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2 Geometry and Trigonometry
There is no Royal Road to geometry. Euclid to King Ptolemy Geometry and trigonometry are Greek words for measurement of the earth and measurement of triangles respectively. The beginnings of geometry can be traced back four millennia to ancient Mesopotamia and Egypt. Euclid put the subject on a logical basis in about 300 BC in his Elements, one of the most successful textbooks ever written. He relied on constructive proofs using only an unmarked ruler (straightedge) and a compass. The subject was transformed in the seventeenth century by the introduction of Cartesian coordinates (the x-y plane in Fig. 1.4 on p. 12) and the development of analytic geometry in which shapes can be represented in a numerical or algebraic way; for example, the circumference of a circle of unit radius centered at the origin consists of the points {x, y} such that x
2 + y2 = 1. Trigono-metry may be regarded as a branch of geometry in which the emphasis is on the properties of angles.
Plane geometry is about shapes on a flat surface. Spherical geometry and trigono-metry study these subjects on the surface of a sphere; they are important because the earth is almost spherical and they have applications in astronomy, geodesy and navigation. Much of Euclidean geometry does not hold on the surface of a sphere (for example the angles of a triangle sum to more than 180°) but it is a good approximation locally.
Euclidean Geometry
Euclid’s Elements is a textbook gathering together existing knowledge about geometry and number theory in a highly logical way. Euclid first considered plane geometry on a flat, two-dimensional space and he began Book 1 with 23 definitions, five postulates and five common notions before deriving a large number of propositions from them using only straightedge and compass for drawing lines and circles (or arcs of circles) respectively.
Parallel and perpendicular lines
Euclid’s postulates and common notions are axioms taken to be self-evident. The fifth postulate is the only axiom that is not completely self-evident. It states that “if a straight line falling on two straight lines make the sum of the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side.” Its meaning is that, when α + β < 180° in Fig. 2.1 the lines meet eventually on the right and when α + β > 180° they meet on the left. Only when α + β = 180° are they parallel and never meet.
Chapter 2
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Figure 2.1. Euclid’s fifth postulate.
Figure 2.2 shows a pair of parallel lines and a transversal line crossing them. β and δ
are called vertical angles. They are equal because α + β =α +δ = 180! whence β = δ . Parallelism is irrelevant. β and η are called corresponding angles. They are equal because of parallelism by Euclid’s fifth postulate because γ +η =180!. δ and η are called alternate angles. They are equal from the two previous results.
Figure 2.2. Vertical, corresponding and alternate angles.
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Geometry and Trigonometry
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One may need to construct by straightedge and compass a line parallel to a given line passing through a point not on that line. The following is one solution (Fig. 2.3). Given a line AB and a point C not on the line, draw a line through C to intersect AB at D. With the compass width set less than the length of CD, place its point on D and draw an arc between DC and DB, intersecting these lines at E and F respectively. With the point on C draw a similar arc from the extended line CD intersecting the latter at G. Set the width of the compass to the distance between E and F, move the point of the compass to G and draw a third arc intersecting the second arc at H. Draw a line through C and H; it will be parallel to AB. This works for the following reason. Draw the lines between E and F and between G and H. Then DE = CG, DF = CH and EF = GH. Thus the triangles EDF and GCH are congruent since they have three equal equivalent sides, so that ∠EDF = ∠GCH. (Congruent triangles are discussed in the next subsection; ∠ means an angle.) These are corresponding angles (see Fig. 2.2) that are only equal when the two lines are parallel by Euclid’s fifth postulate.
Figure 2.3. Construction of a line through C parallel to AB.
Euclid defined a right angle as follows: “When a straight line set up on a straight line
makes the adjacent angles equal to one another, each of the angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.” (See Fig. 2.4.)
Figure 2.4. Euclid’s definition of a right angle.
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Chapter 2
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He made the right angle the unit for measuring angles, for example he wrote of half a right angle. In modern terminology, 1 right angle = 90°.
