[{"data":1,"prerenderedAt":5852},["ShallowReactive",2],{"blog-\u002Fblog\u002Ftienstra-resection-danger-circle":3,"blog-surround-\u002Fblog\u002Ftienstra-resection-danger-circle":5845},{"id":4,"title":5,"author":6,"body":7,"category":5805,"description":5806,"extension":5807,"image":5808,"meta":5809,"navigation":5837,"path":5838,"publishedAt":5839,"readTime":5840,"seo":5841,"sitemap":5842,"stem":5843,"__hash__":5844},"blog\u002Fblog\u002F23.tienstra-resection-danger-circle.md","Tienstra Resection: Understanding the Danger Circle and When to Reposition","Gordon McInnes",{"type":8,"value":9,"toc":5778},"minimark",[10,15,150,154,168,172,190,194,340,344,519,1735,1768,1923,2583,2820,2824,3278,3282,3760,4909,4913,5025,5296,5300,5564,5568,5648,5652,5689,5693,5752,5756,5759,5763],[11,12,14],"h2",{"id":13},"overview","Overview",[16,17,18,22,23,70,71,137,138,141,142,145,146,149],"p",{},[19,20,21],"strong",{},"Resection"," is the process of determining the coordinates of an unknown observer station (",[24,25,28,50],"span",{"className":26},[27],"katex",[24,29,32],{"className":30},[31],"katex-mathml",[33,34,36],"math",{"xmlns":35},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[37,38,39,46],"semantics",{},[40,41,42],"mrow",{},[43,44,45],"mi",{},"P",[47,48,45],"annotation",{"encoding":49},"application\u002Fx-tex",[24,51,55],{"className":52,"ariaHidden":54},[53],"katex-html","true",[24,56,59,64],{"className":57},[58],"base",[24,60],{"className":61,"style":63},[62],"strut","height:0.6833em;",[24,65,45],{"className":66,"style":69},[67,68],"mord","mathnormal","margin-right:0.1389em;",") by measuring the horizontal angles subtended by lines of sight to three known control points (",[24,72,74,101],{"className":73},[27],[24,75,77],{"className":76},[31],[33,78,79],{"xmlns":35},[37,80,81,98],{},[40,82,83,86,90,93,95],{},[43,84,85],{},"A",[87,88,89],"mo",{"separator":54},",",[43,91,92],{},"B",[87,94,89],{"separator":54},[43,96,97],{},"C",[47,99,100],{"encoding":49},"A, B, C",[24,102,104],{"className":103,"ariaHidden":54},[53],[24,105,107,111,114,118,123,127,130,133],{"className":106},[58],[24,108],{"className":109,"style":110},[62],"height:0.8778em;vertical-align:-0.1944em;",[24,112,85],{"className":113},[67,68],[24,115,89],{"className":116},[117],"mpunct",[24,119],{"className":120,"style":122},[121],"mspace","margin-right:0.1667em;",[24,124,92],{"className":125,"style":126},[67,68],"margin-right:0.0502em;",[24,128,89],{"className":129},[117],[24,131],{"className":132,"style":122},[121],[24,134,97],{"className":135,"style":136},[67,68],"margin-right:0.0715em;",") ",[24,139,140],{},"1, 2",". Among the hundreds of historical procedures to solve this \"three-point problem,\" the ",[19,143,144],{},"Tienstra method"," is widely recognized by practitioners as the most compact and elegant analytical solution ",[24,147,148],{},"1, 3",".",[11,151,153],{"id":152},"why-this-matters","Why This Matters",[16,155,156,157,160,161,164,165,149],{},"Unlike traditional resection methods that require solving intermediate bearings, distances, or triangle lengths, the Tienstra formula provides coordinates directly using a center-of-gravity type calculation ",[24,158,159],{},"1, 4",". However, it is governed by a critical geometric constraint: the ",[19,162,163],{},"Danger Circle",". If your station falls on the circumcircle of the three control points, the math fails, and your position becomes indeterminate ",[24,166,167],{},"2, 5",[11,169,171],{"id":170},"background","Background",[16,173,174,175,178,179,182,183,186,187,149],{},"The formula is named after ",[19,176,177],{},"J.M. Tienstra"," (1895–1951), a professor at the Delft University of Technology who popularized the use of ",[19,180,181],{},"barycentric coordinates"," for resection ",[24,184,185],{},"3, 6",". While the mathematical roots of the problem date back to Willebrord Snellius in 1615 and later Laurent Pothenot, Tienstra's contribution was a simplified, direct weighting system that is ideally suited for modern field computation ",[24,188,189],{},"1, 6",[11,191,193],{"id":192},"theory","Theory",[16,195,196,197,225,226,273,274,302,303,306,307,336,337,149],{},"The Tienstra method treats the unknown point ",[24,198,200,213],{"className":199},[27],[24,201,203],{"className":202},[31],[33,204,205],{"xmlns":35},[37,206,207,211],{},[40,208,209],{},[43,210,45],{},[47,212,45],{"encoding":49},[24,214,216],{"className":215,"ariaHidden":54},[53],[24,217,219,222],{"className":218},[58],[24,220],{"className":221,"style":63},[62],[24,223,45],{"className":224,"style":69},[67,68]," as the \"weighted mean\" of the three control points ",[24,227,229,249],{"className":228},[27],[24,230,232],{"className":231},[31],[33,233,234],{"xmlns":35},[37,235,236,246],{},[40,237,238,240,242,244],{},[43,239,85],{},[87,241,89],{"separator":54},[43,243,92],{},[87,245,89],{"separator":54},[47,247,248],{"encoding":49},"A, B,",[24,250,252],{"className":251,"ariaHidden":54},[53],[24,253,255,258,261,264,267,270],{"className":254},[58],[24,256],{"className":257,"style":110},[62],[24,259,85],{"className":260},[67,68],[24,262,89],{"className":263},[117],[24,265],{"className":266,"style":122},[121],[24,268,92],{"className":269,"style":126},[67,68],[24,271,89],{"className":272},[117]," and ",[24,275,277,290],{"className":276},[27],[24,278,280],{"className":279},[31],[33,281,282],{"xmlns":35},[37,283,284,288],{},[40,285,286],{},[43,287,97],{},[47,289,97],{"encoding":49},[24,291,293],{"className":292,"ariaHidden":54},[53],[24,294,296,299],{"className":295},[58],[24,297],{"className":298,"style":63},[62],[24,300,97],{"className":301,"style":136},[67,68]," ",[24,304,305],{},"6",". By assigning a specific mathematical weight (",[24,308,310,324],{"className":309},[27],[24,311,313],{"className":312},[31],[33,314,315],{"xmlns":35},[37,316,317,322],{},[40,318,319],{},[43,320,321],{},"K",[47,323,321],{"encoding":49},[24,325,327],{"className":326,"ariaHidden":54},[53],[24,328,330,333],{"className":329},[58],[24,331],{"className":332,"style":63},[62],[24,334,321],{"className":335,"style":136},[67,68],") to each control point based on the geometry of the setup, the final coordinates are derived without the need for iterative trials or complex trig sequences ",[24,338,339],{},"1",[11,341,343],{"id":342},"mathematical-principles","Mathematical Principles",[16,345,346,347,514,515,518],{},"The coordinates of the unknown point ",[24,348,350,388],{"className":349},[27],[24,351,353],{"className":352},[31],[33,354,355],{"xmlns":35},[37,356,357,385],{},[40,358,359,361,365,373,375,382],{},[43,360,45],{},[87,362,364],{"stretchy":363},"false","(",[366,367,368,371],"msub",{},[43,369,370],{},"E",[43,372,16],{},[87,374,89],{"separator":54},[366,376,377,380],{},[43,378,379],{},"N",[43,381,16],{},[87,383,384],{"stretchy":363},")",[47,386,387],{"encoding":49},"P(E_p, N_p)",[24,389,391],{"className":390,"ariaHidden":54},[53],[24,392,394,398,401,405,462,465,468,510],{"className":393},[58],[24,395],{"className":396,"style":397},[62],"height:1.0361em;vertical-align:-0.2861em;",[24,399,45],{"className":400,"style":69},[67,68],[24,402,364],{"className":403},[404],"mopen",[24,406,408,412],{"className":407},[67],[24,409,370],{"className":410,"style":411},[67,68],"margin-right:0.0576em;",[24,413,416],{"className":414},[415],"msupsub",[24,417,421,453],{"className":418},[419,420],"vlist-t","vlist-t2",[24,422,425,448],{"className":423},[424],"vlist-r",[24,426,430],{"className":427,"style":429},[428],"vlist","height:0.1514em;",[24,431,433,438],{"style":432},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[24,434],{"className":435,"style":437},[436],"pstrut","height:2.7em;",[24,439,445],{"className":440},[441,442,443,444],"sizing","reset-size6","size3","mtight",[24,446,16],{"className":447},[67,68,444],[24,449,452],{"className":450},[451],"vlist-s","​",[24,454,456],{"className":455},[424],[24,457,460],{"className":458,"style":459},[428],"height:0.2861em;",[24,461],{},[24,463,89],{"className":464},[117],[24,466],{"className":467,"style":122},[121],[24,469,471,475],{"className":470},[67],[24,472,379],{"className":473,"style":474},[67,68],"margin-right:0.109em;",[24,476,478],{"className":477},[415],[24,479,481,502],{"className":480},[419,420],[24,482,484,499],{"className":483},[424],[24,485,487],{"className":486,"style":429},[428],[24,488,490,493],{"style":489},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[24,491],{"className":492,"style":437},[436],[24,494,496],{"className":495},[441,442,443,444],[24,497,16],{"className":498},[67,68,444],[24,500,452],{"className":501},[451],[24,503,505],{"className":504},[424],[24,506,508],{"className":507,"style":459},[428],[24,509],{},[24,511,384],{"className":512},[513],"mclose"," are calculated as follows ",[24,516,517],{},"1, 