W Azimuth, Angles, & Bearings Bearings to Azimuths In the Southern Hemisphere the Bearings are measured from South to East or West S 71°31’40”E S 29°25’36”W Azimuth, Angles, & Bearings Bearings to Azimuths In the Northeast quadrant, the Bearing and Azimuth are the same. N N 45°30’30”E = AZ 45°30’30” Azimuth, Angles, & Bearings Bearings to Azimuths In the Southeast quadrant, subtract the Bearing from 180° to get the Azimuth. 180° - S 51°25’13”E = AZ 128°34’47” Azimuth, Angles, & Bearings Bearings to Azimuths In the Southwest quadrant, add the Bearing to 180° to get the Azimuth. S 46°20’30”W + 180° = AZ 226°20’30” Azimuth, Angles, & Bearings Bearings to Azimuths In the Northwest quadrant, subtract the Bearing from 360° to get the Azimuth. 360° - N 51°25’41”W = AZ 308°34’19” Azimuth, Angles, & Bearings Angles: To find the Angle between two Azimuths, subtract the smaller Azimuth from the larger Azimuth. 325°50’30” Larger Azimuth 215°20’10” Smaller Azimuth 110°30’20” Angle Azimuth, Angles, & Bearings Angles: If both bearings are in the same quadrant, subtract the smaller bearing from the larger bearing. S 82°35’40”E S 25°15’10”E 57°20’30” Azimuth, Angles, & Bearings Angles: If both angles are in the same hemisphere (NE and NW) or (SE and SW), add the two bearings together to find the angle N 30°15’26”E N 21°10’14”W 51°25’40” Azimuth, Angles, & Bearings Angles: If one bearing is in the NE and the other is in the SE or (NW and SW), add the two together and subtract the sum from 180° 180°-(N15°50’25”W+S 20°10’15”W)=143°59’20” Coordinate Geometry COGO Coordinate Geometry The science of coordinate geometry states that if two perpendicular directions are known such as an X and Y plane (North and East). Coordinate Geometry The location of any point can be found with respect to the origin of the coordinate system, Coordinate Geometry or with respect to some other known point on the coordinate system. Coordinate Geometry This is accomplished by finding the deference between the X and Y coordinates (North and East) of a known and unknown point and adding that deference to the known point. Coordinate Geometry The magnitude and direction (Azimuth and distance) can also be found between two points if the coordinates of the two points are known. Coordinate Geometry C East B DEast B& D SineA = CscA = B& D DEast DNorth North B& D CosA = SecA = B& D DNorth DEast DNorth TanA = CotA = DNorth D A East This will give you the angle from Pt. A to Pt. B Coordinate Geometry Pythagorean Theorem Dist = DNorth2 + DEast 2 This will give you the distance from Pt.A to Pt.B Coordinate Geometry Example 1 Known: •The coordinates for point A •The bearing and distance from point A to point B Coordinate Geometry Example 1 Point A coordinates N 10,000.00 E 5,000.00 The bearing from Point A to point B N 36°47’16”E The distance from Point A to Point B 1,327.56 feet Coordinate Geometry Example 1: East B North A N 10,000.00 E 5,000.00 Coordinate Geometry Warning! You must convert the degrees, minutes, and seconds of your bearing to decimal degrees before you find the trig function Coordinate Geometry Example 1 Find North: Cos. Bearing x Distance = D North Cos. N 36°47’16”E x 1,327.56’ = 1,063.19’ Coordinate Geometry Example 1: Find East Sine Bearing x Distance = D East Sine N 36°47’16”E x 1,327.56’ = 795.01’ Coordinate Geometry Example 1: Because point B is Northeast of point A you must add your calculated distances (both North and East) to the coordinates of A to find the coordinates of point B Coordinate Geometry Example 1: North A + North = North B East A + East = East B Coordinate Geometry Example 1: N 10,000.00’ + 1,063.19’ = 11,063.19’ E 5,000.00’ + 795.01’ = 5,795.01’ Point B North = 11,063.19’ East = 5,795.01’ Coordinate Geometry Example 2: Given: Coordinates of point A N 10,000.00 E 5,000.00 Coordinates of point B N 10,978.69’ E 3,924.71’ Coordinate Geometry Example 2: Point B N 10,978.69’ E 3,924.71’ Note: Point B is Point A Northwest of N 10,000.00 Point A E 5,000.00 Coordinate Geometry Example 2: First Find the deference in North between point A and point B Point B = 10,978.69’ Point A = 10,000.00’ Deference = 978.69’ Coordinate Geometry Example 2: Second Find the deference in East between point A and point B Point A = 5,000.00’ Point B = 3,924.71’ Deference=1,075.29’ Coordinate Geometry Example 2: Third Find the distance between point A and point B Dist = D North2 + D East2 Dist = 978.692 + 1,075.292 The distance from A to B = 1,453.99’ Coordinate Geometry Example 2: Fourth Find the bearing from point A to point B 1,075.29 D East Tan A = Tan A = D North 978.69 Coordinate Geometry Example 2: Fifth The angle from point A to point B is 47°41’34” Because point B is Northwest of point A the bearing is N 47°41’34”W The Law of Sines a b c = = Sin A Sin B Sin C C A B c The Law of Sines The law of Sines can be used to solve several Surveying problems, such as finding the center of section The Law of Sines Example 1: Given Coordinates for all 4 section quarter corners Need to find The center quarter corner The Law of Sines a Points a, b, c, & d Have known b coordinates d c The Law of Sines First a Inverse between points c and d d b c The Law of Sines a This gives a bearing and distance from c to d d b c The Law of Sines Next a Inverse between a & c d b And inverse between d & b c The Law of Sines After inversing a you will have a bearing and distance Dist & Bear Bear & Dist d b between a & c as well as d & b c The Law of Sines Because the bearings of all three lines are known the angles between them can be calculated.
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