1、航海學 (2)天文航海內 容 球面三角與船位誤差理論基礎 測天定位 天測羅經差天文航海的優點 使用的設備是六分儀，簡單可靠； 不發射任何電波，隱蔽性好； 觀測目標是天體，不受任何人控制； 可能是大洋航行中唯一的全球導航系統。球面三角與船位誤差理論基礎球面三角與船位誤差理論基礎 第一章 球面三角 第二章 內插法 第二章 船位誤差基礎 第二章 等精度觀測平差 第一章 球面三角球面三角球面三角, ,主要研究球面上由三個大圓主要研究球面上由三個大圓弧相交圍成的球面三角形及其性質、解弧相交圍成的球面三角形及其性質、解算等問題算等問題 。 1.1 球面幾何 1.2 球面三角形 1.3 球面三角形的邊角函數

2、關系 1.1 球面幾何1.1.1 球、球面 在空間與一定點等距離的點的軌跡稱為球面球面。包圍在球面中的實體稱為球,這一定點稱為球心球心。球心和球面上任意一點間的距離稱為球半徑球半徑R R。過球心與球面相交的直線段稱為球直徑球直徑。 同球的半徑和直徑都相等。同理,半徑或直徑相等的球全等。所以,球面又可定義為半圓周繞它的直徑旋轉一周的旋轉面。1.1.2 球面上的圓 任意一平面和球面相截的截痕是圓圓。 當平面通過球心時，所截成的圓稱為大大圓，圓，它的一段圓周叫大圓弧大圓弧。 截面不通過球心的圓稱為小圓小圓，它的一段圓周叫小圓弧小圓弧。PPLQLQOCrR1.1.3 大圓的性質 1大圓的圓心與球心重合

3、。2大圓的直徑等于球直徑，半徑等于球半徑。3同球或等球上的大圓的大小相等。4大圓等分球面和球體。5同球上的兩個大圓平面一定相交，交線是它們的直徑，并且兩大圓互相平分。 6過球面上不在同一直徑兩端上的兩個點，能作且僅能作一個大圓，卻能作無數個小圓。若在同一直徑兩端上的兩個點，則能作無數個大圓而不能作小圓。7小于180的大圓弧（劣弧）是球面上兩點間的最短球面距離。因此，兩點間的球面距離應用大圓弧度量。 1.1.4 軸、極、極距、極線 垂直與任意圓面的球直徑稱為該圓（大圓或小圓）的軸軸。軸的兩個端點稱為極極。垂直于同一軸可有數個平行圓，其中只有一個通過球心的是大圓，其余的都是小圓。從極到圓(大圓或小

4、圓)弧上任一點沿大圓弧的球面距離叫極距極距,又叫球球面半徑面半徑。同一個圓的極距或球面半徑都相等。極距為90的大圓弧又稱為該極的極線極線。 大圓弧是它的極的極線。反之，極線必定是大圓弧。1.1.5 球面角及其度量 球面上兩大圓弧相交構成的角稱為球面球面角角，其交點叫球面角的頂點，兩大圓弧稱為球面角的邊。 球面角的三種度量方法：1切于頂點大圓弧的切線夾角CPD；2頂點的極線被其兩邊大圓弧所截的弧長AB；3大圓弧AB所對的球心角AOB。 1.2 球面三角形球面三角形 1.2.1 定義定義 在球面上由三個大圓弧圍成的三角形稱為球面三角形球面三角形。圍成三角形的大圓弧稱為球面三角形的邊邊。由大圓弧相交

5、所成的球面角稱為球面三角形的角角。 三個角A、B、C和三個邊a、b、c合稱為球面三角形的六要素。航海上討論的球面三角形的六要素均大于0，而小于180，又稱其為歐拉球面三角形。 1.2.2 球面三角形分類球面三角形分類 球面三角形分為直角、直邊、等腰、等邊、初等和任意三角形。1球面直角三角形和球面直邊三角形 至少有一個角為90的三角形稱為球面直角三角形。 至少有一個邊為90的三角形稱為球面直邊三角形。 2球面等腰三角形和球面等邊三角形 有兩邊或兩角相等的三角形稱為球面等腰三角形。若三邊或三角都相等的三角形稱為球面等邊三角形。3球面初等三角形 三個邊相對其球半徑甚小的三角形稱為球面小三角形。只有一