7",":",[16,520,521,1144],{},[24,522,524,621],{"className":523},[27],[24,525,527],{"className":526},[31],[33,528,529],{"xmlns":35},[37,530,531,618],{},[40,532,533,539,542],{},[366,534,535,537],{},[43,536,370],{},[43,538,16],{},[87,540,541],{},"=",[543,544,545,594],"mfrac",{},[40,546,547,554,561,564,571,578,580,587],{},[366,548,549,551],{},[43,550,321],{},[552,553,339],"mn",{},[366,555,556,558],{},[43,557,370],{},[43,559,560],{},"a",[87,562,563],{},"+",[366,565,566,568],{},[43,567,321],{},[552,569,570],{},"2",[366,572,573,575],{},[43,574,370],{},[43,576,577],{},"b",[87,579,563],{},[366,581,582,584],{},[43,583,321],{},[552,585,586],{},"3",[366,588,589,591],{},[43,590,370],{},[43,592,593],{},"c",[40,595,596,602,604,610,612],{},[366,597,598,600],{},[43,599,321],{},[552,601,339],{},[87,603,563],{},[366,605,606,608],{},[43,607,321],{},[552,609,570],{},[87,611,563],{},[366,613,614,616],{},[43,615,321],{},[552,617,586],{},[47,619,620],{"encoding":49},"E_p = \\frac{K_1 E_a + K_2 E_b + K_3 E_c}{K_1 + K_2 + K_3}",[24,622,624,682],{"className":623,"ariaHidden":54},[53],[24,625,627,631,671,675,679],{"className":626},[58],[24,628],{"className":629,"style":630},[62],"height:0.9694em;vertical-align:-0.2861em;",[24,632,634,637],{"className":633},[67],[24,635,370],{"className":636,"style":411},[67,68],[24,638,640],{"className":639},[415],[24,641,643,663],{"className":642},[419,420],[24,644,646,660],{"className":645},[424],[24,647,649],{"className":648,"style":429},[428],[24,650,651,654],{"style":432},[24,652],{"className":653,"style":437},[436],[24,655,657],{"className":656},[441,442,443,444],[24,658,16],{"className":659},[67,68,444],[24,661,452],{"className":662},[451],[24,664,666],{"className":665},[424],[24,667,669],{"className":668,"style":459},[428],[24,670],{},[24,672],{"className":673,"style":674},[121],"margin-right:0.2778em;",[24,676,541],{"className":677},[678],"mrel",[24,680],{"className":681,"style":674},[121],[24,683,685,689],{"className":684},[58],[24,686],{"className":687,"style":688},[62],"height:1.3393em;vertical-align:-0.4451em;",[24,690,692,696,1141],{"className":691},[67],[24,693],{"className":694},[404,695],"nulldelimiter",[24,697,699],{"className":698},[543],[24,700,702,1132],{"className":701},[419,420],[24,703,705,1129],{"className":704},[424],[24,706,709,855,866],{"className":707,"style":708},[428],"height:0.8942em;",[24,710,712,716],{"style":711},"top:-2.655em;",[24,713],{"className":714,"style":715},[436],"height:3em;",[24,717,719],{"className":718},[441,442,443,444],[24,720,722,768,772,812,815],{"className":721},[67,444],[24,723,725,728],{"className":724},[67,444],[24,726,321],{"className":727,"style":136},[67,68,444],[24,729,731],{"className":730},[415],[24,732,734,759],{"className":733},[419,420],[24,735,737,756],{"className":736},[424],[24,738,741],{"className":739,"style":740},[428],"height:0.3173em;",[24,742,744,748],{"style":743},"top:-2.357em;margin-left:-0.0715em;margin-right:0.0714em;",[24,745],{"className":746,"style":747},[436],"height:2.5em;",[24,749,753],{"className":750},[441,751,752,444],"reset-size3","size1",[24,754,339],{"className":755},[67,444],[24,757,452],{"className":758},[451],[24,760,762],{"className":761},[424],[24,763,766],{"className":764,"style":765},[428],"height:0.143em;",[24,767],{},[24,769,563],{"className":770},[771,444],"mbin",[24,773,775,778],{"className":774},[67,444],[24,776,321],{"className":777,"style":136},[67,68,444],[24,779,781],{"className":780},[415],[24,782,784,804],{"className":783},[419,420],[24,785,787,801],{"className":786},[424],[24,788,790],{"className":789,"style":740},[428],[24,791,792,795],{"style":743},[24,793],{"className":794,"style":747},[436],[24,796,798],{"className":797},[441,751,752,444],[24,799,570],{"className":800},[67,444],[24,802,452],{"className":803},[451],[24,805,807],{"className":806},[424],[24,808,810],{"className":809,"style":765},[428],[24,811],{},[24,813,563],{"className":814},[771,444],[24,816,818,821],{"className":817},[67,444],[24,819,321],{"className":820,"style":136},[67,68,444],[24,822,824],{"className":823},[415],[24,825,827,847],{"className":826},[419,420],[24,828,830,844],{"className":829},[424],[24,831,833],{"className":832,"style":740},[428],[24,834,835,838],{"style":743},[24,836],{"className":837,"style":747},[436],[24,839,841],{"className":840},[441,751,752,444],[24,842,586],{"className":843},[67,444],[24,845,452],{"className":846},[451],[24,848,850],{"className":849},[424],[24,851,853],{"className":852,"style":765},[428],[24,854],{},[24,856,858,861],{"style":857},"top:-3.23em;",[24,859],{"className":860,"style":715},[436],[24,862],{"className":863,"style":865},[864],"frac-line","border-bottom-width:0.04em;",[24,867,869,872],{"style":868},"top:-3.4159em;",[24,870],{"className":871,"style":715},[436],[24,873,875],{"className":874},[441,442,443,444],[24,876,878,918,960,963,1003,1046,1049,1089],{"className":877},[67,444],[24,879,881,884],{"className":880},[67,444],[24,882,321],{"className":883,"style":136},[67,68,444],[24,885,887],{"className":886},[415],[24,888,890,910],{"className":889},[419,420],[24,891,893,907],{"className":892},[424],[24,894,896],{"className":895,"style":740},[428],[24,897,898,901],{"style":743},[24,899],{"className":900,"style":747},[436],[24,902,904],{"className":903},[441,751,752,444],[24,905,339],{"className":906},[67,444],[24,908,452],{"className":909},[451],[24,911,913],{"className":912},[424],[24,914,916],{"className":915,"style":765},[428],[24,917],{},[24,919,921,924],{"className":920},[67,444],[24,922,370],{"className":923,"style":411},[67,68,444],[24,925,927],{"className":926},[415],[24,928,930,952],{"className":929},[419,420],[24,931,933,949],{"className":932},[424],[24,934,937],{"className":935,"style":936},[428],"height:0.1645em;",[24,938,940,943],{"style":939},"top:-2.357em;margin-left:-0.0576em;margin-right:0.0714em;",[24,941],{"className":942,"style":747},[436],[24,944,946],{"className":945},[441,751,752,444],[24,947,560],{"className":948},[67,68,444],[24,950,452],{"cla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= \\frac{K_1 N_a + K_2 N_b + K_3 N_c}{K_1 + K_2 + 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= \\frac{1}{\\cot A - \\cot 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= \\frac{1}{\\cot B - \\cot \\beta}",[24,2201,2203,2258],{"className":2202,"ariaHidden":54},[53],[24,2204,2206,2209,2249,2252,2255],{"className":2205},[58],[24,2207],{"className":2208,"style":1984},[62],[24,2210,2212,2215],{"className":2211},[67],[24,2213,321],{"className":2214,"style":136},[67,68],[24,2216,2218],{"className":2217},[415],[24,2219,2221,2241],{"className":2220},[419,420],[24,2222,2224,2238],{"className":2223},[424],[24,2225,2227],{"className":2226,"style":2003},[428],[24,2228,2229,2232],{"style":2006},[24,2230],{"className":2231,"style":437},[436],[24,2233,2235],{"className":2234},[441,442,443,444],[24,2236,570],{"className":2237},[67,444],[24,2239,452],{"className":2240},[451],[24,2242,2244],{"className":2243},[424],[24,2245,2247],{"className":2246,"style":2025},[428],[24,2248],{},[24,2250],{"className":2251,"style":674},[121],[24,2253,541],{"className":2254},[678],[24,2256],{"className":2257,"style":674},[121],[24,2259,2261,2265],{"className":2260},[58],[24,2262],{"className":2263,"style":2264},[62],"height:1.3262em;vertical-align:-0.4811em;",[24,2266,2268,2271,2367],{"className":2267},[67],[24,2269],{"className":2270},[404,695],[24,2272,2274],{"className":2273},[543],[24,2275,2277,2358],{"className":2276},[419,420],[24,2278,2280,2355],{"className":2279},[424],[24,2281,2283,2333,2341],{"className":2282,"style":2062},[428],[24,2284,2285,2288],{"style":711},[24,2286],{"className":2287,"style":715},[436],[24,2289,2291],{"className":2290},[441,442,443,444],[24,2292,2294,2306,2309,2312,2315,2327,2330],{"className":2293},[67,444],[24,2295,2297,2300,2303],{"className":2296},[2077,444],[24,2298,593],{"className":2299},[444],[24,2301,2084],{"className":2302},[444],[24,2304,2088],{"className":2305},[444],[24,2307],{"className":2308,"style":2092},[121,444],[24,2310,92],{"className":2311,"style":126},[67,68,444],[24,2313,1965],{"className":2314},[771,444],[24,2316,2318,2321,2324],{"className":2317},[2077,444],[24,2319,593],{"className":2320},[444],[24,2322,2084],{"className":2323},[444],[24,2325,2088],{"className":2326},[444],[24,2328],{"className":2329,"style":2092},[121,444],[24,2331,1878],{"className":2332,"style":1910},[67,68,444],[24,2334,2335,2338],{"style":857},[24,2336],{"className":2337,"style":715},[436],[24,2339],{"className":2340,"style":865},[864],[24,2342,2343,2346],{"style":2127},[24,2344],{"className":2345,"style":715},[436],[24,2347,2349],{"className":2348},[441,442,443,444],[24,2350,2352],{"className":2351},[67,444],[24,2353,339],{"className":2354},[67,444],[24,2356,452],{"className":2357},[451],[24,2359,2361],{"className":2360},[424],[24,2362,2365],{"className":2363,"style":2364},[428],"height:0.4811em;",[24,2366],{},[24,2368],{"className":2369},[513,695],[1927,2371,2372],{},[24,2373,2375,2415],{"className":2374},[27],[24,2376,2378],{"className":2377},[31],[33,2379,2380],{"xmlns":35},[37,2381,2382,2412],{},[40,2383,2384,2390,2392],{},[366,2385,2386,2388],{},[43,2387,321],{},[552,2389,586],{},[87,2391,541],{},[543,2393,2394,2396],{},[552,2395,339],{},[40,2397,2398,2400,2402,2404,2406,2408,2410],{},[43,2399,1957],{},[87,2401,1960],{},[43,2403,97],{},[87,2405,1965],{},[43,2407,1957],{},[87,2409,1960],{},[43,2411,1883],{},[47,2413,2414],{"encoding":49},"K_3 = \\frac{1}{\\cot