集美大学航海学2教案：天文航海 (2)

1、航海学 (2) 天文航海 内 容 球面三角与船位误差理论基础 测天定位 天测罗经差 天文航海的优点 使用的设备是六分仪，简单可靠； 不发射任何电波，隐蔽性好； 观测目标是天体，不受任何人控制； 可能是大洋航行中唯一的全球导航系统。 球面三角与船位误差理论基础球面三角与船位误差理论基础 第一章 球面三角 第二章 内插法 第二章 船位误差基础 第二章 等精度观测平差 第一章 球面三角 球面三角球面三角, ,主要研究球面上由三个大圆主要研究球面上由三个大圆 弧相交围成的球面三角形及其性质、解弧相交围成的球面三角形及其性质、解 算等问题算等问题 。 1.1 球面几何 1.2 球面三角形 1.3 球面三

2、角形的边角函数关系 1.1 球面几何 1.1.1 球、球面 在空间与一定点等距离的点的轨迹称为 球面球面。包围在球面中的实体称为球,这一 定点称为球心球心。球心和球面上任意一点 间的距离称为球半径球半径R R。过球心与球面相 交的直线段称为球直径球直径。 同球的半径和直径都相等。同理,半径 或直径相等的球全等。所以,球面又可 定义为半圆周绕它的直径旋转一周的旋 转面。 1.1.2 球面上的圆 任意一平面和球面相截的截痕是圆圆。 当平面通过球心时，所截成的圆称为大大 圆，圆，它的一段圆周叫大圆弧大圆弧。 截面不通过球心的圆称为小圆小圆，它的一 段圆周叫小圆弧小圆弧。 P P L Q L Q O

3、C r R 1.1.3 大圆的性质 1大圆的圆心与球心重合。 2大圆的直径等于球直径，半径等于球 半径。 3同球或等球上的大圆的大小相等。 4大圆等分球面和球体。 5同球上的两个大圆平面一定相交，交 线是它们的直径，并且两大圆互相平分。 6过球面上不在同一直径两端上的两个 点，能作且仅能作一个大圆，却能作无 数个小圆。若在同一直径两端上的两个 点，则能作无数个大圆而不能作小圆。 7小于180的大圆弧（劣弧）是球面上 两点间的最短球面距离。因此，两点间 的球面距离应用大圆弧度量。 1.1.4 轴、极、极距、极线 垂直与任意圆面的球直径称为该圆（大 圆或小圆）的轴轴。轴的两个端点称为极极。 垂直于

4、同一轴可有数个平行圆，其中只 有一个通过球心的是大圆，其余的都是 小圆。从极到圆(大圆或小圆)弧上任一 点沿大圆弧的球面距离叫极距极距,又叫球球 面半径面半径。同一个圆的极距或球面半径都 相等。极距为90的大圆弧又称为该极 的极线极线。 大圆弧是它的极的极线。反之，极线必 定是大圆弧。 1.1.5 球面角及其度量 球面上两大圆弧相交构成的角称为球面球面 角角，其交点叫球面角的顶点，两大圆弧 称为球面角的边。 球面角的三种度量方法： 1切于顶点大圆弧的切线夹角CPD； 2顶点的极线被其两边大圆弧所截的弧 长AB； 3大圆弧AB所对的球心角AOB。 1.2 球面三角形球面三角形 1.2.1 定义定

5、义 在球面上由三个大圆弧围成的三角形称 为球面三角形球面三角形。围成三角形的大圆弧称 为球面三角形的边边。由大圆弧相交所成 的球面角称为球面三角形的角角。 三个角A、B、C和三个边a、b、c合称 为球面三角形的六要素。航海上讨论的 球面三角形的六要素均大于0，而小 于180，又称其为欧拉球面三角形。 1.2.2 球面三角形分类球面三角形分类 球面三角形分为直角、直边、等腰、 等边、初等和任意三角形。 1球面直角三角形和球面直边三角形 至少有一个角为90的三角形称为球 面直角三角形。 至少有一个边为90的三角形称为球 面直边三角形。 2球面等腰三角形和球面等边三角形 有两边或两角相等的三角形称为

6、球面等腰三 角形。若三边或三角都相等的三角形称为球 面等边三角形。 3球面初等三角形 三个边相对其球半径甚小的三角形称为球 面小三角形。只有一个角及其对边相对球半 径甚小的三角形称为球面窄三角形。两者统 称为球面初等三角形。 4球面任意三角形 不具备上述特殊条件的球面三角形 1.2.3 极线球面三角形极线球面三角形 球面三角形ABC三个顶点的极线所构成 的球面三角形ABC称为原球面三角形 ABC的极线球面三角形。 1原三角形与其极线三角形是互为极线三角形。 原球面三角形ABC的三个边，也就是其极线 球面三角形AB C三个顶点的极线。换句话 说，若画极线三角形ABC的极线三角形，则 所得到的就是

