If you went looking for π, welcome back. In an effort to be specific about concepts and to communicate clearly using unambiguous symbols, the following conventions are identified and used throughout the topic as consistently as possible even though they may differ from one discipline to the next or from one culture to another.
Numbers
Whole numbers are integers, and numerical values between integers are called real numbers. A line such as that shown in Figure 3.1 can be used to represent all numbers—both integer and real. If some point on the line is assigned the value “zero,” points to the left of zero are negative numbers, and points to the right of zero are positive numbers. Numbers are composed of the Arabic digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, and the Hindu-Arabic number system used in modern mathematics is decimally constructed by decades where the column of a digit implies its value times 1, 10, 100, 1,000, and so on.
Fractions
A ratio of one integer divided by any other (except 0) is known as a fraction. Many examples exist, but successive division by 2 is an intuitive example that serves very well when cutting a pie or a pizza. Fractions of 1/2, 1/4, 1/8, and so on are familiar to everyone and are used, for example, when driving a car and judging the amount of fuel remaining in the tank. Successive division by 2 is also appropriate in other cases, but when carried too far it becomes cumbersome and using decimal equivalents is easier. In the case of the U.S. Public Land Survey System (USPLSS), it is no coincidence that 1 square mile nominally contains 640 acres. Successive division of area by quarters (two divisions of length by two) is convenient down to a parcel of 10 acres. Although more could be said about repetitive division, it is noted that some disciplines in the United States still use fractions (e.g., architects, carpenters, millwrights, and ironworkers).
Decimal
Another prevalent practice for counting objects utilizes decades of 10, presumably based upon prehistoric humans having ten fingers. With the invention of “zero” by mathematicians in India about 600 a.d., the decimal system was developed in its present form, borrowed by the Arabs about 700 a.d., and subsequently adopted by
European merchants.
FIGURE 3.1 The Real Number Line
The decimal form conveniently handles numbers (both positive and negative, and both integer and real) of any size from the very large to the very small, and is used worldwide.Spatial data users are probably more interested in the development of the meter as a decimal standard of length. The goal of the French Academy of Science in the
1790s was to devise a standard of length that could be duplicated in nature and that was decimally divided. The arc distance from the Earth’s equator to the North Pole was determined as accurately as possible by means of a geodetic survey, and the result—5,130,766 toises—was defined to be exactly 10 million meters (Smith 1986;
Alder 2002).
The decimal system is used in the International System of units (SI) adopted by the Eleventh General Conference on Weights and Measures in 1960 (Chen 1995,2455). The SI system is a coherent system of units included in, or derived from, the seven independent SI base units of the meter, kilogram, second, ampere, degrees Kelvin, mole, and candela. One advantage of the decimal system is names for units that differ by a magnitude of 1,000, as shown in Table 3.1. Perhaps the names are best recognized when referring to computer speeds (megahertz), data storage (gigabytes),or very short periods of time (nanoseconds). While the SI system defines decimal subdivision for length (meters), time (seconds), and angles (radian), standard practice in many parts of the world still uses the sexagesimal division of angles
(degrees, minutes, and seconds) and time (hours, minutes, and seconds).
Sexagesimal System
About 5,000 years ago, the Babylonians related 360° in a circle to 365 days in a year.
Given that six circles will fit exactly around a seventh, a subdivision of 360° into six sectors of 60° each is plausible. Tooley ([1949] 1990) credits the Babylonians with subdividing both the sky into degrees and the day into hours. The sexagesimal system of minutes and seconds was applied to each, allowing stars of the night sky to be plotted in a consistent proportional manner. Wilford (1981) credits Ptolemy with subdividing the degree into 60 minutes and each minute into 60 seconds, while Smith (1986) credits the Chinese with first using zero and a circle sexagesimally divided into degrees, minutes, and seconds. Regardless of the origin of the practice, the second is now the defining unit of time, and the sexagesimal system (60 seconds = 1 minute, 60 minutes = 1 hour, and 24 hours = 1 day) is used worldwide. The sexagesimal system is also widely used in angular measurement, but the radian (defined as an angle whose arc length equals its radius) is the defining unit for rotation in the SI system.
|
1,000,000,000,000. |
tera- |
one trillion |
|
1,000,000,000. |
giga- |
one billion |
|
1,000,000. |
mega- |
one million |
|
1,000. |
kilo- |
one thousand |
|
1. |
unit |
one |
|
0.001 |
milli- |
one thousandth |
|
0.000 001 |
micro- |
one millionth |
|
0.000 000 001 |
nano- |
one billionth |
|
0.000 000 000 001 |
pico- |
one trillionth |
Binary System
Computer science professionals use combinations of zeros and ones (0’s and 1’s) to represent numbers, letters, and other symbols within a computer’s memory. A string of eight bits (0’s and 1’s) is called a byte and can be used to represent up to 256 different items. The American Standard Code for Information Interchange (ASCII) uses combinations of seven of the eight bits to represent uppercase and lowercase letters of the English alphabet, digits 0-9, and other symbols as text characters. If ASCII characters were used to represent real numbers in a computer, it would be quite costly in terms of memory requirements. Therefore, numeric data are stored using a base 2 (binary) system that accommodates real numbers and integers, both positive and negative. Rather than exploring the details of the binary system, the point here is to recognize that numerical values, whether integer or real, are coded differently than text. In most cases, the user need not be concerned with computer binary operations because numeric input in decimal form is immediately converted to binary, and numeric output, unless specified otherwise by the user, is displayed or printed in conventional decimal format. It is interesting to note, however, that there are similarities between fractions (dividing by two) and binary arithmetic. For an interesting tongue-in-cheek discussion of the advantages of binary computations for survey measurements, see Stanfel (1994).