A method for constructing a perpendicular with straightedge and compass to a line from a point C not lying on it is shown in Fig. 2.5. With the compass width set to more than the distance from C to the line, place its point on C and draw an arc on the line either side of C cutting the line at A and B. From each of these points draw an arc the other side of the line below C, intersecting each other at D. Draw a straight line through C and D. It will be the required perpendicular for reasons similar to the justification of the method for drawing a parallel line in Fig. 2.3. Similar constructions can be found to draw the perpendicular bisector of a line AB, to draw a perpendicular line from a point C on a given line or to bisect an angle.
Figure 2.5. Construction of a perpendicular from C to a line.
Triangles
Euclid defined a triangle as a figure contained by three straight lines, an equilateral triangle as having three sides equal, an isosceles triangle as having two of its sides alone equal, and a scalene triangle as having its three sides unequal.
To show that the sum of the angles in a triangle is 180°, consider the triangle ABC in Fig. 2.6. Draw the line CE parallel to AB (see Fig. 2.3). Then ∠BAC = ∠ACE (alternate angles) and ∠ABC = ∠DCE (corresponding angles, see Fig. 2.2). Hence
∠BAC+∠ABC+∠ACB = ∠ACE +∠DCE +∠ACB = 180!.
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Figure 2.6. The angles in a triangle sum to 180°.
Two triangles are congruent if they have the same shape and size. The second triangle
is congruent to the first if it can be superimposed upon it after translation (vertical or horizontal movement), rotation or reflection (forming a mirror image); these operations are called rigid motions that preserve the distances between points In other words, when the second triangle is cut out from the paper on which it is drawn, it is congruent to the first triangle if it can be superimposed on the first triangle, if necessary after turning it over. Two triangles are similar if they have the same shape but not necessarily the same size; in other words, they would be congruent if one of them were enlarged. The triangle EDF in Fig. 2.7 is similar to ABC above because it can be superimposed on it after rotation to the right, reflection, translation upwards and to the left, and slight enlargement.
Figure 2.7. Triangle similar to ABC above.
Equivalent angles and sides in congruent triangles are equal. Equivalent angles in
similar triangles are equal, while equivalent sides are proportional. For example, equivalent sides in ABC are about 5.6% longer than those in EDF. The equal angles are ∠ACB = ∠DFE , ∠BAC = ∠DEF and ∠ABC = ∠EDF. (The word “equivalent” has been used instead of the more usual word “corresponding” because the latter has been used in a different context in Fig. 2.2. In two triangles with the same shape, equivalent sides are sides opposite equal angles.)
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Chapter 2
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Three sufficient conditions for two triangles to be congruent are the following. 1. Three sides equal. If the lengths of all three sides of a triangle are known, imagine
that you are given three knitting needles with these lengths. There is only one way of making a triangle with them (rigid motions aside) with their tips meeting. Hence all triangles with sides of these given lengths are congruent.
2. Two sides and the included angle equal. Take two knitting needles with the given lengths and put their tips together with the given angle between them. This deter-mines the position and length of the third side so that all such triangles are congruent.
3. Two angles and an equivalent side equal. If a pair of triangles has two equal angles the third angle must also be equal since they sum to 180°. Thus they have the same shape and will also have the same size if a pair of equivalent sides is equal. The comparable conditions for two triangles to be similar are the following.
1. Three sides proportional. If three sides of two triangles are proportional, then the triangles are similar. This follows from the first condition for congruent triangles.
2. Two sides proportional and the included angle equal. If two sides of a pair of triangles are proportional and their included angles are equal, then the triangles are similar. This follows from the second condition for congruent triangles.
3. Two angles equal. If a pair of triangles has two equal angles the third angle must also be equal and they are the same shape. These naïve proofs are not Euclidean. In Book 6 of the Elements Euclid defined
similar triangles as “such as have their angles severally equal and the sides about the equal angles proportional” and went on to prove rigorously the above conditions, together with a fourth condition not considered here.
The most famous theorem about triangles is Pythagoras’ theorem that in a right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides. Its first proof is attributed to the Greek mathematician Pythagoras in the sixth century BC, though it was known before then. More than one hundred proofs are known, the simplest of which uses similar triangles.