7、原球面三角形ABC。所以，它 们之间的关系是相互的。 2原球面三角形的边与其极线三角形对应角互 补 1.3 球面三角形的边角函数关系球面三角形的边角函数关系 1.3.1 任意球面三角形任意球面三角形 1.3.2 球面直角和直边三角形球面直角和直边三角形 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 边的余弦公式 记忆口诀：一边的余弦等于其它两边余弦一边的余弦等于其它两边余弦 的乘积，加上这两边正弦及其夹角余弦的乘积，加上这两边正弦及其夹角余弦 的乘积。的乘积。 )cos()sin()sin()cos()cos()cos(Acbcba 边的余弦公式应用：已知两边及其

8、夹角求 对边；已知三边求三角。 By transposing the formula, we get a form in which it may be used, given three sides, to find any angle. )sin()sin( )cos()cos()cos( )cos( cb cba A 角的余弦公式 记忆口诀：一角的余弦等于其它两角余弦一角的余弦等于其它两角余弦 的乘积冠以负号加上这两角正弦及其夹的乘积冠以负号加上这两角正弦及其夹 边余弦的乘积。边余弦的乘积。 cosAcosA-cosBcosC-cosBcosCsinBsinCcosasinBsinCco

9、sa 角的余弦公式应用：已知两角及其夹边求 对角；已知三角求三边。 1.3.1.2 正弦公式正弦公式 记忆口诀：边的正弦与其对角的记忆口诀：边的正弦与其对角的 正弦成比例。正弦成比例。 )sin( )sin( )sin( )sin( )sin( )sin( C c B b A a 正弦公式应用：已知两角及其一对边，求 另一边；已知两边及其一对角，求另一 角。 Its big disadvantage is the ambiguity about the actual value of the part found, since )180sin()sin( 0 AA In short, when

10、 we have taken out the anti- sine and obtained (say) 42, the question arises-is the answer 42 or 138? This difficulty 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

11、possibility, some progress can be made on the first two by remembering that, in any triangle, A-B and a-b must be of the 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 requiremen

12、t; thus both are solutions of the data as given. )sin( )sin( )sin( )sin( B b A a 1.3.1.3 余切公式（四联公式）余切公式（四联公式） 记忆口诀：外边余切内边正弦乘积等于外 角余切内角正弦乘积加上内边内角余弦积。 CbCctgAbctgacoscossinsin A C B a b c 余切公式用于：在球面三角形中，已知相 连的四个要素中的三个，求另一个要素。 即已知两边及夹角求相连的角或已知两 角及夹边求相连的边。 1.3.1.4 解球面任意三角形 根据球面三角形已知要素求解其余要素 的方法称为解球面三角形。

13、 已 知求应用公式说 明 两边夹角 a、b、C 第三边及其它两角 c、A、B 边的余弦公式求c 四联公式求A、B 有 一 确 定 解 两角夹边 A、B、c 第三角及其它两边 C、a、b 角的余弦公式求C； 四联公式求a、b 三边a、b、c 三角A、B、C边的余弦公式 三角A、B、C 三边a、b、c角的余弦公式 两边及其一边 对角 a、b、A 求另一角和其它两 边 c、B、C 正弦公式求B; 四联公式 一解 两解 或 无解 两角及其一角 对边 A、B、a 求另一角和其它两 边 C、b、c 正弦公式求b; 四联公式 使用计算器解算球面三角时注意事项 计算器有三种角度单位供选择即： 度DEG、弧度R

14、AD、公制度GRAD。 球面三角中，角或边都是以六十进制的 度、分及分的小数给出的，利用计算器 计算，在DEG状态下，一定要将角或边 转换成以十进制的度为单位输入。 Examples, in a triangle PAB, P=6630, a=4700, b=6700, find A. We shall use the four-parts formula. A=5232.6 pbpctgAbctgacoscossinsin 1.3.2 球面直角和直边三角形球面直角和直边三角形 球面直角三角形公式 有一个或一个以上的角为直角的球面三 角形称为球面直角三角形。 设球面直角三角形ABC中，C90。

15、 因为sin901，cos900，可由球 面任意三角形的基本公式，导出相应的 球面直角三角形十个公式： sinasinAsinc coscctgActgB sinactgBtgb cosAsinBcosa sinbsinBsinc cosActgctgb sinbctgAtga cosBsinAcosb cosccosacosb cosBctgctga 球面直角三角形公式的纳比尔记忆法则 在球面三角形ABC中，C90。先画“大” 字图形，大字上部竖线代表直角C，相 邻两侧为夹直角的两边a和b，大字下面 三个空格依次填入相对应元素边或角的 余数（a边对90A，b边对90B， C角对90c)。 任