C - \\cot \\gamma}",[24,2416,2418,2473],{"className":2417,"ariaHidden":54},[53],[24,2419,2421,2424,2464,2467,2470],{"className":2420},[58],[24,2422],{"className":2423,"style":1984},[62],[24,2425,2427,2430],{"className":2426},[67],[24,2428,321],{"className":2429,"style":136},[67,68],[24,2431,2433],{"className":2432},[415],[24,2434,2436,2456],{"className":2435},[419,420],[24,2437,2439,2453],{"className":2438},[424],[24,2440,2442],{"className":2441,"style":2003},[428],[24,2443,2444,2447],{"style":2006},[24,2445],{"className":2446,"style":437},[436],[24,2448,2450],{"className":2449},[441,442,443,444],[24,2451,586],{"className":2452},[67,444],[24,2454,452],{"className":2455},[451],[24,2457,2459],{"className":2458},[424],[24,2460,2462],{"className":2461,"style":2025},[428],[24,2463],{},[24,2465],{"className":2466,"style":674},[121],[24,2468,541],{"className":2469},[678],[24,2471],{"className":2472,"style":674},[121],[24,2474,2476,2479],{"className":2475},[58],[24,2477],{"className":2478,"style":2264},[62],[24,2480,2482,2485,2580],{"className":2481},[67],[24,2483],{"className":2484},[404,695],[24,2486,2488],{"className":2487},[543],[24,2489,2491,2572],{"className":2490},[419,420],[24,2492,2494,2569],{"className":2493},[424],[24,2495,2497,2547,2555],{"className":2496,"style":2062},[428],[24,2498,2499,2502],{"style":711},[24,2500],{"className":2501,"style":715},[436],[24,2503,2505],{"className":2504},[441,442,443,444],[24,2506,2508,2520,2523,2526,2529,2541,2544],{"className":2507},[67,444],[24,2509,2511,2514,2517],{"className":2510},[2077,444],[24,2512,593],{"className":2513},[444],[24,2515,2084],{"className":2516},[444],[24,2518,2088],{"className":2519},[444],[24,2521],{"className":2522,"style":2092},[121,444],[24,2524,97],{"className":2525,"style":136},[67,68,444],[24,2527,1965],{"className":2528},[771,444],[24,2530,2532,2535,2538],{"className":2531},[2077,444],[24,2533,593],{"className":2534},[444],[24,2536,2084],{"className":2537},[444],[24,2539,2088],{"className":2540},[444],[24,2542],{"className":2543,"style":2092},[121,444],[24,2545,1883],{"className":2546,"style":1920},[67,68,444],[24,2548,2549,2552],{"style":857},[24,2550],{"className":2551,"style":715},[436],[24,2553],{"className":2554,"style":865},[864],[24,2556,2557,2560],{"style":2127},[24,2558],{"className":2559,"style":715},[436],[24,2561,2563],{"className":2562},[441,442,443,444],[24,2564,2566],{"className":2565},[67,444],[24,2567,339],{"className":2568},[67,444],[24,2570,452],{"className":2571},[451],[24,2573,2575],{"className":2574},[424],[24,2576,2578],{"className":2577,"style":2364},[428],[24,2579],{},[24,2581],{"className":2582},[513,695],[16,2584,2585],{},[2586,2587,2588,2589,2619,2620,2667,2668,2697,2698,2742,2743,2697,2773,137,2817,149],"em",{},"(Note: ",[24,2590,2592,2606],{"className":2591},[27],[24,2593,2595],{"className":2594},[31],[33,2596,2597],{"xmlns":35},[37,2598,2599,2603],{},[40,2600,2601],{},[43,2602,1873],{},[47,2604,2605],{"encoding":49},"\\alpha",[24,2607,2609],{"className":2608,"ariaHidden":54},[53],[24,2610,2612,2616],{"className":2611},[58],[24,2613],{"className":2614,"style":2615},[62],"height:0.4306em;",[24,2617,1873],{"className":2618,"style":1900},[67,68]," is the angle ",[24,2621,2623,2645],{"className":2622},[27],[24,2624,2626],{"className":2625},[31],[33,2627,2628],{"xmlns":35},[37,2629,2630,2642],{},[40,2631,2632,2636,2638,2640],{},[43,2633,2635],{"mathvariant":2634},"normal","∠",[43,2637,92],{},[43,2639,45],{},[43,2641,97],{},[47,2643,2644],{"encoding":49},"\\angle BPC",[24,2646,2648],{"className":2647,"ariaHidden":54},[53],[24,2649,2651,2655,2658,2661,2664],{"className":2650},[58],[24,2652],{"className":2653,"style":2654},[62],"height:0.6922em;",[24,2656,2635],{"className":2657},[67],[24,2659,92],{"className":2660,"style":126},[67,68],[24,2662,45],{"className":2663,"style":69},[67,68],[24,2665,97],{"className":2666,"style":136},[67,68],", ",[24,2669,2671,2685],{"className":2670},[27],[24,2672,2674],{"className":2673},[31],[33,2675,2676],{"xmlns":35},[37,2677,2678,2682],{},[40,2679,2680],{},[43,2681,1878],{},[47,2683,2684],{"encoding":49},"\\beta",[24,2686,2688],{"className":2687,"ariaHidden":54},[53],[24,2689,2691,2694],{"className":2690},[58],[24,2692],{"className":2693,"style":1896},[62],[24,2695,1878],{"className":2696,"style":1910},[67,68]," is ",[24,2699,2701,2721],{"className":2700},[27],[24,2702,2704],{"className":2703},[31],[33,2705,2706],{"xmlns":35},[37,2707,2708,2718],{},[40,2709,2710,2712,2714,2716],{},[43,2711,2635],{"mathvariant":2634},[43,2713,97],{},[43,2715,45],{},[43,2717,85],{},[47,2719,2720],{"encoding":49},"\\angle CPA",[24,2722,2724],{"className":2723,"ariaHidden":54},[53],[24,2725,2727,2730,2733,2736,2739],{"className":2726},[58],[24,2728],{"className":2729,"style":2654},[62],[24,2731,2635],{"className":2732},[67],[24,2734,97],{"className":2735,"style":136},[67,68],[24,2737,45],{"className":2738,"style":69},[67,68],[24,2740,85],{"className":2741},[67,68],", and ",[24,2744,2746,2760],{"className":2745},[27],[24,2747,2749],{"className":2748},[31],[33,2750,2751],{"xmlns":35},[37,2752,2753,2757],{},[40,2754,2755],{},[43,2756,1883],{},[47,2758,2759],{"encoding":49},"\\gamma",[24,2761,2763],{"className":2762,"ariaHidden":54},[53],[24,2764,2766,2770],{"className":2765},[58],[24,2767],{"className":2768,"style":2769},[62],"height:0.625em;vertical-align:-0.1944em;",[24,2771,1883],{"className":2772,"style":1920},[67,68],[24,2774,2776,2796],{"className":2775},[27],[24,2777,2779],{"className":2778},[31],[33,2780,2781],{"xmlns":35},[37,2782,2783,2793],{},[40,2784,2785,2787,2789,2791],{},[43,2786,2635],{"mathvariant":2634},[43,2788,85],{},[43,2790,45],{},[43,2792,92],{},[47,2794,2795],{"encoding":49},"\\angle APB",[24,2797,2799],{"className":2798,"ariaHidden":54},[53],[24,2800,2802,2805,2808,2811,2814],{"className":2801},[58],[24,2803],{"className":2804,"style":2654},[62],[24,2806,2635],{"className":2807},[67],[24,2809,85],{"className":2810},[67,68],[24,2812,45],{"className":2813,"style":69},[67,68],[24,2815,92],{"className":2816,"style":126},[67,68],[24,2818,2819],{},"7, 8",[11,2821,2823],{"id":2822},"field-workflow","Field Workflow",[2825,2826,2827,2831,2920,2924,2959,2963,3099,3103],"steps",{},[1736,2828,2830],{"id":2829},"reconnaissance","Reconnaissance",[16,2832,2833,2834,2888,2889,2917,2918,149],{},"Identify three stable control points (",[24,2835,2837,2858],{"className":2836},[27],[24,2838,2840],{"className":2839},[31],[33,2841,2842],{"xmlns":35},[37,2843,2844,2856],{},[40,2845,2846,2848,2850,2852,2854],{},[43,2847,85],{},[87,2849,89],{"separator":54},[43,2851,92],{},[87,2853,89],{"separator":54},[43,2855,97],{},[47,2857,100],{"encoding":49},[24,2859,2861],{"className":2860,"ariaHidden":54},[53],[24,2862,2864,2867,2870,2873,2876,2879,2882,2885],{"className":2863},[58],[24,2865],{"className":2866,"style":110},[62],[24,2868,85],{"className":2869},[67,68],[24,2871,89],{"className":2872},[117],[24,2874],{"className":2875,"style":122},[121],[24,2877,92],{"className":2878,"style":126},[67,68],[24,2880,89],{"className":2881},[117],[24,2883],{"className":2884,"style":122},[121],[24,2886,97],{"className":2887,"style":136},[67,68],") with known coordinates. Ensure your station ",[24,2890,2892,2905],{"className":2891},[27],[24,2893,2895],{"className":2894},[31],[33,2896,2897],{"xmlns":35},[37,2898,2899,2903],{},[40,2900,2901],{},[43,2902,45],{},[47,2904,45],{"encoding":49},[24,2906,2908],{"className":2907,"ariaHidden":54},[53],[24,2909,2911,2914],{"className":2910},[58],[24,2912],{"className":2913,"style":63},[62],[24,2915,45],{"className":2916,"style":69},[67,68]," is not near the \"Danger Circle\" (see below) ",[24,2919,570],{},[1736,2921,2923],{"id":2922},"angular-observation","Angular Observation",[16,2925,2926,2927,2955,2956,149],{},"Set up the theodolite or total station at point ",[24,2928,2930,2943],{"className":2929},[27],[24,2931,2933],{"className":2932},[31],[33,2934,2935],{"xmlns":35},[37,2936,2937,2941],{},[40,2938,2939],{},[43,2940,45],{},[47,2942,45],{"encoding":49},[24,2944,2946],{"className":2945,"ariaHidden":54},[53],[24,2947,2949,2952],{"className":2948},[58],[24,2950],{"className":2951,"style":63},[62],[24,2953,45],{"className":2954,"style":69},[67,68],". Measure the horizontal angles between the three control points with high precision (multiple sets on both faces) ",[24,2957,2958],{},"2, 9",[1736,2960,2962],{"id":2961},"interior-angle-calculation","Interior Angle Calculation",[16,2964,2965,2966,273,3012,3040,3041,3095,3096,149],{},"Using the known coordinates of ",[24,2967,2969,2988],{"className":2968},[27],[24,2970,2972],{"className":2971},[31],[33,2973,2974],{"xmlns":35},[37,2975,2976,2986],{},[40,2977,2978,2980,2982,2984],{},[43,2979,85],{},[87,2981,89],{"separator":54},[43,2983,92],{},[87,2985,89],{"separator":54},[47,2987,248],{"encoding":49},[24,2989,2991],{"className":2990,"ariaHidden":54},[53],[24,2992,2994,2997,3000,3003,3006,3009],{"className":2993},[58],[24,2995],{"className":2996,"style":110},[62],[24,2998,85],{"className":2999},[67,68],[24,3001,89],{"className":3002},[117],[24,3004],{"className":3005,"style":122},[121],[24,3007,92],{"className":3008,"style":126},[67,68],[24,3010,89],{"className":3011},[117],[24,3013,3015,3028],{"className":3014},[27],[24,3016,3018],{"className":3017},[31],[33,3019,3020],{"xmlns":35},[37,3021,3022,3026],{},[40,3023,3024],{},[43,3025,97],{},[47,3027,97],{"encoding":49},[24,3029,3031],{"className":3030,"ariaHidden":54},[53],[24,3032,3034,3037],{"className":3033},[58],[24,3035],{"className":3036,"style":63},[62],[24,3038,97],{"className":3039,"style":136},[67,68],", calculate