Figure 2.8. Pythagoras’ Theorem
In the right-angled triangle ABC in Fig. 2.8 draw the perpendicular from B inter-
secting AC at D (Fig. 2.5). The triangles ABC and ADB are similar since they have a right angle and the angle at A in common. Hence
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Geometry and Trigonometry
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ABAC
= ADAB
so that AB2 = AC ⋅AD.
The triangles ABC and BDC are similar since they have a right angle and the angle at C in common. Hence
BCAC
= DCBC
so that BC2 = AC ⋅DC.
Summing the two equations on the right hand side gives
AB2 + BC2 = AC(AD+ DC) = AC2 which is Pythagoras’ theorem. Euclid proved Pythagoras’ theorem in Book 1 of the Elements by a rather complicated construction. After considering the properties of similar triangles in Book 6 he showed that the three triangles in Fig. 2.8 are similar but he did not proceed to the derivation of Pythagoras’ theorem from that fact.
Trigonometry
All right triangles with one other angle fixed are similar so that the ratios of the sides are determined by that angle. Trigonometry arose from the study of how these ratios depend on that angle. It was at first limited to acute angles since any other angle in a right triangle must be acute but was later extended to arbitrary angles to allow its application to general periodic phenomena. It has a wide range of uses in surveying, astronomy, physics, engineering and biology.
Right-angled triangle definition of trigonometric functions
Figure 2.9. A right-angled triangle with acute angle θ at ∠ACB.
Fig. 2.9 shows a right-angled triangle with acute angle θ at ∠ACB. The standard trigonometric functions are
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Chapter 2
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sinθ = OppositeHypotenuse
= ABAC
cosθ = AdjacentHypotenuse
= BCAC
tanθ = OppositeAdjacent
= ABBC
Note that tanθ = sinθ /cosθ. Table 2.1 shows a few values of the trigonometric functions. Values for 0° and 90°
are obvious. Values for 45° can be found from Fig. 1.3 (p. 9) that shows an isosceles right triangle in which both the acute angles are 45°. Values for 30° and 60° can be found from Fig. 2.10 that shows a right triangle with those angles that has been reflected about BC to form an equilateral triangle ACD. If AB = 1 then AC = AD = 2 and BC = 3 by Pythagoras’ theorem. Table 2.1. Values of the trigonometric functions. θ 0° 30° 45° 60° 90°
sinθ 0 1/2 1/ 2 3/2 1
cosθ 1 3/2 1/ 2 1/2 0
tanθ 0 1/ 3 1 3 ∞
Figure 2.10. Trigonometric functions for 30° and 60°.
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Geometry and Trigonometry
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A few trigonometric identities will now be considered. The Pythagorean identity
follows from Pythagoras’ theorem (Fig. 2.9):
sin2θ + cos2θ = 1.
The angle sum identities are:
sin(α + β ) = sinα cosβ + cosα sinβcos(α + β ) = cosα cosβ − sinα sinβ.
In Fig. 2.11 the horizontal line AF was drawn with a second line AD at an angle α to it and a third line AB at an angle β to the second line. A perpendicular was dropped from a point B on the third line to intersect the first line at C and another perpendicular to intersect the second line at D. DE was drawn parallel to CF and DF parallel to CE. AFD, DEG and BED are similar triangles because they are all right-angled and because ∠FAD is alternate to ∠EDG which is 90° − ∠EDB.
Figure 2.11. The angle sum identities.
Hence
sin(α + β ) = BC
AB= DF
AB+ BE
AB= DF
ADADAB
+ BEBD
BDAB
= sinα cosβ + cosα sinβ
cos(α + β ) = AC
AB= AF
AB+ CF
AB= AF
ADADAB
− EDBD
BDAB
= cosα cosβ − sinα sinβ.
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Chapter 2
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The double angle identities are special cases of the angle sum identities when the two angles are equal. The identity for cos(2θ) can be put into two alternative forms by using the Pythagorean identity
cos(2θ ) = cos2θ − sin2θ = 2cos2θ −1= 1− 2sin2θ which can be used in turn to obtain the half angle identities:
cos
θ2= 1+ cosθ
2, sin
θ2= 1− cosθ
2.