Conversions
The ability to use numbers is important, but the real meaning of any mathematical operation comes from knowing what physical quantities (i.e., units) are associated with the numbers. And, it is important to understand the use of ratios where the units cancel out and the relative size of the number is really the issue. The value of π is one example; trigonometric ratios comprise another. The solution of most problems involves a reasonable number of something (units) that can be understood. Conversion is the process whereby equivalency is established between seemingly unrelated physical quantities. Mathematical operations also apply to the units. An easy example might be area (m2) = length (m) x width (m). Not as obvious, but not really that unusual, the volume of concrete in a hypothetical sidewalk is length (30 meters) x width (5 feet) x thickness (4 inches). Unit conversions are “exact” and are used as ratios to find the desired answer. Note how units cancel in the separate computations and make it easy to find the volume of concrete in either cubic yards or cubic meters.
Coordinate Systems
An origin and three mutually perpendicular axes that intersect at the origin define a generic Cartesian coordinate system, as shown in Figure 3.2. When studying 2-D phenomena, the X-axis and Y-axis are the ones most commonly used. In a threedimensional system, the direction of the positive Z-axis for a right-handed coordinate system is given by the direction of the thumb on one’s right hand when rotating the positive X-axis (one’s index finger) toward the Y-axis (one’s palm). An example of a left-handed coordinate system is given as north/east/up. The convention in this topic is to use a right-handed rectangular Cartesian coordinate system whenever possible. That includes both X/Y/Z and east/north/up. Note that while labeling the axes of a coordinate system is really the prerogative of the user, the GSDM uses X/Y/Z for geocentric ECEF coordinates and uses e/n/u for local perspective coordinates.
FIGURE 3.2 Three-Dimensional Rectangular Coordinate System
Significant Figures
The rules of significant digits are not arbitrary, but are based upon the principles of error propagation. If the validity of an answer obtained using significant figures is in doubt, the uncertainty of any answer can be verified using the standard deviations of measured quantities and error propagation computations.
• Integer values may have an infinite number of significant figures. For example, when dividing the area of a rectangle in half, 2 is an exact number. But, physical operations are not so precise. If one of two siblings cuts a piece of candy in half, the astute parent permits the second sibling to have first choice of the two pieces (in case one piece is larger).
• Conversions that are exact (12” = 1′ or 27 ft3 = 1 yd3) contain an infinite number of significant figures when used as a ratio.
• With regard to controlling round-off error, it is recommended to carry at least one more digit in the computations than can be justified by the original data. The final answer of any computation should reflect the user’s judgment with respect to significant figures.
Addition and Subtraction
The column (decade) of the least accurate number in a sum (addition) or in a difference (subtraction) determines the number of significant digits in the answer. A zero listed after the decimal point is usually counted as significant. But, zeros that serve only to position the decimal point are generally not significant. Table 3.2 shows examples for addition and subtraction. For an exception, see the area computation in Figure 3.3 where 3,000 ft2 has four significant figures, not one. In such cases, some authors place a bar over the last significant digit.
Multiplication and Division
The number of significant digits in the product or quotient is determined by the term used in the operation containing the least number of significant digits. A product or quotient does not contain more significant digits than either term used to compute it. A simple example is area computed as length (L = 94.87’) x width (W = 31.62’), which gives area = 2,999.7894 ft2. An appropriate answer to four significant digits is 3,000 ft 2.
TABLE 3.2
Significant Figure Examples Using Addition and Subtraction
|
addition |
addition |
subtraction |
subtraction |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
FIGURE 3.3 Rectangle for Area Computation
Area for the same rectangle (Figure 3.3) can be computed using the area-by-coordinates (cross multiplication) equation and the listed state plane coordinates of its corners. A common form of the area-by-coordinates equation is
Area is one-half the difference of the two sums:
What happened? The area equation is correct. The coordinates are good. A ten-digit calculator was used to compute the answer. But, the answer is obviously in error by 500 ft2. The problem is one of significant figures. Two issues (innocent mistakes) are as follows:
• The Y coordinate contains only eight significant figures, yet each XY product lists ten significant figures. This is mistake number one and invalidates the computation.
• But, a second mistake (and separate issue) is that significant figures are lost in taking the difference of two large numbers of similar magnitudes. In this case, only one significant figure remains after computing the difference of the two sums of products.
Both mistakes can be avoided by working with coordinate differences. By finding a coordinate difference first, the problem of working with large coordinate values is avoided because the cross products are formed using much smaller numbers.Another advantage of equation 3.2 is that it can be extended easily for any number of points. A minimum of three points is required in the first line of equation 3.2. Beyond that, point 4 is brought into line 2, point 5 is brought into line 3, and so on. Note that with the orderly progression of subscripts, a program need only be written for the first line and used in a loop with updated subscripts until all points around the figure are used. Points used must be in graphical sequence to form a closed figure, and no line crossovers are permitted.