the interior angles of the control triangle (",[24,3042,3044,3065],{"className":3043},[27],[24,3045,3047],{"className":3046},[31],[33,3048,3049],{"xmlns":35},[37,3050,3051,3063],{},[40,3052,3053,3055,3057,3059,3061],{},[43,3054,85],{},[87,3056,89],{"separator":54},[43,3058,92],{},[87,3060,89],{"separator":54},[43,3062,97],{},[47,3064,100],{"encoding":49},[24,3066,3068],{"className":3067,"ariaHidden":54},[53],[24,3069,3071,3074,3077,3080,3083,3086,3089,3092],{"className":3070},[58],[24,3072],{"className":3073,"style":110},[62],[24,3075,85],{"className":3076},[67,68],[24,3078,89],{"className":3079},[117],[24,3081],{"className":3082,"style":122},[121],[24,3084,92],{"className":3085,"style":126},[67,68],[24,3087,89],{"className":3088},[117],[24,3090],{"className":3091,"style":122},[121],[24,3093,97],{"className":3094,"style":136},[67,68],") using standard COGO \"Join\" computations ",[24,3097,3098],{},"7, 9",[1736,3100,3102],{"id":3101},"computation","Computation",[16,3104,3105,3106,3134,3135,273,3205,302,3275,149],{},"Calculate the three ",[24,3107,3109,3122],{"className":3108},[27],[24,3110,3112],{"className":3111},[31],[33,3113,3114],{"xmlns":35},[37,3115,3116,3120],{},[40,3117,3118],{},[43,3119,321],{},[47,3121,321],{"encoding":49},[24,3123,3125],{"className":3124,"ariaHidden":54},[53],[24,3126,3128,3131],{"className":3127},[58],[24,3129],{"className":3130,"style":63},[62],[24,3132,321],{"className":3133,"style":136},[67,68]," weights using the cotangent formula, then apply the weighted mean equations to find ",[24,3136,3138,3156],{"className":3137},[27],[24,3139,3141],{"className":3140},[31],[33,3142,3143],{"xmlns":35},[37,3144,3145,3153],{},[40,3146,3147],{},[366,3148,3149,3151],{},[43,3150,370],{},[43,3152,16],{},[47,3154,3155],{"encoding":49},"E_p",[24,3157,3159],{"className":3158,"ariaHidden":54},[53],[24,3160,3162,3165],{"className":3161},[58],[24,3163],{"className":3164,"style":630},[62],[24,3166,3168,3171],{"className":3167},[67],[24,3169,370],{"className":3170,"style":411},[67,68],[24,3172,3174],{"className":3173},[415],[24,3175,3177,3197],{"className":3176},[419,420],[24,3178,3180,3194],{"className":3179},[424],[24,3181,3183],{"className":3182,"style":429},[428],[24,3184,3185,3188],{"style":432},[24,3186],{"className":3187,"style":437},[436],[24,3189,3191],{"className":3190},[441,442,443,444],[24,3192,16],{"className":3193},[67,68,444],[24,3195,452],{"className":3196},[451],[24,3198,3200],{"className":3199},[424],[24,3201,3203],{"className":3202,"style":459},[428],[24,3204],{},[24,3206,3208,3226],{"className":3207},[27],[24,3209,3211],{"className":3210},[31],[33,3212,3213],{"xmlns":35},[37,3214,3215,3223],{},[40,3216,3217],{},[366,3218,3219,3221],{},[43,3220,379],{},[43,3222,16],{},[47,3224,3225],{"encoding":49},"N_p",[24,3227,3229],{"className":3228,"ariaHidden":54},[53],[24,3230,3232,3235],{"className":3231},[58],[24,3233],{"className":3234,"style":630},[62],[24,3236,3238,3241],{"className":3237},[67],[24,3239,379],{"className":3240,"style":474},[67,68],[24,3242,3244],{"className":3243},[415],[24,3245,3247,3267],{"className":3246},[419,420],[24,3248,3250,3264],{"className":3249},[424],[24,3251,3253],{"className":3252,"style":429},[428],[24,3254,3255,3258],{"style":489},[24,3256],{"className":3257,"style":437},[436],[24,3259,3261],{"className":3260},[441,442,443,444],[24,3262,16],{"className":3263},[67,68,444],[24,3265,452],{"className":3266},[451],[24,3268,3270],{"className":3269},[424],[24,3271,3273],{"className":3272,"style":459},[428],[24,3274],{},[24,3276,3277],{},"1, 9",[11,3279,3281],{"id":3280},"step-by-step-example","Step-by-Step Example",[16,3283,3284,3287,3288,2667,3348,2742,3406,3464,3465,3493,3494,2667,3585,2742,3671,302,3757,149],{},[19,3285,3286],{},"Scenario:"," Control triangle points are ",[24,3289,3291,3317],{"className":3290},[27],[24,3292,3294],{"className":3293},[31],[33,3295,3296],{"xmlns":35},[37,3297,3298,3314],{},[40,3299,3300,3302,3304,3307,3309,3312],{},[43,3301,85],{},[87,3303,364],{"stretchy":363},[552,3305,3306],{},"1000",[87,3308,89],{"separator":54},[552,3310,3311],{},"2000",[87,3313,384],{"stretchy":363},[47,3315,3316],{"encoding":49},"A(1000, 2000)",[24,3318,3320],{"className":3319,"ariaHidden":54},[53],[24,3321,3323,3327,3330,3333,3336,3339,3342,3345],{"className":3322},[58],[24,3324],{"className":3325,"style":3326},[62],"height:1em;vertical-align:-0.25em;",[24,3328,85],{"className":3329},[67,68],[24,3331,364],{"className":3332},[404],[24,3334,3306],{"className":3335},[67],[24,3337,89],{"className":3338},[117],[24,3340],{"className":3341,"style":122},[121],[24,3343,3311],{"className":3344},[67],[24,3346,384],{"className":3347},[513],[24,3349,3351,3376],{"className":3350},[27],[24,3352,3354],{"className":3353},[31],[33,3355,3356],{"xmlns":35},[37,3357,3358,3373],{},[40,3359,3360,3362,3364,3367,3369,3371],{},[43,3361,92],{},[87,3363,364],{"stretchy":363},[552,3365,3366],{},"3000",[87,3368,89],{"separator":54},[552,3370,3311],{},[87,3372,384],{"stretchy":363},[47,3374,3375],{"encoding":49},"B(3000, 2000)",[24,3377,3379],{"className":3378,"ariaHidden":54},[53],[24,3380,3382,3385,3388,3391,3394,3397,3400,3403],{"className":3381},[58],[24,3383],{"className":3384,"style":3326},[62],[24,3386,92],{"className":3387,"style":126},[67,68],[24,3389,364],{"className":3390},[404],[24,3392,3366],{"className":3393},[67],[24,3395,89],{"className":3396},[117],[24,3398],{"className":3399,"style":122},[121],[24,3401,3311],{"className":3402},[67],[24,3404,384],{"className":3405},[513],[24,3407,3409,3434],{"className":3408},[27],[24,3410,3412],{"className":3411},[31],[33,3413,3414],{"xmlns":35},[37,3415,3416,3431],{},[40,3417,3418,3420,3422,3424,3426,3429],{},[43,3419,97],{},[87,3421,364],{"stretchy":363},[552,3423,3311],{},[87,3425,89],{"separator":54},[552,3427,3428],{},"3732",[87,3430,384],{"stretchy":363},[47,3432,3433],{"encoding":49},"C(2000, 3732)",[24,3435,3437],{"className":3436,"ariaHidden":54},[53],[24,3438,3440,3443,3446,3449,3452,3455,3458,3461],{"className":3439},[58],[24,3441],{"className":3442,"style":3326},[62],[24,3444,97],{"className":3445,"style":136},[67,68],[24,3447,364],{"className":3448},[404],[24,3450,3311],{"className":3451},[67],[24,3453,89],{"className":3454},[117],[24,3456],{"className":3457,"style":122},[121],[24,3459,3428],{"className":3460},[67],[24,3462,384],{"className":3463},[513],". The measured angles at ",[24,3466,3468,3481],{"className":3467},[27],[24,3469,3471],{"className":3470},[31],[33,3472,3473],{"xmlns":35},[37,3474,3475,3479],{},[40,3476,3477],{},[43,3478,45],{},[47,3480,45],{"encoding":49},[24,3482,3484],{"className":3483,"ariaHidden":54},[53],[24,3485,3487,3490],{"className":3486},[58],[24,3488],{"className":3489,"style":63},[62],[24,3491,45],{"className":3492,"style":69},[67,68]," are ",[24,3495,3497,3522],{"className":3496},[27],[24,3498,3500],{"className":3499},[31],[33,3501,3502],{"xmlns":35},[37,3503,3504,3519],{},[40,3505,3506,3508,3510],{},[43,3507,1873],{},[87,3509,541],{},[3511,3512,3513,3516],"msup",{},[552,3514,3515],{},"110",[87,3517,3518],{},"∘",[47,3520,3521],{"encoding":49},"\\alpha = 110^\\circ",[24,3523,3525,3543],{"className":3524,"ariaHidden":54},[53],[24,3526,3528,3531,3534,3537,3540],{"className":3527},[58],[24,3529],{"className":3530,"style":2615},[62],[24,3532,1873],{"className":3533,"style":1900},[67,68],[24,3535],{"className":3536,"style":674},[121],[24,3538,541],{"className":3539},[678],[24,3541],{"className":3542,"style":674},[121],[24,3544,3546,3550,3554],{"className":3545},[58],[24,3547],{"className":3548,"style":3549},[62],"height:0.6741em;",[24,3551,3553],{"className":3552},[67],"11",[24,3555,3557,3561],{"className":3556},[67],[24,3558,3560],{"className":3559},[67],"0",[24,3562,3564],{"className":3563},[415],[24,3565,3567],{"className":3566},[419],[24,3568,3570],{"className":3569},[424],[24,3571,3573],{"className":3572,"style":3549},[428],[24,3574,3576,3579],{"style":3575},"top:-3.063em;margin-right:0.05em;",[24,3577],{"className":3578,"style":437},[436],[24,3580,3582],{"className":3581},[441,442,443,444],[24,3583,3518],{"className":3584},[771,444],[24,3586,3588,3611],{"className":3587},[27],[24,3589,3591],{"className":3590},[31],[33,3592,3593],{"xmlns":35},[37,3594,3595,3608],{},[40,3596,3597,3599,3601],{},[43,3598,1878],{},[87,3600,541],{},[3511,3602,3603,3606],{},[552,3604,3605],{},"120",[87,3607,3518],{},[47,3609,3610],{"encoding":49},"\\beta = 