The half angle and angle sum identities can be used to extend the values of the trigonometric functions in Table 2.1. Values for 15° can be found from the former and can be used to find values for 75° from the latter; values for 7.5° can then be found from the former and used to extend the table at intervals of 7.5° from the latter; this procedure can be repeated until enough values have been calculated that interpolation will suffice. A more efficient method is to use series expansions (see pp. 66−68). Graphs of the trigonometric functions for acute angles are shown in Fig. 2.12.
Figure 2.12. The trigonometric functions for acute angles.
Circular definition of trigonometric functions
The above definitions of the trigonometric functions based on right angled triangles only work for acute angles (between 0° and 90°). To extend them to obtuse angles (between 90° and 180°) and then to more general angles consider Fig. 2.13. A Cartesian coordinate system has been drawn with the x-axis intersecting the y-axis at the origin {0, 0}. A circle has been drawn with unit radius and with its center at the origin. The point on the circle has coordinates {x, y} = (cos 30°, sin 30°} = { 3/2 , 1/2} = {0.866, 0.5)}and the line from the origin to this point makes an angle of θ =30° with the x-axis on the understanding that angles are measured counterclockwise from the x-axis. (The curly braces in {x, y} denote an ordered pair of numbers; see p. 29.)
This analysis allows the trigonometric functions to be extended to any angles, large or small, positive or negative. Suppose that the point rotates counterclockwise around the circle to meet the small dashed line at 150° where {x, y} = {−0.866, 0.5} so that cos 150° = −0.866, sin 150° = 0.5. Rotating the point by another 60° to 210° in the third quadrant gives cos 210° = −0.866, sin 210° = −0.5, rotating further to 330° in the fourth quadrant gives cos 330° = 0.866, sin 330° = −0.5, and a further rotation of 60° to 390° brings the point back to its original position at 30°. Negative angles correspond to clockwise rotation so that −30° is equivalent to +330°.
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Geometry and Trigonometry
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Figure 2.13. Circular definitions in Cartesian coordinates.
Fig. 2.13 suggests another way of measuring angles in which the size of an angle is
the length of the arc of the unit circle that it subtends. This is called radian measure. The constant π = 3.14259 ! is defined as the ratio of the circumference of a circle to its diameter so that the circumference of the unit circle is 2π radians. A right angle is π/2 radians and the angle θ in Fig. 2.13 is π/6 radians. Degrees are often used in practical applications such as surveying but radians are more often used in mathematics because of the simpler formulae to which they lead. For example, the theorem that (sin θ)/θ tends to 1 as θ tends to zero, which is important in calculus, is only true when θ is measured in radians. From now on it will be assumed that angles are measured in radians unless otherwise stated.
The trigonometric identities proved for acute angles above are valid for all angles, though care must be taken to assign the correct sign to the square root in the half angle identities. Some useful formulae for calculating the trigonometric functions for any angles follow from inspection of Fig. 2.13 and are shown below:
sin(−θ ) = −sinθ cos(−θ ) = cosθ sin(π2 −θ ) = cosθ cos(π2 −θ ) = sinθ
sin(π −θ ) = sinθ cos(π −θ ) = −cosθ sin(θ+ π2 ) = cosθ cos(θ+ π
2 ) = −sinθ
sin(θ +π ) = −sinθ cos(θ +π ) = −cosθ sin(θ + 2πn) = sinθ cos(θ + 2πn) = cosθ
where n is any integer. Graphs of the trigonometric functions for any angle are shown in Fig. 2.14. Note their
periodic nature and note also the discontinuous nature of tanθ, which tends to ∞ as θ tends to π/2 and then reappears from −∞.
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Chapter 2
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Figure 2.14. The trigonometric functions for any angles.
