第3講(三角網(wǎng)條件平差

1、第三章 條件平差3.2 條件方程內(nèi)容安排一、高程控制網(wǎng)條件方程的列立二、導(dǎo)線網(wǎng)條件方程的列立三、測(cè)角網(wǎng)條件方程的列立四、測(cè)邊網(wǎng)條件方程的列立3.2 條件方程多于四個(gè)必要起算數(shù)據(jù)時(shí)，稱為非自由網(wǎng)。三、測(cè)角網(wǎng)條件方程的列立（一）三角網(wǎng)平差的目的求待定點(diǎn)平面坐標(biāo)平差值，并進(jìn)行精度評(píng)定。（二）三角網(wǎng)的種類測(cè)角網(wǎng)、測(cè)邊網(wǎng)、邊角同測(cè)網(wǎng)無(wú)論網(wǎng)型多么復(fù)雜，都是由三角形和大地四邊形相互鄰接或重疊而組成。當(dāng)網(wǎng)中僅具備4個(gè)必要起算數(shù)據(jù)（一點(diǎn)坐標(biāo)、一條邊的方位、一條邊的邊長(zhǎng)或已知兩點(diǎn)坐標(biāo)）時(shí)，稱為自由網(wǎng)。這四個(gè)數(shù)據(jù)稱為必要起算數(shù)據(jù)。三、測(cè)角網(wǎng)條件方程的列立（三）條件方程個(gè)數(shù)的確定條件方程個(gè)數(shù)等于多余觀測(cè)個(gè)數(shù)r=n-t

2、 關(guān)鍵在于確定必要觀測(cè)個(gè)數(shù) t1.當(dāng)網(wǎng)中有2個(gè)或2個(gè)以上已知點(diǎn)時(shí) t=2倍待定點(diǎn)數(shù)2.當(dāng)網(wǎng)中少于2個(gè)已知點(diǎn)時(shí)（1）測(cè)角網(wǎng) t=2倍總點(diǎn)數(shù)-4（2）測(cè)邊或邊角網(wǎng) t=2倍總點(diǎn)數(shù)-3測(cè)角網(wǎng)：n=12 t=2X6-4=8 r=4測(cè)邊網(wǎng)：n=9 t=2X6-3=9 r=0邊角網(wǎng)：n=21 t=9 r=12測(cè)角網(wǎng)：n=23 t=12 r=11測(cè)邊網(wǎng)：n=14 t=12 r=2邊角網(wǎng)：n=37 t=12 r=25（四）條件方程的列立條件方程的種類：1、圖形條件（內(nèi)角和條件）2、水平條件（圓周條件）3、極條件4、方位角條件5、邊長(zhǎng)條件6、坐標(biāo)條件三、測(cè)角網(wǎng)條件方程的列立每個(gè)三角形內(nèi)角平差值和等于180018

3、0321LLL0321wvvv)180(321LLLw（四）條件方程的列立1、圖形條件(n=15 t=8 r=7 哪7個(gè)？)中點(diǎn)多邊形中心點(diǎn)角度平差值之和等于36003601512963LLLLL01512963wvvvvv)360(1512963LLLLLw（四）條件方程的列立2、水平條件14131110875421sinsinsinsinsinsinsinsinsinsinLLLLLLLLLLSSBPBP（四）條件方程的列立3、極條件中點(diǎn)多邊形和大地四邊形存在極條件。中點(diǎn)多邊形：從中心點(diǎn)的任一邊開(kāi)始，依次推算其它邊的長(zhǎng)度，最后回到起始邊，則起始邊長(zhǎng)度的平差值應(yīng)該與推算值相等。（從極點(diǎn)出發(fā)各

4、邊之比為1）14131110875421sinsinsinsinsinsinsinsinsinsinLLLLLLLLLLSSBPBP01sinsinsinsinsinsinsinsinsinsin14118521310741LLLLLLLLLL列立規(guī)律：列出從極點(diǎn)P出發(fā)的各條邊之比，把邊長(zhǎng)比換為正弦的比，即可列出（四）條件方程的列立3、極條件1sinsinsinsinsinsinsinsinsinsin1sinsinsinsinsinsinsinsinsinsin1411852131074114118521310741LLLLLLLLLLLLLLLLLLLL 22141185213107411