120^\\circ",[24,3612,3614,3632],{"className":3613,"ariaHidden":54},[53],[24,3615,3617,3620,3623,3626,3629],{"className":3616},[58],[24,3618],{"className":3619,"style":1896},[62],[24,3621,1878],{"className":3622,"style":1910},[67,68],[24,3624],{"className":3625,"style":674},[121],[24,3627,541],{"className":3628},[678],[24,3630],{"className":3631,"style":674},[121],[24,3633,3635,3638,3642],{"className":3634},[58],[24,3636],{"className":3637,"style":3549},[62],[24,3639,3641],{"className":3640},[67],"12",[24,3643,3645,3648],{"className":3644},[67],[24,3646,3560],{"className":3647},[67],[24,3649,3651],{"className":3650},[415],[24,3652,3654],{"className":3653},[419],[24,3655,3657],{"className":3656},[424],[24,3658,3660],{"className":3659,"style":3549},[428],[24,3661,3662,3665],{"style":3575},[24,3663],{"className":3664,"style":437},[436],[24,3666,3668],{"className":3667},[441,442,443,444],[24,3669,3518],{"className":3670},[771,444],[24,3672,3674,3697],{"className":3673},[27],[24,3675,3677],{"className":3676},[31],[33,3678,3679],{"xmlns":35},[37,3680,3681,3694],{},[40,3682,3683,3685,3687],{},[43,3684,1883],{},[87,3686,541],{},[3511,3688,3689,3692],{},[552,3690,3691],{},"130",[87,3693,3518],{},[47,3695,3696],{"encoding":49},"\\gamma = 130^\\circ",[24,3698,3700,3718],{"className":3699,"ariaHidden":54},[53],[24,3701,3703,3706,3709,3712,3715],{"className":3702},[58],[24,3704],{"className":3705,"style":2769},[62],[24,3707,1883],{"className":3708,"style":1920},[67,68],[24,3710],{"className":3711,"style":674},[121],[24,3713,541],{"className":3714},[678],[24,3716],{"className":3717,"style":674},[121],[24,3719,3721,3724,3728],{"className":3720},[58],[24,3722],{"className":3723,"style":3549},[62],[24,3725,3727],{"className":3726},[67],"13",[24,3729,3731,3734],{"className":3730},[67],[24,3732,3560],{"className":3733},[67],[24,3735,3737],{"className":3736},[415],[24,3738,3740],{"className":3739},[419],[24,3741,3743],{"className":3742},[424],[24,3744,3746],{"className":3745,"style":3549},[428],[24,3747,3748,3751],{"style":3575},[24,3749],{"className":3750,"style":437},[436],[24,3752,3754],{"className":3753},[441,442,443,444],[24,3755,3518],{"className":3756},[771,444],[24,3758,3759],{},"10",[3761,3762,3763,3890,4272,4475,4871],"ol",{},[1927,3764,3765,3768,3769,3823,3824,3887,3888,149],{},[19,3766,3767],{},"Triangle Geometry:"," Assume internal angles ",[24,3770,3772,3793],{"className":3771},[27],[24,3773,3775],{"className":3774},[31],[33,3776,3777],{"xmlns":35},[37,3778,3779,3791],{},[40,3780,3781,3783,3785,3787,3789],{},[43,3782,85],{},[87,3784,89],{"separator":54},[43,3786,92],{},[87,3788,89],{"separator":54},[43,3790,97],{},[47,3792,100],{"encoding":49},[24,3794,3796],{"className":3795,"ariaHidden":54},[53],[24,3797,3799,3802,3805,3808,3811,3814,3817,3820],{"className":3798},[58],[24,3800],{"className":3801,"style":110},[62],[24,3803,85],{"className":3804},[67,68],[24,3806,89],{"className":3807},[117],[24,3809],{"className":3810,"style":122},[121],[24,3812,92],{"className":3813,"style":126},[67,68],[24,3815,89],{"className":3816},[117],[24,3818],{"className":3819,"style":122},[121],[24,3821,97],{"className":3822,"style":136},[67,68]," are all ",[24,3825,3827,3846],{"className":3826},[27],[24,3828,3830],{"className":3829},[31],[33,3831,3832],{"xmlns":35},[37,3833,3834,3843],{},[40,3835,3836],{},[3511,3837,3838,3841],{},[552,3839,3840],{},"60",[87,3842,3518],{},[47,3844,3845],{"encoding":49},"60^\\circ",[24,3847,3849],{"className":3848,"ariaHidden":54},[53],[24,3850,3852,3855,3858],{"className":3851},[58],[24,3853],{"className":3854,"style":3549},[62],[24,3856,305],{"className":3857},[67],[24,3859,3861,3864],{"className":3860},[67],[24,3862,3560],{"className":3863},[67],[24,3865,3867],{"className":3866},[415],[24,3868,3870],{"className":3869},[419],[24,3871,3873],{"className":3872},[424],[24,3874,3876],{"className":3875,"style":3549},[428],[24,3877,3878,3881],{"style":3575},[24,3879],{"className":3880,"style":437},[436],[24,3882,3884],{"className":3883},[441,442,443,444],[24,3885,3518],{"className":3886},[771,444]," (equilateral) ",[24,3889,3759],{},[1927,3891,3892,3965],{},[19,3893,3894,3895,518],{},"Calculate ",[24,3896,3898,3916],{"className":3897},[27],[24,3899,3901],{"className":3900},[31],[33,3902,3903],{"xmlns":35},[37,3904,3905,3913],{},[40,3906,3907],{},[366,3908,3909,3911],{},[43,3910,321],{},[552,3912,339],{},[47,3914,3915],{"encoding":49},"K_1",[24,3917,3919],{"className":3918,"ariaHidden":54},[53],[24,3920,3922,3925],{"className":3921},[58],[24,3923],{"className":3924,"style":1984},[62],[24,3926,3928,3931],{"className":3927},[67],[24,3929,321],{"className":3930,"style":136},[67,68],[24,3932,3934],{"className":3933},[415],[24,3935,3937,3957],{"className":3936},[419,420],[24,3938,3940,3954],{"className":3939},[424],[24,3941,3943],{"className":3942,"style":2003},[428],[24,3944,3945,3948],{"style":2006},[24,3946],{"className":3947,"style":437},[436],[24,3949,3951],{"className":3950},[441,442,443,444],[24,3952,339],{"className":3953},[67,444],[24,3955,452],{"className":3956},[451],[24,3958,3960],{"className":3959},[424],[24,3961,3963],{"className":3962,"style":2025},[428],[24,3964],{},[24,3966,3968,4042],{"className":3967},[27],[24,3969,3971],{"className":3970},[31],[33,3972,3973],{"xmlns":35},[37,3974,3975,4039],{},[40,3976,3977,3983,3985,3987,3990,3992,3994,3996,4002,4004,4006,4008,4014,4016,4018,4020,4022,4024,4027,4029,4032,4034,4036],{},[366,3978,3979,3981],{},[43,3980,321],{},[552,3982,339],{},[87,3984,541],{},[552,3986,339],{},[43,3988,3989],{"mathvariant":2634},"\u002F",[87,3991,364],{"stretchy":363},[43,3993,1957],{},[87,3995,1960],{},[3511,3997,3998,4000],{},[552,3999,3840],{},[87,4001,3518],{},[87,4003,1965],{},[43,4005,1957],{},[87,4007,1960],{},[3511,4009,4010,4012],{},[552,4011,3515],{},[87,4013,3518],{},[87,4015,384],{"stretchy":363},[87,4017,541],{},[552,4019,339],{},[43,4021,3989],{"mathvariant":2634},[87,4023,364],{"stretchy":363},[552,4025,4026],{},"0.57735",[87,4028,563],{},[552,4030,4031],{},"0.36397",[87,4033,384],{"stretchy":363},[87,4035,541],{},[552,4037,4038],{},"1.0623",[47,4040,4041],{"encoding":49},"K_1 = 1 \u002F (\\cot 60^\\circ - \\cot 110^\\circ) = 1 \u002F (0.57735 + 0.36397) = 1.0623",[24,4043,4045,4100,4161,4217,4241,4262],{"className":4044,"ariaHidden":54},[53],[24,4046,4048,4051,4091,4094,4097],{"className":4047},[58],[24,4049],{"className":4050,"style":1984},[62],[24,4052,4054,4057],{"className":4053},[67],[24,4055,321],{"className":4056,"style":136},[67,68],[24,4058,4060],{"className":4059},[415],[24,4061,4063,4083],{"className":4062},[419,420],[24,4064,4066,4080],{"className":4065},[424],[24,4067,4069],{"className":4068,"style":2003},[428],[24,4070,4071,4074],{"style":2006},[24,4072],{"className":4073,"style":437},[436],[24,4075,4077],{"className":4076},[441,442,443,444],[24,4078,339],{"className":4079},[67,444],[24,4081,452],{"className":4082},[451],[24,4084,4086],{"className":4085},[424],[24,4087,4089],{"className":4088,"style":2025},[428],[24,4090],{},[24,4092],{"className":4093,"style":674},[121],[24,4095,541],{"className":4096},[678],[24,4098],{"className":4099,"style":674},[121],[24,4101,4103,4106,4110,4113,4116,4119,4122,4151,4155,4158],{"className":4102},[58],[24,4104],{"className":4105,"style":3326},[62],[24,4107,4109],{"className":4108},[67],"1\u002F",[24,4111,364],{"className":4112},[404],[24,4114,1957],{"className":4115},[2077],[24,4117],{"className":4118,"style":122},[121],[24,4120,305],{"className":4121},[67],[24,4123,4125,4128],{"className":4124},[67],[24,4126,3560],{"className":4127},[67],[24,4129,4131],{"className":4130},[415],[24,4132,4134],{"className":4133},[419],[24,4135,4137],{"className":4136},[424],[24,4138,4140],{"className":4139,"style":3549},[428],[24,4141,4142,4145],{"style":3575},[24,4143],{"className":4144,"style":437},[436],[24,4146,4148],{"className":4147},[441,442,443,444],[24,4149,3518],{"className":4150},[771,444],[24,4152],{"className":4153,"style":4154},[121],"margin-right:0.2222em;",[24,4156,1965],{"className":4157},[771],[24,4159],{"className":4160,"style":4154},[121],[24,4162,4164,4167,4170,4173,4176,4205,4208,4211,4214],{"className":4163},[58],[24,4165],{"className":4166,"style":3326},[62],[24,4168,1957],{"className":4169},[2077],[24,4171],{"className":4172,"style":122},[121],[24,4174,3553],{"className":4175},[67],[24,4177,4179,4182],{"className":4178},[67],[24,4180,3560],{"className":4181},[67],[24,4183,4185],{"className":4184},[415],[24,4186,4188],{"className":4187},[419],[24,4189,4191],{"className":4190},[424],[24,4192,4194],{"className":4193,"style":3549},[428],[24,4195,4196,4199],{"style":3575},[24,4197],{"className":4198,"style":437},[436],[24,4200,4202],{"className":4201},[441,442,443,444],[24,4203,3518],{"className":4204},[771,444],[24,4206,384],{"className":4207},[513],[24,4209],{"className":4210,"style":674},[121],[24,4212,541],{"className":4213},[678],[24,4215],{"className":4216,"style":674},[121],[24,4218,4220,4223,4226,4229,4232,4235,4238],{"className":4219},[58],[24,4221],{"className":4222,"style":3326},[62],[24,4224,4109],{"className":4225},[67],[24,4227,364],{"className":4228},[404],[24,4230,4026],{"className":4231},[67],[24,4233],{"className":4234,"style":4154},[121],[24,4236,563],{"className":4237},[771],[24,4239],{"className":4240,"style":4154},[121],[24,4242,4244,4247,4250,4253,4256,4259],{"className":4243