The inverse trigonometric functions are arcsin x, arccos x and arctan x which give the
angle whose sine (or cosine or tangent) is x. Because the trigonometric functions are periodic, these inverse functions are multi-valued; for example, sin(nπ/6) = 0.5 when n = −11, −7, 1, 5, 13 and infinitely many other values. To obtain a single value, the principal value of these inverse functions is usually defined as the value between –π/2 and π/2 for arcsin x and arctan x and as the value between 0 and π for arccos x. The results are shown in Fig. 2.15.
Figure 2.15. The inverse trigonometric functions.
[The inverse function arctan(x) is inadequate in some contexts. The point {x, y} in Cartesian coordinates can be written in polar coordinates as {r cos θ, r sin θ} where r is its distance
from the origin, r = x2 + y2 , and θ is the angle made with the x-axis by the line joining the origin to the point (Fig. 2.13). Since y/x = tan θ, it is tempting to calculate θ = arctan(y/x) but this may be misleading. When {x, y} = {0.866, 0.5} as in Fig. 2.13 this gives the correct result π/6 radians (30°) but if {x, y} = {−0.866, −0.5} it gives the same result instead of the correct result 7π/6 radians (210°) since there is no way of deciding from the ratio y/x whether the point is in the first or the third quadrant. To avoid this problem the function arctan(x, y) has been defined as a function of two variables as follows. If x ≠ 0 calculate arctan(y/x). When x > 0 this is the result; when x < 0, add π if y ≥ 0 and subtract π if y < 0. When x = 0, arctan(x, y) = π/2 if y > 0 and −π/2 if y < 0. This gives a result between −π and π. Add 2π when the result is negative to obtain a result between 0 and 2π. Arctan(0, 0) is undefined.]
Applications of trigonometry
The periodic nature of the sine and cosine functions has made them invaluable in the study of periodic phenomena in the natural sciences. More traditional problems are the solution of triangles and computation of their area that have applications in geography, surveying and
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Geometry and Trigonometry
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navigation. The area of a rectangle with sides of lengths a and b is ab. Drawing a diagonal gives two congruent triangles each with area ab/2. Hence the area of a right triangle is ab/2 where a and b are the lengths of the sides adjoining the right angle.
The area of a parallelogram with sides of length a and b and with an angle γ between them is absinγ . (It does not matter whether this is the acute or the obtuse angle in the parallelogram since they sum to π and sinγ = sin(π −γ )). Consider the parallelogram ABCD in Fig. 2.16. Draw the perpendicular from D meeting the extension of BA at F and the perpendicular from C meeting AB at E so that CE and DF have length . The triangles AFD and BEC are congruent so that the area of the parallelogram is the same as that of the rectangle DFEC, which is Drawing the diagonal from A to C gives two
congruent triangles ABC and ADC each having area 12 absinγ . Thus the area of a triangle is
half the product of the lengths of any two sides and the sine of the angle between them. There are other formulae for calculating the area of a triangle knowing the lengths of all three sides or two angles and a side.
Figure 2.16. Area of a parallelogram.
For the solution of triangles consider the generic triangle in Fig. 2.17. The problem is
to find the remaining angles and sides given some of them. The solution uses two laws, the law of sines and the law of cosines.
Figure 2.17. A generic triangle.
h = bsinγ
ah = absinγ .
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Chapter 2
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The law of sines states that
sinα
a= sinβ
b= sinγ
c.
The area of the triangle is
Area = bcsinα
2= acsinβ
2= absinγ
2.
Multiplying throughout by 2/abc gives the law of sines. The law of cosines is a generalization of Pythagoras’ theorem stating that
a2 = b2 + c2 − 2bccosα
with two similar equations after interchanging symbols. In Fig. 2.18 the perpendicular has been dropped from A in Fig. 2.17, intersecting BC
at D. Note that BD = ccosβ and that DC = bcosγ so that a = ccosβ + bcosγ . (This remains true when β or γ is obtuse although the perpendicular drops outside the triangle.) Multi-plication by a leads to the equation
a2 = accosβ + abcosγ .
Dropping perpendiculars from B and C leads to the equivalent equations
b2 = bccosα + abcosγ
c2 = accosβ + bccosα .