6、個角及其對邊相對球半徑甚小的三角形稱為球面窄三角形。兩者統稱為球面初等三角形。4球面任意三角形 不具備上述特殊條件的球面三角形1.2.3 極線球面三角形極線球面三角形 球面三角形ABC三個頂點的極線所構成的球面三角形ABC稱為原球面三角形ABC的極線球面三角形。 1原三角形與其極線三角形是互為極線三角形。 原球面三角形ABC的三個邊，也就是其極線球面三角形AB C三個頂點的極線。換句話說，若畫極線三角形ABC的極線三角形，則所得到的就是原球面三角形ABC。所以，它們之間的關系是相互的。2原球面三角形的邊與其極線三角形對應角互補 1.3 球面三角形的邊角函數關系球面三角形的邊角函數關系 1.3.

7、1 任意球面三角形任意球面三角形 1.3.2 球面直角和直邊三角形球面直角和直邊三角形 1.3.3 球面初等三角形球面初等三角形 1.3.1.1 余弦公式余弦公式 邊的余弦公式記憶口訣：一邊的余弦等于其它兩邊余弦一邊的余弦等于其它兩邊余弦的乘積，加上這兩邊正弦及其夾角余弦的乘積，加上這兩邊正弦及其夾角余弦的乘積。的乘積。 )cos()sin()sin()cos()cos()cos(Acbcba邊的余弦公式應用：已知兩邊及其夾角求對邊；已知三邊求三角。 By transposing the formula, we get a form in which it may be used, given

8、 three sides, to find any angle. )sin()sin()cos()cos()cos()cos(cbcbaA角的余弦公式記憶口訣：一角的余弦等于其它兩角余弦一角的余弦等于其它兩角余弦的乘積冠以負號加上這兩角正弦及其夾的乘積冠以負號加上這兩角正弦及其夾邊余弦的乘積。邊余弦的乘積。cosAcosA-cosBcosC-cosBcosCsinBsinCcosasinBsinCcosa 角的余弦公式應用：已知兩角及其夾邊求對角；已知三角求三邊。 1.3.1.2 正弦公式正弦公式 記憶口訣：邊的正弦與其對角的記憶口訣：邊的正弦與其對角的正弦成比例。正弦成比例。 )sin()s

9、in()sin()sin()sin()sin(CcBbAa正弦公式應用：已知兩角及其一對邊，求另一邊；已知兩邊及其一對角，求另一角。 Its big disadvantage is the ambiguity about the actual value of the part found, since)180sin()sin(0AA In short, when we have taken out the anti-sine and obtained (say) 42, the question arises-is the answer 42 or 138? This difficulty

10、arises whenever an angle is found through its sine. There will sometimes be two solutions, sometimes one, and sometimes no solution at all. Disregarding the third rather theoretical possibility, some progress can be made on the first two by remembering that, in any triangle, A-B and a-b must be of t

11、he same sign.For example, a triangle PAB, in which b=2621, B=5222, A=10444.To find a, we have Sin(a)=0.542039 a=3249.4or a=14710.6 AB, ab. Both values for a satisfy this requirement; thus both are solutions of the data as given. )sin()sin()sin()sin(BbAa1.3.1.3 余切公式（四聯公式余切公式（四聯公式）記憶口訣：外邊余切內邊正弦乘積等于外角余

12、切內角正弦乘積加上內邊內角余弦積。 CbCctgAbctgacoscossinsinACBabc余切公式用于：在球面三角形中，已知相連的四個要素中的三個，求另一個要素。即已知兩邊及夾角求相連的角或已知兩角及夾邊求相連的邊。 1.3.1.4 解球面任意三角形 根據球面三角形已知要素求解其余要素的方法稱為解球面三角形。已 知求應用公式說 明兩邊夾角a、b、C第三邊及其它兩角c、A、B邊的余弦公式求c四聯公式求A、B 有一確定解兩角夾邊A、B、c第三角及其它兩邊C、a、b角的余弦公式求C；四聯公式求a、b三邊a、b、c三角A、B、C邊的余弦公式三角A、B、C三邊a、b、c角的余弦公式兩邊及其一邊對角

13、a、b、A求另一角和其它兩邊c、B、C正弦公式求B;四聯公式一解兩解或無解兩角及其一角對邊A、B、a求另一角和其它兩邊C、b、c正弦公式求b;四聯公式使用計算器解算球面三角時注意事項計算器有三種角度單位供選擇即： 度DEG、弧度RAD、公制度GRAD。球面三角中，角或邊都是以六十進制的度、分及分的小數給出的，利用計算器計算，在DEG狀態下，一定要將角或邊轉換成以十進制的度為單位輸入。 Examples, in a triangle PAB, P=6630, a=4700, b=6700, find A.We shall use the four-parts formula. A=5232.6