16、一要素的正弦等于相邻两要素正切乘任一要素的正弦等于相邻两要素正切乘 积或相对两要素余弦乘积积或相对两要素余弦乘积 Napiers Rules are usually stated as follows: Sine middle part = product of tangents of adjacent parts Sine middle part = product of cosines of opposite parts There is one final point to be noted, namely, that in writing down the equations by m

17、eans 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 triangles with one important modification, which must be noted: In a quadrantal triangle, if both adj

18、acents 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 quadrant, the other three parts are written down as complements. For example, given b and C, find A B

19、y Napiers Rules : ctgCtgAbcos tgCbtgAcos 1.5.3 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 affection. (“of the same affection”, i.e. both greater than 90, or both less than 90) This follows

20、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. ctgAbctgasin Properties -cont (2) In any right angled triangle, we must have either, all three sides less than 90 or, two sides gre

21、ater 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. baccoscoscos 1.6 Small Spherical Triangle & Narrow Spherical Triangle A spherical triangle with th

22、ree 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 of the sphere is called . In a narrow spherical triangle ABC with side a be smaller than the o

23、ther two sides, known c, B and a, find (c-b) and A. Narrow Spherical Triangle Babccos)( 1 c Ba A sin sin 1 ctgcB a bcbc 2 2 12 sin 2 )()( cctgcB a AAcos2sin 2 2 12 Chapter 2 Interpolation 2.1 Introduction 2.2 Single Interpolation 2.3 Double Interpolation 2.1 Introduction If one quantity varies with

24、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. y x 2.1 Introduction -cont1 To find the value of either quantity corresponding to a given value of the other,

25、 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. y x 2.1 Introduction -cont2 Information of this kind can also be tabulated

26、. 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. The finding of declination for a time between two whole hours

27、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 both given values are between tabulated values, is needed. 2.1 In

28、troduction -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 sometimes be avoided. Here are two kinds of method. Try to think of

29、 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 the required quantity the respondent. For any value of the arg

30、ument within these intervals, the respondent can be extracted from the table without interpolation. 9.9 10.2 10.6 11.0 11.4 11.8 -5.6 -5.7 -5.8 -5.9 -6.0 Height Dip If the height of eye is between 10.2 and 10.6 meters, the dip will be 5.7. If the height of eye is equal to 10.6 meters, the dip will b

31、e 5.7. Example of a critical table 2.1 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 values, the respondent corresponding to the

32、 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 x2 y xx1 2.2.1 Proportional Interpolation The accurate determination of intermediate va

33、lues 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 Function x1y1 x2y2 )( 12 12 1 1 yy xx xx

34、 yy x 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 interpolation can be carried

35、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 Interpolation.4 However, in most nav

36、igational 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 circle of equal altitude. 2.

37、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. )( 0 00 xxyyy xy=x3y=3x2 113 2812 32727 46448 For example, given a table as follows, obtain 2.73. x0=3, y0=27, y0=27 y=27+2

38、7*(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=x3 y=3x2 113 2812 32727 46448 2.3 Double Interpolation In a double-entry table, it may be necessary to interpolate for each entering argument. If one entering argument is an

39、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 single inter

40、polation 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.Declination 21.522 4531.232.0 4631.832.6 The base value is 32.6, for declinat

41、ion 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 .21 228 .21o oo oo o )6 .320 .32( 4645 467 .45 o oo oo o Lat.Declination 21.522 4531.232.0 4631.832.

42、6 Chapter 3 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 pract

43、iced, navigation 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 con

44、t1 Thus, when 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 I

45、ntroduction cont2 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, s

46、hould seldom 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

47、a price. The navigator 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

48、a recent or present 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.

49、3.1.2 Definitions 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

50、a reciprocal bearing. 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 t

51、he actual value at 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

52、 if the horizon is 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 co

53、mpensation can be made 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

54、 observed, the reading 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

55、error will be too great. 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 tab

56、le 3.1. A close approximation 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 reac

57、hes zero at +10 and 10. ErrorNo. of obs.Percent of obs. -1000.0 -910.2 -820.4 -740.8 -69 -5173.4 -4285.6 -3408.0 -25310.6 -16312.6 06613.2 16312.6 25310.6 3408.0 4285.6 5173.4 691.8 74 0.8 820.4 910.2 1000.0 500100.0 Figure 3.1 The high of the curve at any point represents the percentage of observat

58、ions that can be expected to have the error 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

59、 of an observation having an error of 3 is 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 perc

60、ent of the observations can be expected to 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 squ