Adding the second and the third of these three equations and subtracting the first leads to the law of cosines stated above; the two similar equations are obtained by adding the first and second of the three equations and subtracting the third or by adding the first and third and subtracting the second.
Figure 2.18. The law of cosines.
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Geometry and Trigonometry
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We now return to the solution of triangles using Fig. 2.17. If the lengths of all three sides a, b and c are known the angles can be calculated from the law of cosines
α = arccos
b2 + c2 − a2
2bc
with similar formulae for the other two angles. If the lengths of two sides a and b and the angle between them γ are known the third side can be determined from the law of cosines
c = a2 + b2 − 2abcosγ
and then the other angles can be determined as above. If a side and two angles are known, first determine the third angle from the fact that they sum to π and then determine the other sides by the law of sines; for example, if c is known, then
a = c sinα
sinγ, b = c sinβ
sinγ.
If the lengths of two sides and an angle not included between them are known, the solution is more delicate; the reader is referred to the discussion of “Solution of triangles” in Wikipedia.
The calculation of π
As a final example of the use of trigonometry we consider the evaluation of π, defined as the ratio of the circumference to the diameter of a circle. Archimedes found upper and lower bounds for π around 250 BC by calculating the perimeters of the smallest regular hexagon circumscribing a circle and of the largest regular hexagon inscribed within it and then successively doubling the number of sides of the regular polygons to tighten the bounds.
The simplest regular polygon with an even number of sides is the square. The smallest square that can circumscribe the unit circle (with radius 1 and diameter 2) has sides equal to 2 so that its perimeter is 8. The unit circle has by definition a circumference of 2π so that the upper bound for π is half the perimeter of the square, its semiperimeter, which is 4. The largest square that can be inscribed within the unit circle has a diagonal of 2, the diameter of the circle, so that by Pythagoras’ theorem it has sides of length 2 and semiperimeter 2 2 = 2.82843. Hence 2.82843 < π < 4.
Consider now a regular polygon with k sides. Fig. 2.19 shows the smallest circumscribing polygon of the unit circle and the largest polygon that can be inscribed in it, using the hexagon as an example. The smallest circumscribing polygon is composed of k isosceles triangles each with the angle at the center of the circle equal to 2π/k and with the side opposite this angle of length a tangent to the circle in its center. The perpendicular from the center bisects this side where it touches the circle, from which it follows that a/2 = tan(π/k). The semiperimeter that is an upper bound for π is ka/2 = k tan(π/k).
The largest polygon that can be inscribed within the circle is composed of k isosceles triangles each with the angle at the center of the circle equal to 2π/k, with the sides next to this angle equal to 1 and with the side opposite it equal to b, say. It follows from the law of sines that
Chapter 2
28
Figure 2.19. Calculation of upper and lower bounds for π from outer and inner polygons.
bsin(2π /k)
= 1sin(π /2−π /k)
= 1cos(π /k)
so that
b = sin(2π /k)
cos(π /k)= 2sin(π /k)cos(π /k)
cos(π /k)= 2sin(π /k).
The semiperimeter that is a lower bound for π is kb/2 = k sin(π/k). Thus the upper and lower bounds for π from circumscribing and inscribed regular
polygons with k sides are k tan(π/k) and k sin(π/k) respectively. They can easily be calculated for a regular hexagon with six sides since the trigonometric functions are known explicitly for an argument of π/6 radians = 30° (see Table 2.1). They can then be calculated by successive doubling of k by using the half angle identities. Some values of these bounds to five decimal places are shown in Table 2.2.
Table 2.2. Lower and upper bounds for π from regular polygons with k sides.
k Lower Upper k Lower Upper
bound bound bound bound 6 3 3.46410 384 3.14156 3.14166 24 3.13263 3.15966 1536 3.14159 3.14160 96 3.14103 3.14271 6144 3.14159 3.14159
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Geometry and Trigonometry
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To find the area of the circle, consider the triangle in the inscribed polygon in Fig. 2.19, which has area 1/2 sin(2π/k) for a k-sided polygon so that the area of the polygon is (k/2)sin(2π /k) = ′k sin(π / ′k ) where ′k = k /2 . As ′k →∞ this quantity tends to the area of the circle and also tends to π since sin(x)/x → 1 as x → 0 (p. 49). Thus the area of a circle with unit radius is π so that the area of a circle with radius r is πr2.