14、pbpctgAbctgacoscossinsin1.3.2 球面直角和直邊三角形球面直角和直邊三角形 球面直角三角形公式 有一個或一個以上的角為直角的球面三角形稱為球面直角三角形。 設球面直角三角形ABC中，C90。因為sin901，cos900，可由球面任意三角形的基本公式，導出相應的球面直角三角形十個公式： sinasinAsinc coscctgActgB sinactgBtgb cosAsinBcosa sinbsinBsinc cosActgctgb sinbctgAtga cosBsinAcosb cosccosacosb cosBctgctga 球面直角三角形公式的納比爾記憶法則

15、在球面三角形ABC中，C90。先畫“大”字圖形，大字上部豎線代表直角C，相鄰兩側為夾直角的兩邊a和b，大字下面三個空格依次填入相對應元素邊或角的余數（a邊對90A，b邊對90B，C角對90c)。 任一要素的正弦等于相鄰兩要素正切乘任一要素的正弦等于相鄰兩要素正切乘積或相對兩要素余弦乘積積或相對兩要素余弦乘積 Napiers Rules are usually stated as follows:Sine middle part = product of tangents of adjacent partsSine middle part = product of cosines of oppo

16、site parts There is one final point to be noted, namely, that in writing down the equations by means of these rules, the two parts next to the right angle are written down as they are, the others (away from the right angle) are written down as complements.(2) Napiers Rules apply to quadrantal triang

17、les with one important modification, which must be noted: In a quadrantal triangle, if both adjacents or both opposites are both sides or both angles, put in a minus sign (i.e. put a minus sign in front of the product). In the triangle ABC, let AB be the quadrant. Then A and B are next to the quadra

18、nt, the other three parts are written down as complements. For example, given b and C, find A By Napiers Rules :ctgCtgAbcostgCbtgAcos1.5.3 Properties of Right angled and Quadrantal Triangles (1) In any right angled or quadrantal triangle, an angle and its opposite side are always of the same affecti

19、on. (“of the same affection”, i.e. both greater than 90, or both less than 90) This follows from the formula Since the sine is +ve for all values up to 180, it follows that ctg(a) and ctg(A) must be of the same sign, i.e. a and A must be of the same affection.ctgAbctgasinProperties -cont (2) In any

20、right angled triangle, we must have either, all three sides less than 90 or, two sides greater and one less than 90. This follows from the formula Since cos(c) must have the same sign as the product cos(a)cos(b). For this to happen we must have either all three consines +ve or only one +ve. baccosco

21、scos1.6 Small Spherical Triangle & Narrow Spherical Triangle A spherical triangle with three sides much smaller than the radius of the sphere is called . In this case we will treat this small spherical triangle as a plan triangle. A spherical triangle with one side much smaller than the radius o

22、f the sphere is called . In a narrow spherical triangle ABC with side a be smaller than the other two sides, known c, B and a, find (c-b) and A. Narrow Spherical TriangleBabccos)(1cBaAsinsin1ctgcBabcbc2212sin2)()(cctgcBaAAcos2sin2212Chapter 2 Interpolation 2.1 Introduction 2.2 Single Interpolation 2

23、.3 Double Interpolation 2.1 Introduction If one quantity varies with changing values of a second quantity, and the mathematical relationship of the two is known, a curve can be drawn to represent the values of one corresponding to various values of the other. yx2.1 Introduction -cont1 To find the va

24、lue of either quantity corresponding to a given value of the other, one finds that point on the curve defined by the given values and reads the answer on the scale relating to the other quantity. This assumes that for each value of one quantity, there is only one value of the other quantity.yx2.1 In

25、troduction -cont2 Information of this kind can also be tabulated. Each entry represents one point on the curve. The finding of a value between tabulated entries is called . 2.1 Introduction -cont3 Thus, the Nautical Almanac tables values of declination of the Sun for each hour of Greenwich Mean Time

26、. The finding of declination for a time between two whole hours requires interpolation. Since there is only one argument, is involved.2.1 Introduction -cont4 There is one table that gives the distance traveled in various times at certain speeds. In this table, there are two entering arguments. If bo