},[58],[24,4245],{"className":4246,"style":3326},[62],[24,4248,4031],{"className":4249},[67],[24,4251,384],{"className":4252},[513],[24,4254],{"className":4255,"style":674},[121],[24,4257,541],{"className":4258},[678],[24,4260],{"className":4261,"style":674},[121],[24,4263,4265,4269],{"className":4264},[58],[24,4266],{"className":4267,"style":4268},[62],"height:0.6444em;",[24,4270,4038],{"className":4271},[67],[1927,4273,4274,4417,4418,273,4446,4474],{},[19,4275,3894,4276,4346,4347,518],{},[24,4277,4279,4297],{"className":4278},[27],[24,4280,4282],{"className":4281},[31],[33,4283,4284],{"xmlns":35},[37,4285,4286,4294],{},[40,4287,4288],{},[366,4289,4290,4292],{},[43,4291,321],{},[552,4293,570],{},[47,4295,4296],{"encoding":49},"K_2",[24,4298,4300],{"className":4299,"ariaHidden":54},[53],[24,4301,4303,4306],{"className":4302},[58],[24,4304],{"className":4305,"style":1984},[62],[24,4307,4309,4312],{"className":4308},[67],[24,4310,321],{"className":4311,"style":136},[67,68],[24,4313,4315],{"className":4314},[415],[24,4316,4318,4338],{"className":4317},[419,420],[24,4319,4321,4335],{"className":4320},[424],[24,4322,4324],{"className":4323,"style":2003},[428],[24,4325,4326,4329],{"style":2006},[24,4327],{"className":4328,"style":437},[436],[24,4330,4332],{"className":4331},[441,442,443,444],[24,4333,570],{"className":4334},[67,444],[24,4336,452],{"className":4337},[451],[24,4339,4341],{"className":4340},[424],[24,4342,4344],{"className":4343,"style":2025},[428],[24,4345],{}," & ",[24,4348,4350,4368],{"className":4349},[27],[24,4351,4353],{"className":4352},[31],[33,4354,4355],{"xmlns":35},[37,4356,4357,4365],{},[40,4358,4359],{},[366,4360,4361,4363],{},[43,4362,321],{},[552,4364,586],{},[47,4366,4367],{"encoding":49},"K_3",[24,4369,4371],{"className":4370,"ariaHidden":54},[53],[24,4372,4374,4377],{"className":4373},[58],[24,4375],{"className":4376,"style":1984},[62],[24,4378,4380,4383],{"className":4379},[67],[24,4381,321],{"className":4382,"style":136},[67,68],[24,4384,4386],{"className":4385},[415],[24,4387,4389,4409],{"className":4388},[419,420],[24,4390,4392,4406],{"className":4391},[424],[24,4393,4395],{"className":4394,"style":2003},[428],[24,4396,4397,4400],{"style":2006},[24,4398],{"className":4399,"style":437},[436],[24,4401,4403],{"className":4402},[441,442,443,444],[24,4404,586],{"className":4405},[67,444],[24,4407,452],{"className":4408},[451],[24,4410,4412],{"className":4411},[424],[24,4413,4415],{"className":4414,"style":2025},[428],[24,4416],{}," (Follow the same process using ",[24,4419,4421,4434],{"className":4420},[27],[24,4422,4424],{"className":4423},[31],[33,4425,4426],{"xmlns":35},[37,4427,4428,4432],{},[40,4429,4430],{},[43,4431,1878],{},[47,4433,2684],{"encoding":49},[24,4435,4437],{"className":4436,"ariaHidden":54},[53],[24,4438,4440,4443],{"className":4439},[58],[24,4441],{"className":4442,"style":1896},[62],[24,4444,1878],{"className":4445,"style":1910},[67,68],[24,4447,4449,4462],{"className":4448},[27],[24,4450,4452],{"className":4451},[31],[33,4453,4454],{"xmlns":35},[37,4455,4456,4460],{},[40,4457,4458],{},[43,4459,1883],{},[47,4461,2759],{"encoding":49},[24,4463,4465],{"className":4464,"ariaHidden":54},[53],[24,4466,4468,4471],{"className":4467},[58],[24,4469],{"className":4470,"style":2769},[62],[24,4472,1883],{"className":4473,"style":1920},[67,68],").",[1927,4476,4477,4550,149],{},[19,4478,4479,4480,4549],{},"Final Easting (",[24,4481,4483,4500],{"className":4482},[27],[24,4484,4486],{"className":4485},[31],[33,4487,4488],{"xmlns":35},[37,4489,4490,4498],{},[40,4491,4492],{},[366,4493,4494,4496],{},[43,4495,370],{},[43,4497,16],{},[47,4499,3155],{"encoding":49},[24,4501,4503],{"className":4502,"ariaHidden":54},[53],[24,4504,4506,4509],{"className":4505},[58],[24,4507],{"className":4508,"style":630},[62],[24,4510,4512,4515],{"className":4511},[67],[24,4513,370],{"className":4514,"style":411},[67,68],[24,4516,4518],{"className":4517},[415],[24,4519,4521,4541],{"className":4520},[419,420],[24,4522,4524,4538],{"className":4523},[424],[24,4525,4527],{"className":4526,"style":429},[428],[24,4528,4529,4532],{"style":432},[24,4530],{"className":4531,"style":437},[436],[24,4533,4535],{"className":4534},[441,442,443,444],[24,4536,16],{"className":4537},[67,68,444],[24,4539,452],{"className":4540},[451],[24,4542,4544],{"className":4543},[424],[24,4545,4547],{"className":4546,"style":459},[428],[24,4548],{},"):",[24,4551,4553,4615],{"className":4552},[27],[24,4554,4556],{"className":4555},[31],[33,4557,4558],{"xmlns":35},[37,4559,4560,4612],{},[40,4561,4562,4568,4570,4572,4574,4577,4579,4581,4587,4589,4591,4593,4599,4601,4603,4605,4607,4610],{},[366,4563,4564,4566],{},[43,4565,370],{},[43,4567,16],{},[87,4569,541],{},[87,4571,364],{"stretchy":363},[552,4573,4038],{},[87,4575,4576],{},"×",[552,4578,3306],{},[87,4580,563],{},[366,4582,4583,4585],{},[43,4584,321],{},[552,4586,570],{},[87,4588,4576],{},[552,4590,3366],{},[87,4592,563],{},[366,4594,4595,4597],{},[43,4596,321],{},[552,4598,586],{},[87,4600,4576],{},[552,4602,3311],{},[87,4604,384],{"stretchy":363},[43,4606,3989],{"mathvariant":2634},[87,4608,4609],{},"∑",[43,4611,321],{},[47,4613,4614],{"encoding":49},"E_p = (1.0623 \\times 1000 + K_2 \\times 3000 + K_3 \\times 2000) \u002F \\sum K",[24,4616,4618,4673,4694,4713,4768,4786,4841],{"className":4617,"ariaHidden":54},[53],[24,4619,4621,4624,4664,4667,4670],{"className":4620},[58],[24,4622],{"className":4623,"style":630},[62],[24,4625,4627,4630],{"className":4626},[67],[24,4628,370],{"className":4629,"style":411},[67,68],[24,4631,4633],{"className":4632},[415],[24,4634,4636,4656],{"className":4635},[419,420],[24,4637,4639,4653],{"className":4638},[424],[24,4640,4642],{"className":4641,"style":429},[428],[24,4643,4644,4647],{"style":432},[24,4645],{"className":4646,"style":437},[436],[24,4648,4650],{"className":4649},[441,442,443,444],[24,4651,16],{"className":4652},[67,68,444],[24,4654,452],{"className":4655},[451],[24,4657,4659],{"className":4658},[424],[24,4660,4662],{"className":4661,"style":459},[428],[24,4663],{},[24,4665],{"className":4666,"style":674},[121],[24,4668,541],{"className":4669},[678],[24,4671],{"className":4672,"style":674},[121],[24,4674,4676,4679,4682,4685,4688,4691],{"className":4675},[58],[24,4677],{"className":4678,"style":3326},[62],[24,4680,364],{"className":4681},[404],[24,4683,4038],{"className":4684},[67],[24,4686],{"className":4687,"style":4154},[121],[24,4689,4576],{"className":4690},[771],[24,4692],{"className":4693,"style":4154},[121],[24,4695,4697,4701,4704,4707,4710],{"className":4696},[58],[24,4698],{"className":4699,"style":4700},[62],"height:0.7278em;vertical-align:-0.0833em;",[24,4702,3306],{"className":4703},[67],[24,4705],{"className":4706,"style":4154},[121],[24,4708,563],{"className":4709},[771],[24,4711],{"className":4712,"style":4154},[121],[24,4714,4716,4719,4759,4762,4765],{"className":4715},[58],[24,4717],{"className":4718,"style":1984},[62],[24,4720,4722,4725],{"className":4721},[67],[24,4723,321],{"className":4724,"style":136},[67,68],[24,4726,4728],{"className":4727},[415],[24,4729,4731,4751],{"className":4730},[419,420],[24,4732,4734,4748],{"className":4733},[424],[24,4735,4737],{"className":4736,"style":2003},[428],[24,4738,4739,4742],{"style":2006},[24,4740],{"className":4741,"style":437},[436],[24,4743,4745],{"className":4744},[441,442,443,444],[24,4746,570],{"className":4747},[67,444],[24,4749,452],{"className":4750},[451],[24,4752,4754],{"className":4753},[424],[24,4755,4757],{"className":4756,"style":2025},[428],[24,4758],{},[24,4760],{"className":4761,"style":4154},[121],[24,4763,4576],{"className":4764},[771],[24,4766],{"className":4767,"style":4154},[121],[24,4769,4771,4774,4777,4780,4783],{"className":4770},[58],[24,4772],{"className":4773,"style":4700},[62],[24,4775,3366],{"className":4776},[67],[24,4778],{"className":4779,"style":4154},[121],[24,4781,563],{"className":4782},[771],[24,4784],{"className":4785,"style":4154},[121],[24,4787,4789,4792,4832,4835,4838],{"className":4788},[58],[24,4790],{"className":4791,"style":1984},[62],[24,4793,4795,4798],{"className":4794},[67],[24,4796,321],{"className":4797,"style":136},[67,68],[24,4799,4801],{"className":4800},[415],[24,4802,4804,4824],{"className":4803},[419,420],[24,4805,4807,4821],{"className":4806},[424],[24,4808,4810],{"className":4809,"style":2003},[428],[24,4811,4812,4815],{"style":2006},[24,4813],{"className":4814,"style":437},[436],[24,4816,4818],{"className":4817},[441,442,443,444],[24,4819,586],{"className":4820},[67,444],[24,4822,452],{"className":4823},[451],[24,4825,4827],{"className":4826},[424],[24,4828,4830],{"className":4829,"style":2025},[428],[24,4831],{},[24,4833],{"className":4834,"style":4154},[121],[24,4836,4576],{"className":4837},[771],[24,4839],{"className":4840,"style":4154},[121],[24,4842,4844,4847,4850,4853,4856,4859,4865,4868],{"className":4843},[58],[24,4845],{"className":4846,"style":3326},[62],[24,4848,3311],{"className":4849},[67],[24,4851,384],{"className":4852},[513],[24,4854,3989],{"className":4855},[67],[24,4857],{"className":4858,"style":122},[121],[24,4860,4609],{"className":4861,"style":4864},[2077,4862,4863],"op-symbol","small-op","position:relative;top:0em;",[24,4866],{"className":4867,"style":122},[121],[24,4869,321],{"className":4870,"style":136},[67,68],[1927,4872,4873,4876,4877,4905,4906,149],{},[19,4874,4875],{},"Result:"," This