Analytic Geometry
In the geometry of Euclid a point may be regarded as a very small mark (“A point is that which has no part”) made on a plain sheet of paper. A straitedge (an unmarked ruler) may be used to draw a straight line (“A line is breadthless length”) between two points and a compass to draw a circle, but no measurement is allowed except that the compass may be used to measure equal distances at different positions. A marked ruler, a protractor and a set square (triangle) are not allowed. These constraints placed severe limitations on the development of geometry.
Fermat and Descartes transformed the subject in the seventeenth century by inventing the idea of representing a point by a pair of numbers, the x and y coordinates in the plane in the Cartesian coordinate system named after Descartes, replacing the plain paper of Euclidean geometry by graph paper. This allowed arithmetic and algebra to be applied to geometry.
A point p in the plane is represented by a pair of real numbers, p = {x, y}, showing its x and y coordinates. Note that x = r cosθ and y = r sinθ where r is the distance of the point from the origin and θ is the angle made by the line from the origin to the point with the x-axis (see Figs. 1.4 and 2.13). In general, a vector is an ordered list of elements. The number of elements in the list is called its length. In this book the elements in the list will always be real numbers and vectors will be written as lower case bold letters. The vector p = {x, y} ={r cosθ , r sinθ} with length 2 is a point vector representing the position of a point in two-dimensional space and can be shown graphically as a dot at the appropriate position. These remarks can be extended to three dimensions in which case the vector will have length 3. In physics, vectors are used to represent quantities with both magnitude and direction (given by r and θ for a two-dimensional point vector) and are often represented by arrows pointing in the appropriate direction. Vectors are discussed further in Chapter 6.
The algebra of transformations will be considered before turning to the geometry of the straight line, based on linear equations in x and y, and then the conic sections, the circle, the ellipse, the parabola and the hyperbola, based on quadratic equations in x and y.
Transformations
It was stated earlier that two triangles are congruent, having the same shape and size, if one of them can be obtained from the other by translation, rotation or reflection, and the same is true for any other geometrical figures. It is useful to see how these operations are defined in analytic geometry. If the point p = {x, y} is translated by v = {a, b}, meaning that it is moved
Chapter 2
30
by an amount a to the right (or to the left if a is negative), and by an amount b up (or down if b is negative), then it becomes
′p ={x + a, y + b}= p+ v. (Addition of vectors is performed element by element.)
The second transformation preserving both shape and size is rotation. Suppose that the point p ={x, y}={r cosθ , r sinθ} is rotated counterclockwise about the origin by an angle φ so that it becomes ′p ={r cos(θ +φ), r sin(θ +φ)}. By the angle sum identities (p. 21)
′p ={(cosφ)x − (sinφ)y, (sinφ)x + (cosφ)y}.
If the 2×2 rotation matrix Rφ is defined as
Rφ =
cosφ − sinφsinφ cosφ⎡
⎣⎢
⎤
⎦⎥
this result can be stated succinctly as
′p = Rφ i p
where i denotes the product of the matrix and the vector. Matrices are discussed in Chapter 6.
This formula rotates a point about the origin. It is frequently useful to rotate a point p about another point q not at the origin. This can be done by translating both points by –q, rotating the new p about the origin and then translating it by q. In other words
′p = Rφ i (p− q)+ q.
This technique was used in drawing the circumscribing hexagon in Fig. 2.19. p1,!, p6 were
defined as {x, 1}, {−x, 1}, {−3x, 1}, ! , {−9x, 1} where x = tan(π/6); p3,!,p6 were rotated
by π/3 radians about p2 ; the new p4,!,p6 were rotated by π/3 radians about the new p3 ; and so on. This procedure produced the vertices of the hexagon that were then connected by lines. A similar technique was used to draw the inscribed hexagon.