27、th given values are between tabulated values, is needed.2.1 Introduction -cont5 In Pub. 229, azimuth angle varies with a change in any of the three variables latitude, declination, and local hour angle. With intermediate values of all three, is needed.2.1 Introduction -cont6 Interpolation can someti

28、mes be avoided. Here are two kinds of method. Try to think of them? -2.1 Introduction -cont7 A table having a single entering argument can be arranged as . Interpolation is avoided through dividing the argument into intervals so chosen that successive intervals corresponds to successive values of th

29、e required quantity the respondent. For any value of the argument within these intervals, the respondent can be extracted from the table without interpolation.9.910.210.611.011.411.8-5.6-5.7-5.8-5.9-6.0Height DipIf the height of eye is between 10.2 and 10.6 meters, the dip will be 5.7.If the height

30、of eye is equal to 10.6 meters, the dip will be 5.7.Example of a critical table2.1 Introduction -cont8 The lower and upper limits (critical values) of the argument correspond to half-way values of the respondent and, by convention, are chosen so that when the argument is equal to one of the critical

31、 values, the respondent corresponding to the proceeding (upper) interval is to be used. 2.1 Introduction -cont9 Another way of avoiding interpolation would be .2.2 Single Interpolation 2.2.1 Proportional Interpolation 2.2.2 Interpolation by Rate of Change x2yxx12.2.1 Proportional Interpolation The a

32、ccurate determination of intermediate values requires knowledge of the nature of the change between tabulated values. The simplest relationship is linear, the change in the tabulated value being directly proportional to the change in the entering argument.2.2.1 Proportional Interpolation.1 Entries F

33、unctionx1y1 x2y2)(121211yyxxxxyyx y=?2.2.1 Proportional Interpolation.2 When the curve representing the values of a table is a straight line. The process of finding intermediate values in the manner described above is called . If tabulated values of such a line are exact (not approximations), the in

34、terpolation can be carried to any degree of precision without scarifying accuracy. 2.2.1 Proportional Interpolation.3 Many of the tables of navigation are not linear. To be strictly accurate in interpolating in such a table, one should consider the curvature of the line. 2.2.1 Proportional Interpola

35、tion.4 However, in most navigational tables the point on the curve selected for tabulation are sufficiently close that the portion of the curve between entries without introducing a significant error. This is similar to considering the line of position from a celestial observation as a part of the c

36、ircle of equal altitude.2.2.2 Interpolation by Rate of Change If we know the derivative of a function, the tangent line can be treated as the curve. Then where x is nearer to x0 than to any other entry.)(000 xxyyyxy=x3y=3x211328123272746448For example, given a table as follows, obtain 2.73. x0=3, y0

37、=27, y0=27 y=27+27*(2.7-3) =18.9The true value is 19.7 If we take x0=2, y0=8, y0=12 y=8+12*2.7-2) =16.4The error is bigger than that above.xy=x3y=3x2113281232727464482.3 Double Interpolation In a double-entry table, it may be necessary to interpolate for each entering argument. If one entering argum

38、ent is an exact tabulated value, the function can be found by single interpolation. However, if neither entering argument is a tabulated value, double interpolation is needed. Combined method: Select , preferably that nearest the given tabulated entering arguments. Next, find , with its sign, for si

39、ngle interpolation of this base value both horizontally and vertically. Finally, algebraically to the base value. For example, given an extract table from Amplitudes, latitude is 45.7 and declination is 21.8, find the amplitude. Lat.Declination21.5224531.232.04631.832.6 The base value is 32.6, for d

40、eclination 22 (21.8 is nearer 22 than 21.5) and latitude 46. The correction for declination is =-0.3. The correction for latitude is=-0.2. The interpolated value is then 32.6-0.3-0.2=32.1. )6 .328 .31(225 .21228 .21oooooo)6 .320 .32(4645467 .45ooooooLat.Declination21.5224531.232.04631.832.6Chapter 3

41、 Navigational Errors 3.1 Error of Observation 3.2 Random Error and its Criteria 3.3 Probability Distribution of Random Error 3.4 Arithmetic Mean and Least Square Method 3.5 Error Propagation 3.1 Error of Observation 3.1.1 Introduction 3.1.2 Definitions 3.1.1 Introduction As commonly practiced, navig

42、ation is not an exact science. A number of approximations, which would be unacceptable in careful scientific work, are used by the navigator, because greater accuracy may not be consistent with the requirements or time available, or because there is not alternative.3.1.1 Introduction cont1 Thus, whe