5、114118521310741cotsinsinsinsinsinsinsinsinsinsincotsinsinsinsinsinsinsinsinsinsinvLLLLLLLLLLLvLLLLLLLLLLL 55141185213107414414118521310741cotsinsinsinsinsinsinsinsinsinsincotsinsinsinsinsinsinsinsinsinsinvLLLLLLLLLLLvLLLLLLLLLLL 88141185213107417714118521310741cotsinsinsinsinsinsinsinsinsinsincotsin

6、sinsinsinsinsinsinsinsinsinvLLLLLLLLLLLvLLLLLLLLLLL 111114118521310741101014118521310741cotsinsinsinsinsinsinsinsinsinsincotsinsinsinsinsinsinsinsinsinsinvLLLLLLLLLLLvLLLLLLLLLLL0cotsinsinsinsinsinsinsinsinsinsincotsinsinsinsinsinsinsinsinsinsin141414118521310741131314118521310741 vLLLLLLLLLLLvLLLLL

7、LLLLLL01sinsinsinsinsinsinsinsinsinsin14118521310741LLLLLLLLLL極條件的線性化0cotcotcotcot cotcotcotcotcotcot1414131311111010887755442211wvLvLvLvLvLvLvLvLvLvL 13107411411852sinsinsinsinsinsinsinsinsinsin1LLLLLLLLLLwSin 變 cot分子取+ 分母取- 常數(shù)項(xiàng)顛倒01sinsinsinsinsinsinsinsinsinsin14118521310741LLLLLLLLLL極條件的線性化記憶規(guī)律01

8、sin)sin(sinsinsin)sin(74314287LLLLLLLL01sin)sin(sin)sin(sinsin74318742LLLLLLLL0)()( )()(887778744343432211wvLLctgvctgLLLctgvLLctgctgLvLLctgvctgLvctgL )sin(sinsinsin)sin(sin187427431LLLLLLLLw01DBDCDADCDADB（四）條件方程的列立3、極條件大地四邊形：取一頂點(diǎn)（D）為極點(diǎn)，從極點(diǎn)出發(fā)的各條邊之比等于1。把邊長(zhǎng)比換為角度正弦比。0EFEFTT180312963LLLLTTABEF0180312963E

9、FABTTLLLL012963Twvvvv)1803(12963EFABTTTLLLLw（四）條件方程的列立4、方位角條件（n=12 t=4 r=8 哪8個(gè)？)平差值條件方程：從一個(gè)已知方位角推算另一個(gè)已知方位角，推算值應(yīng)該與已知值相等。0EFEFSS1185210741sinsinsinsinsinsinsinsinLLLLLLLLSSABEF01sinsinsinsinsinsinsinsin1185210741LLLLSLLLLSEFAB0 11111010887755442211SwvctgLvctgLvctgLvctgLvctgLvctgLvctgLvctgL 1074111852s

10、insinsinsinsinsinsinsin1LLLLSLLLLSwABEFS（四）條件方程的列立5、邊長(zhǎng)條件從一條已知邊推算另一已知邊，推算值等于已知值線性化方法同極條件0EExx0EEyyCECEBCBCBCEBCBETSTSxxxxxcoscos21sinsinLLSSABBC1185212741sinsinsinsinsinsinsinsinLLLLLLLLSSABCE1803LTTABBC180210963LLLLTTABCE（四）條件方程的列立6、坐標(biāo)條件將上述公式代入XE式，用泰勒公式線性化得)(265.2061000)(EEEExxxxxw )(265.206EEyyyw0

11、)()()()( )()( )()(1096311111212887755442211yCECECEBECECECEBEwvxxvxxvxxvxxvctgLvctgLyyvctgLvctgLyyvctgLvctgLyyvctgLvctgLyy0 )()()()( )()( )()(1096311111212887755442211xCECECEBECECECEBEwvyyvyyvyyvyyvctgLvctgLxxvctgLvctgLxxvctgLvctgLxxvctgLvctgLxx（四）條件方程的列立6、坐標(biāo)條件請(qǐng)大家尋找記憶規(guī)律例題： 如圖如圖3 31414所示三角網(wǎng)中，有所示三角網(wǎng)中，有