direct calculation yields the ",[24,4878,4880,4893],{"className":4879},[27],[24,4881,4883],{"className":4882},[31],[33,4884,4885],{"xmlns":35},[37,4886,4887,4891],{},[40,4888,4889],{},[43,4890,45],{},[47,4892,45],{"encoding":49},[24,4894,4896],{"className":4895,"ariaHidden":54},[53],[24,4897,4899,4902],{"className":4898},[58],[24,4900],{"className":4901,"style":63},[62],[24,4903,45],{"className":4904,"style":69},[67,68]," coordinates without any distance measurements required ",[24,4907,4908],{},"4, 11",[11,4910,4912],{"id":4911},"formula-breakdown-the-danger-circle","Formula Breakdown: The Danger Circle",[16,4914,4915,4916,4918,4919,4947,4948,273,4994,302,5022,149],{},"The ",[19,4917,163],{}," occurs when the unknown point ",[24,4920,4922,4935],{"className":4921},[27],[24,4923,4925],{"className":4924},[31],[33,4926,4927],{"xmlns":35},[37,4928,4929,4933],{},[40,4930,4931],{},[43,4932,45],{},[47,4934,45],{"encoding":49},[24,4936,4938],{"className":4937,"ariaHidden":54},[53],[24,4939,4941,4944],{"className":4940},[58],[24,4942],{"className":4943,"style":63},[62],[24,4945,45],{"className":4946,"style":69},[67,68]," lies on the circumcircle passing through ",[24,4949,4951,4970],{"className":4950},[27],[24,4952,4954],{"className":4953},[31],[33,4955,4956],{"xmlns":35},[37,4957,4958,4968],{},[40,4959,4960,4962,4964,4966],{},[43,4961,85],{},[87,4963,89],{"separator":54},[43,4965,92],{},[87,4967,89],{"separator":54},[47,4969,248],{"encoding":49},[24,4971,4973],{"className":4972,"ariaHidden":54},[53],[24,4974,4976,4979,4982,4985,4988,4991],{"className":4975},[58],[24,4977],{"className":4978,"style":110},[62],[24,4980,85],{"className":4981},[67,68],[24,4983,89],{"className":4984},[117],[24,4986],{"className":4987,"style":122},[121],[24,4989,92],{"className":4990,"style":126},[67,68],[24,4992,89],{"className":4993},[117],[24,4995,4997,5010],{"className":4996},[27],[24,4998,5000],{"className":4999},[31],[33,5001,5002],{"xmlns":35},[37,5003,5004,5008],{},[40,5005,5006],{},[43,5007,97],{},[47,5009,97],{"encoding":49},[24,5011,5013],{"className":5012,"ariaHidden":54},[53],[24,5014,5016,5019],{"className":5015},[58],[24,5017],{"className":5018,"style":63},[62],[24,5020,97],{"className":5021,"style":136},[67,68],[24,5023,5024],{},"2, 12",[5026,5027,5029],"callout",{"type":5028},"danger",[16,5030,5031,5032,5126,5127,5191,5192,302,5256,5259,5260,5293,5294,149],{},"Mathematically, on this circle, the sum of the angles ",[24,5033,5035,5065],{"className":5034},[27],[24,5036,5038],{"className":5037},[31],[33,5039,5040],{"xmlns":35},[37,5041,5042,5062],{},[40,5043,5044,5046,5048,5050,5052,5054,5058,5060],{},[87,5045,364],{"stretchy":363},[43,5047,1873],{},[87,5049,563],{},[43,5051,1878],{},[87,5053,563],{},[5055,5056,5057],"mtext",{},"Internal Angle ",[43,5059,92],{},[87,5061,384],{"stretchy":363},[47,5063,5064],{"encoding":49},"(\\alpha + \\beta + \\text{Internal Angle } B)",[24,5066,5068,5089,5107],{"className":5067,"ariaHidden":54},[53],[24,5069,5071,5074,5077,5080,5083,5086],{"className":5070},[58],[24,5072],{"className":5073,"style":3326},[62],[24,5075,364],{"className":5076},[404],[24,5078,1873],{"className":5079,"style":1900},[67,68],[24,5081],{"className":5082,"style":4154},[121],[24,5084,563],{"className":5085},[771],[24,5087],{"className":5088,"style":4154},[121],[24,5090,5092,5095,5098,5101,5104],{"className":5091},[58],[24,5093],{"className":5094,"style":1896},[62],[24,5096,1878],{"className":5097,"style":1910},[67,68],[24,5099],{"className":5100,"style":4154},[121],[24,5102,563],{"className":5103},[771],[24,5105],{"className":5106,"style":4154},[121],[24,5108,5110,5113,5120,5123],{"className":5109},[58],[24,5111],{"className":5112,"style":3326},[62],[24,5114,5117],{"className":5115},[67,5116],"text",[24,5118,5057],{"className":5119},[67],[24,5121,92],{"className":5122,"style":126},[67,68],[24,5124,384],{"className":5125},[513]," equals ",[24,5128,5130,5149],{"className":5129},[27],[24,5131,5133],{"className":5132},[31],[33,5134,5135],{"xmlns":35},[37,5136,5137,5146],{},[40,5138,5139],{},[3511,5140,5141,5144],{},[552,5142,5143],{},"180",[87,5145,3518],{},[47,5147,5148],{"encoding":49},"180^\\circ",[24,5150,5152],{"className":5151,"ariaHidden":54},[53],[24,5153,5155,5158,5162],{"className":5154},[58],[24,5156],{"className":5157,"style":3549},[62],[24,5159,5161],{"className":5160},[67],"18",[24,5163,5165,5168],{"className":5164},[67],[24,5166,3560],{"className":5167},[67],[24,5169,5171],{"className":5170},[415],[24,5172,5174],{"className":5173},[419],[24,5175,5177],{"className":5176},[424],[24,5178,5180],{"className":5179,"style":3549},[428],[24,5181,5182,5185],{"style":3575},[24,5183],{"className":5184,"style":437},[436],[24,5186,5188],{"className":5187},[441,442,443,444],[24,5189,3518],{"className":5190},[771,444]," or ",[24,5193,5195,5214],{"className":5194},[27],[24,5196,5198],{"className":5197},[31],[33,5199,5200],{"xmlns":35},[37,5201,5202,5211],{},[40,5203,5204],{},[3511,5205,5206,5209],{},[552,5207,5208],{},"360",[87,5210,3518],{},[47,5212,5213],{"encoding":49},"360^\\circ",[24,5215,5217],{"className":5216,"ariaHidden":54},[53],[24,5218,5220,5223,5227],{"className":5219},[58],[24,5221],{"className":5222,"style":3549},[62],[24,5224,5226],{"className":5225},[67],"36",[24,5228,5230,5233],{"className":5229},[67],[24,5231,3560],{"className":5232},[67],[24,5234,5236],{"className":5235},[415],[24,5237,5239],{"className":5238},[419],[24,5240,5242],{"className":5241},[424],[24,5243,5245],{"className":5244,"style":3549},[428],[24,5246,5247,5250],{"style":3575},[24,5248],{"className":5249,"style":437},[436],[24,5251,5253],{"className":5252},[441,442,443,444],[24,5254,3518],{"className":5255},[771,444],[24,5257,5258],{},"2, 13",". In this configuration, the denominators in the Tienstra weights approach zero, and the coordinates become indeterminate (",[24,5261,5263,5281],{"className":5262},[27],[24,5264,5266],{"className":5265},[31],[33,5267,5268],{"xmlns":35},[37,5269,5270,5278],{},[40,5271,5272,5274,5276],{},[552,5273,3560],{},[43,5275,3989],{"mathvariant":2634},[552,5277,3560],{},[47,5279,5280],{"encoding":49},"0\u002F0",[24,5282,5284],{"className":5283,"ariaHidden":54},[53],[24,5285,5287,5290],{"className":5286},[58],[24,5288],{"className":5289,"style":3326},[62],[24,5291,5280],{"className":5292},[67],"). Your position is mathematically \"lost\" ",[24,5295,167],{},[11,5297,5299],{"id":5298},"practical-tips","Practical Tips",[1924,5301,5302,5416,5425],{},[1927,5303,5304,5307,5308,302,5336,5339,5340,273,5386,302,5414,149],{},[19,5305,5306],{},"Inside is Safest:"," To physically guarantee you avoid the Danger Circle, always site your station ",[24,5309,5311,5324],{"className":5310},[27],[24,5312,5314],{"className":5313},[31],[33,5315,5316],{"xmlns":35},[37,5317,5318,5322],{},[40,5319,5320],{},[43,5321,45],{},[47,5323,45],{"encoding":49},[24,5325,5327],{"className":5326,"ariaHidden":54},[53],[24,5328,5330,5333],{"className":5329},[58],[24,5331],{"className":5332,"style":63},[62],[24,5334,45],{"className":5335,"style":69},[67,68],[19,5337,5338],{},"inside"," the triangle formed by ",[24,5341,5343,5362],{"className":5342},[27],[24,5344,5346],{"className":5345},[31],[33,5347,5348],{"xmlns":35},[37,5349,5350,5360],{},[40,5351,5352,5354,5356,5358],{},[43,5353,85],{},[87,5355,89],{"separator":54},[43,5357,92],{},[87,5359,89],{"separator":54},[47,5361,248],{"encoding":49},[24,5363,5365],{"className":5364,"ariaHidden":54},[53],[24,5366,5368,5371,5374,5377,5380,5383],{"className":5367},[58],[24,5369],{"className":5370,"style":110},[62],[24,5372,85],{"className":5373},[67,68],[24,5375,89],{"className":5376},[117],[24,5378],{"className":5379,"style":122},[121],[24,5381,92],{"className":5382,"style":126},[67,68],[24,5384,89],{"className":5385},[117],[24,5387,5389,5402],{"className":5388},[27],[24,5390,5392],{"className":5391},[31],[33,5393,5394],{"xmlns":35},[37,5395,5396,5400],{},[40,5397,5398],{},[43,5399,97],{},[47,5401,97],{"encoding":49},[24,5403,5405],{"className":5404,"ariaHidden":54},[53],[24,5406,5408,5411],{"className":5407},[58],[24,5409],{"className":5410,"style":63},[62],[24,5412,97],{"className":5413,"style":136},[67,68],[24,5415,570],{},[1927,5417,5418,5421,5422,149],{},[19,5419,5420],{},"4th Point Redundancy:"," Always observe a fourth control point if available. This allows you to compute a second resection as a check and provides a statistical measure of error ",[24,5423,5424],{},"2, 6",[1927,5426,5427,5430,5431,5560,5561,149],{},[19,5428,5429],{},"Check Your Sums:"," Ensure your measured angles ",[24,5432,5434,5464],{"className":5433},[27],[24,5435,5437],{"className":5436},[31],[33,5438,5439],{"xmlns":35},[37,5440,5441,5461],{},[40,5442,5443,5445,5447,5449,5451,5453,5455],{},[43,5444,1873],{},[87,5446,563],{},[43,5448,1878],{},[87,5450,563],{},[43,5452,1883],{},[87,5454,541],{},[3511,5456,5457,5459],{},[552,5458,5208],{},[87,5460,3518],{},[47,5462,5463],{"encoding":49},"\\alpha + \\beta + \\gamma = 360^\\circ",[24,5465,5467,5486,5504,5522],{"className":5466,"ariaHidden":54},[53],[24,5468,5470,5474,5477,5480,5483],{"className":5469},[58],[24,5471],{"className":5472,"style":5473},[62],"height:0.6667em;vertical-align:-0.0833em;",[24,5475,1873],{"className":5476,"style":1900},[67,68],[24,5478],{"className":5479,"style":4154},[121],[24,5481,563],{"className":5482},[771],[24,5484],{"className":5485,"style":4154},[121],[24,5487,5489,5492,5495,5498,5501],{"className":5488},[58],[24,5490],{"className":5491,"style":1896},[62],[24,5493,1878],{"className":5494,"style":1910},[67,68],[24,5496],{"className":5497,"style":4154},[121],[24,5499,563],{"className":5500},[771],[24,5502],{"className":5503,"style":4154},[121],[24,5505,5507,5510,5513,5516,5519],{"className":5506},[58],[24,5508],{"className":5509,"style":2769},[62],[24,5511,1883],{"className":5512,"style":1920},[67,68],[24,5514],{"className":5515,"style":674},[121],[24,5517,541],{"className":5518},[678],[24,5520],{"className":5521,"style":674},[121],[24,5523,5525,5528,5531],{"className":5524},[58],[24,5526],{"className":5527,"style":3549},[62],[24,5529,5226],{"className":5530},[67],[24,5532,5534,5537],{"className":5533},[67],[24,5535,3560],{"className":5536},[67],[24,5538,5540],{"className":5539},[415],[24,5541,5543],{"className":5542},[419],[24,5544,5546],{"className":5545},[424],[24,5547,5549],{"className":5548,"style":3549},[428],[24,5550,5551,5554],{"style":3575},[24,5552],{"className":5553,"style":437},[436],[24,5555,5557],{"className":5556},[441,442,443,444],[24,5558,3518],{"className":5559},[771,444],". Any deviation indicates a gross observation error or a misidentified target ",[24,5562,5563],{},"9",[11,5565,5567],{"id":5566},"common-mistakes","Common Mistakes",[1924,5569,5570,5636],{},[1927,5571,5572,5575,5576,5604,5605,5633,5634,149],{},[19,5573,5574],{},"Sign Confusion:"," When ",[24,5577,5579,5592],{"className":5578},[27],[24,5580,5582],{"className":5581},[31],[33,5583,5584],{"xmlns":35},[37,5585,5586,5590],{},[40,5587,5588],{},[43,5589,45],{},[47,5591,45],{"encoding":49},[24,5593,5595],{"className":5594,"ariaHidden":54},[53],[24,5596,5598,5601],{"className":5597},[58],[24,5599],{"className":5600,"style":63},[62],[24,5602,45],{"className":5603,"style":69},[67,68]," is outside the control triangle, one or more ",[24,5606,5608,5621],{"className":5607},[27],[24,5609,5611],{"className":5610},[31],[33,5612,5613],{"xmlns":35},[37,5614,5615,5619],{},[40,5616,5617],{},[43,5618,321],{},[47,5620,321],{"encoding":49},[24,5622,5624],{"className":5623,"ariaHidden":54},[53],[24,5625,5627,5630],{"className":5626},[58],[24,5628],{"className":5629,"style":63},[62],[24,5631,321],{"className":5632,"style":136},[67,68]," values may become negative. Ensure you maintain the correct algebraic signs throughout the summation ",[24,5635,339],{},[1927,5637,5638,5641,5642,5645,5646,149],{},[19,5639,5640],{},"Near-Circle Geometry:"," Even if you aren't ",[2586,5643,5644],{},"exactly"," on the Danger Circle, being near it results in very small denominators, which massively magnifies small angular measurement errors ",[24,5647,570],{},[11,5649,5651],{"id":5650},"best-practices","Best Practices",[16,5653,5654,5655,5683,5684,302,5687,149],{},"If your reconnaissance shows that you must set up outside the control triangle, perform a quick field check: if the angle at ",[24,5656,5658,5671],{"className":5657},[27],[24,5659,5661],{"className":5660},[31],[33,5662,5663],{"xmlns":35},[37,5664,5665,5669],{},[40,5666,5667],{},[43,5668,45],{},[47,5670,45],{"encoding":49},[24,5672,5674],{"className":5673,"ariaHidden":54},[53],[24,5675,5677,5680],{"className":5676},[58],[24,5678],{"className":5679,"style":63},[62],[24,5681,45],{"className":5682,"style":69},[67,68]," to the two \"outside\" points is roughly equal to the internal angle of the triangle at the \"middle\" point, you are near the danger circle and ",[19,5685,5686],{},"must reposition",[24,5688,570],{},[11,5690,5692],{"id":5691},"faq","FAQ",[5694,5695,5696,5735,5743],"accordion",{},[5697,5698,5700],"accordion-item",{"label":5699},"Why use Tienstra instead of the Cotangent method?",[16,5701,5702,5703,5731,5732,149],{},"The Tienstra formula is more \"elegant\" because it uses the same weights (",[24,5704,5706,5719],{"className":5705},[27],[24,5707,5709],{"className":5708},[31],[33,5710,5711],{"xmlns":35},[37,5712,5713,5717],{},[40,5714,5715],{},[43,5716,321],{},[47,5718,321],{"encoding":49},[24,5720,5722],{"className":5721,"ariaHidden":54},[53],[24,5723,5725,5728],{"className":5724},[58],[24,5726],{"className":5727,"style":63},[62],[24,5729,321],{"className":5730,"style":136},[67,68],") for both Easting and Northing, reducing the potential for calculation errors compared to the traditional cotangent method ",[24,5733,5734],{},"1, 14",[5697,5736,5738],{"label":5737},"Can I use Tienstra with a Total Station?",[16,5739,5740,5741,149],{},"Yes. While most Total Stations have built-in \"Free Station\" routines (which often use Least Squares), understanding Tienstra allows you to manually verify a position if the internal software is suspect ",[24,5742,570],{},[5697,5744,5746],{"label":5745},"Does this work in 3D?",[16,5747,5748,5749,149],{},"Tienstra is a 2D (horizontal) resection method. Vertical position is typically determined separately through trigonometrical levelling after the horizontal position is fixed ",[24,5750,5751],{},"15",[11,5753,5755],{"id":5754},"conclusion","Conclusion",[16,5757,5758],{},"The Tienstra method remains a masterclass in geodetic mathematics, offering a direct path from angular field data to coordinate mapping. By understanding the barycentric principles and respecting the \"trap\" of the Danger Circle, surveyors can utilize this method to establish high-precision site control in even the most challenging environments.",[11,5760,5762],{"id":5761},"references","References",[16,5764,5765,5766,5769,5770,5773,5774,5777],{},"Schofield, W. (2001). ",[2586,5767,5768],{},"Engineering Surveying",". 5th ed. Butterworth-Heinemann.\nAllan, A. L. (1969). ",[2586,5771,5772],{},"The Tienstra Method of Resection",".\nPorta, J. M., & Thomas, F. (2009). ",[2586,5775,5776],{},"Concise Proof of Tienstra’s Formula",". Journal of Surveying Engineering.",{"title":5779,"searchDepth":5780,"depth":5780,"links":5781},"",2,[5782,5783,5784,5785,5786,5791,5797,5798,5799,5800,5801,5802,5803,5804],{"id":13,"depth":5780,"text":14},{"id":152,"depth":5780,"text":153},{"id":170,"depth":5780,"text":171},{"id":192,"depth":5780,"text":193},{"id":342,"depth":5780,"text":343,"children":5787},[5788],{"id":1738,"depth":5789,"text":5790},3,"Defining the Weights (KKK)",{"id":2822,"depth":5780,"text":2823,"children":5792},[5793,5794,5795,5796],{"id":2829,"depth":5789,"text":2830},{"id":2922,"depth":5789,"text":2923},{"id":2961,"depth":5789,"text":2962},{"id":3101,"depth":5789,"text":3102},{"id":3280,"depth":5780,"text":3281},{"id":4911,"depth":5780,"text":4912},{"id":5298,"depth":5780,"text":5299},{"id":5566,"depth":5780,"text":5567},{"id":5650,"depth":5780,"text":5651},{"id":5691,"depth":5780,"text":5692},{"id":5754,"depth":5780,"text":5755},{"id":5761,"depth":5780,"text":5762},"Technical Deep-Dive","The Tienstra method is one of the most elegant solutions in surveying mathematics — but it carries a geometric trap that has caused field problems for generations of surveyors. Understanding the danger circle is not optional.","md","https:\u002F\u002Fimages.unsplash.com\u002Fphoto-1503387762-592deb58ef4e?w=800&q=80",{"authorCredentials":5810,"authorBio":5811,"commentCount":5812,"tags":5813,"relatedTools":5819,"relatedArticles":5821,"sources":5824},"FInstCES","Principal engineer with 35 years in engineering surveying. Specialist in structural monitoring and high-precision setting out.",23,[5814,5815,5816,5817,5818],"tienstra","resection","COGO","danger-circle","geometry",[5820],"tienstra-resection",[5822,5823],"coordinate-computation","traversing-control-networks",[5825,5829,5833],{"title":5826,"type":5827,"note":5828},"Engineering Surveying (Fifth Edition)","Book","Source for general resection theory and the Circle of Danger.",{"title":5830,"type":5831,"note":5832},"THE TIENSTRA METHOD OF RESECTION","PDF","Primary source for the barycentric formula and mathematical proof.",{"title":5834,"type":5835,"note":5836},"Tienstra formula - Wikipedia","Webpage","Source for the standard coordinate weighting equations.",true,"\u002Fblog\u002Ftienstra-resection-danger-circle","2026-01-15",18,{"title":5,"description":5806},{"loc":5838},"blog\u002F23.tienstra-resection-danger-circle","vfTDzY5VqLXyWF5-KefXoBjAx9vnDBViLbypT6I3pt8",[5846,5851],{"title":5847,"path":5848,"stem":5849,"description":5850,"children":-1},"Inertial Surveying Systems (ISS): High-Speed Positioning Without Satellites","\u002Fblog\u002Finertial-surveying-systems-iss","blog\u002F28.inertial-surveying-systems-iss","A technical look at Inertial Surveying Systems, including the use of accelerometers, gyroscopes, and the critical Zero Velocity Update (ZUPT) procedure.",null,1783851944426]