43、n the navigator interpolate in sight reduction or lattice tables, he is assumes a linear (constant rate) change between tabulated values. When he measures distance by radar, or depth by echo sounder, he assumes that the radio or sound wave has constant speed under all conditions.3.1.1 Introduction c

44、ont2 These are only a few of the approximations commonly applied by a navigator. There are so many that there is a natural tendency for some of them to cancel others. Thus, under favorable conditions, a position at sea, determined from celestial observation by an experienced observer, should seldom

45、be in error by more than 2 miles. However, if the various small errors in a particular observation all have the same sign, the error might be several times amount, without any mistake having been made by the navigator.3.1.1 Introduction cont3 Greater accuracy could be attained, but at a price. The n

46、avigator is a practical individual. In the course of ordinary navigation, he could rather spend 10 minutes determining a position having a probable error of plus or minus 2 miles, than to spend several hours learning where he was to an accuracy of a few meters. But if he can determine a recent or pr

47、esent position to greater accuracy, the decrease in error is attractive to him.3.1.1 Introduction cont4 An understanding of the kinds of errors involved in navigation, and of the elementary principles of probability, should be of assistance to a navigator in interpreting his results.3.1.2 Definition

48、s is the difference between a specific value and the correct or standard value. As used here, it does not include mistakes, but is related to lack of perfection. is a blunder, such as an incorrect reading of an instrument, the taking of a wrong value from a table, or the plotting of a reciprocal bea

49、ring.3.1.2 Definitions cont1 is something established by custom, agreement, or authority as a basis for comparison. Frequently, a standard is so chosen that it serves as a model which approximates a mean or average condition. However, the distinction between the standard value and the actual value a

50、t any time should not be forgotten.3.1.2 Definitions cont2 is the degree of conformance with the correct value, while is the degree of refinement of a value. Thus, an altitude determined by marine sextant might be stated to the nearest 0.1, and yet be accurate only to the nearest 1 if the horizon is

51、 indistinct.3.1.2 Definitions cont3 are those which follow some law by which they can be predicted. The accuracy with which a systematic error can be predicted depends upon the accuracy which the governing law is understood. An error which can be predicted can be eliminated, or compensation can be m

52、ade for it. The simplest form of systematic error is one of unchanging magnitude and sign. This is called . 3.2 Random Error and its Criteria are chance errors, unpredictable in magnitude or sign. They are governed by the laws of probability. If the altitude of a celestial body is observed, the read

53、ing may be (1) too great, (2) correct, or (3) too small. If a number of observations are made, and there is no systematic error, the probability of a positive error is exactly equal to the probability of a negative error. This does not mean that every second observation having an error will be too g

54、reat. However, the greater the number of observations, the greater is the probability that the percentage of positive errors will equal the percentage of negative ones, and that their magnitudes will correspond. Suppose that 500 observations are made, with the results shown in table 3.1. A close app

55、roximation of the plot of these errors is shown in figure 3.1. The plot has been modified slightly to constitute the normal curve of random errors, which is the same as the actual curve except that the normal curve approaches zero as the error increase, while the actual curve reaches zero at +10 and

56、 10. ErrorNo. of obs.Percent of obs.-1000.0-910.2-820.4-740.8-69-5173.4-4285.6 -3408.0-25310.6-16312.606613.216312.625310.63408.04285.65173.4691.8740.8820.4910.21000.0 500100.0Figure 3.1 The high of the curve at any point represents the percentage of observations that can be expected to have the err

57、or indicated at that point. The probability of any similar observation having any given error is the proportion of the number of observations having this error to the total number of observations, or the percentage expected as a decimal. Thus, the probability of an observation having an error of 3 i

58、s 40/500=8%. If the error under the curve represents 100 percent of the observations, half the area represents 50 percent of the observations. The value of the error at the limits is often called the “50 percent error”, or probable error, meaning that 50 percent of the observations can be expected t

59、o have less error, and 50 percent greater error. Similarly, the limits which contain the central 95 percent of the area denote the 95 percent error. The percentage of error is found mathematically. For a normal curve, each error is squared, the sum of the squares is divided by one less than the numb

60、er of observations, and the square root of the quotient is determined. This value is called the standard deviation or standard error (, the Greek letter sigma). In the illustration, the standard deviation is 99. 2499100.)8(2)9(1)10(02222 The standard deviation is 68.27% error. The probability of the occ