12、2 2個(gè)個(gè)已知點(diǎn)已知點(diǎn)A A、B B，2 2個(gè)待定點(diǎn)個(gè)待定點(diǎn)C C、D D和和8 8個(gè)角個(gè)角度觀測(cè)值。度觀測(cè)值。A A點(diǎn)和點(diǎn)和B B點(diǎn)的坐標(biāo)、角度觀點(diǎn)的坐標(biāo)、角度觀測(cè)值見(jiàn)表測(cè)值見(jiàn)表3 33 3。試按條件平差法計(jì)算；。試按條件平差法計(jì)算；各待定點(diǎn)坐標(biāo)平差值；各待定點(diǎn)坐標(biāo)平差值；C C點(diǎn)坐標(biāo)點(diǎn)坐標(biāo)平差值的點(diǎn)位中誤差；平差值的點(diǎn)位中誤差；C C點(diǎn)至點(diǎn)至D D點(diǎn)間點(diǎn)間邊長(zhǎng)平差值的中誤差和邊長(zhǎng)相對(duì)中誤邊長(zhǎng)平差值的中誤差和邊長(zhǎng)相對(duì)中誤差；差；C C點(diǎn)至點(diǎn)至D D點(diǎn)間方位角平差值的中點(diǎn)間方位角平差值的中誤差。誤差。解解：本題中n=8，t=4，r=41sinsinsin)sin()sin(sin0180018

13、0018072576324876565434321LLLLLLLLLLLLLLLLLLLL00 . 700 .1100 .10876565434321vvvvvvvvvvvv03 . 9cot)cot()cot(cotcot)cot()cot(cot777667655443322322vLLLvLLvLvLvLLvLLL1)條件方程和平差值函數(shù)式 4個(gè)條件方程，包括3個(gè)圖形條件和1個(gè)極條件，閉合差以秒(“)為單位，平差值條件方程為)cos(sin)sin(1432LLLLSXXABABAC432124. 005. 005. 087. 0LdLdLdLdXdC)sin(sin)sin(1432L

14、LLLSYYABABAC432197. 020. 020. 022. 0LdLdLdLdYdC平差值函數(shù)式和權(quán)函數(shù)式，如下權(quán)函數(shù)式為C點(diǎn)Y坐標(biāo)平差值的函數(shù)式為權(quán)函數(shù)式為C點(diǎn)X坐標(biāo)平差值的函數(shù)式為6431sinsinsinsinLLLLSSABCD643119. 059. 067. 029. 0LdLdLdLdSdCD1808761LLLLABCD8761LdLdLdLddCDC點(diǎn)至D點(diǎn)間邊長(zhǎng)平差值的函數(shù)式為權(quán)函數(shù)式為C點(diǎn)至D點(diǎn)間邊長(zhǎng)平差值的函數(shù)式為權(quán)函數(shù)式為EPQ1043. 157. 079. 111. 123. 002. 1011110000001111000000111184A029. 9

15、7111090. 779. 348. 19 . 179. 342048. 12429 . 10244321kkkk2)權(quán)的確定取各角度為等權(quán)觀測(cè)值，權(quán)均為1，各觀測(cè)角度兩兩獨(dú)立，協(xié)因數(shù)陣為單位陣，即 條件方程系數(shù)陣為組成法方程為(單位陣)T13751. 00131. 478678. 404128. 0K)(0 . 42 . 47 . 05 . 00 . 58 . 42 . 00TVTL 6 .0657302 .2026493 .5921705 .3414290 .1001422 .1622388 .3628380 .5707613)解算法方程利用改正數(shù)方程求得計(jì)算平差值，并代入平差值條件式檢核，如下經(jīng)檢驗(yàn)滿足所有條件方程。4)改正數(shù)5)平差值mYmXmYmXDDCC6179.6158875000.43766012121.6166873320.4377352)(54. 4)(446.820rPVVT)(65.