The third transformation preserving shape and size is reflection. If the point p = {x, y} is reflected in the x-axis then y changes sign so that ′p ={x,− y}. If it is reflected in the y-axis then x changes sign so that ′p ={−x, y}.
Another important transformation stretches or compresses the positions of points. If ′p ={kx,my} , then it is stretched or compressed in the x-axis when 1 < k or 0 < k < 1 and likewise in the y-axis when 1 < m or 0 < m < 1. The term scaling includes either stretching or compression and the constants k and m are called scale factors. If all points in a figure are scaled by the same factor in each direction (isotropic scaling with k = m), the new figure will have the same shape as the original figure and will be similar to it; if all points in a figure are scaled by different factors in each direction (anisotropic scaling with k ≠ m) the new figure will have a different shape from the original figure.
Geometry and Trigonometry
31
Straight lines
A straight line consists of all the points satisfying the linear equation ax + by + c = 0
where a, b and c are constants. If b ≠ 0 this can be put in slope-intercept form y =α + βx
where α is the intercept on the y-axis when x = 0 and β is the slope of the line. Fig. 2.20 shows segments of the lines y = −1+ x and y = 2− x. The two lines intersect at the point {1.5, 0.5} since −1+ x = 2− x gives x = 1.5 whence y = 0.5 from either equation. If the two lines have the same slope the condition for them to intersect is that the difference between the intercepts is zero. When they have the same intercept this is always true, which reflects the fact that the two lines coincide; when they have different intercepts it is never true, which reflects the fact that they are parallel. A line can also be defined by two points on it, say
p0 ={x0, y0} and p1 ={x1, y1}. The set of points tp0 + (1− t)p1 with 0 ≤ t ≤1 is the line segment between the two points; the set of points with t taking any real value is the line extended indefinitely in both directions.
Figure 2.20. Two lines in Cartesian coordinates.
In three dimensions the linear equation ax + by + cz + d = 0
defines a plane. A line is the intersection between two planes so that two linear equations are needed to define a line in this way. A line can also be defined from two points on it as in the two-dimensional case. Lines in three dimensions do not usually intersect.
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Chapter 2
32
The circle and the ellipse
A circle consists of all the points at the same distance (the radius of the circle) from its center. The unit circle, with center at the origin and unit radius, illustrated in Fig. 2.13 on p. 23, consists of the points {x, y} satisfying the equation
x2 + y2 = 1.
The general circle, with its center translated from the origin to {a, b} and scaled isotropically by the factor r (so that its radius is r), consists of the points satisfying the equation
(x − a)2 + ( y − b)2 = r2 that may also be written
(x − a)2
r2 + ( y − b)2
r2 = 1.
Rotation leaves a circle unchanged because of its symmetry. It will be seen from Fig. 2.13 that the point on the unit circle corresponding to the
angle θ is {x, y} = {cosθ , sinθ} so that the point corresponding to θ in the general circle is
{a + r cosθ , b+ r sinθ}. This is called the parametric representation of the circle because it allows the points on it to be calculated directly from the parameter θ for 0 ≤ θ < 2π.
The ellipse is an anisotropic scaling of the circle. Consider a unit circle that has been scaled by a factor a on the x-axis and by a factor b on the y-axis. (These factors have no connection with the symbols used above for the translation of the center of the circle.) It will be supposed for simplicity that the center of the ellipse remains at the origin and that a > b so that the ellipse is wider than it is tall. It consists of the points {x, y} such that
x2
a2 + y2
b2 = 1.
Figure 2.21. An ellipse centered at the origin with a = 2.5, b = 1.5.
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Geometry and Trigonometry
33
The parametric representation is {x, y} = {acosθ , bsinθ} for 0 ≤ θ < 2π. The ellipse centered at the origin with a = 2.5, b = 1.5 is shown in Fig. 2.21. This ellipse is symmetrical about either the x-axis or the y-axis since it does not change when reflected about either of these lines. The segment of the x-axis of length 2a between {−a, 0} and {a, 0} is called the major axis of the ellipse and the segment of the y-axis of length 2b between {0, −b} and {0, b} is called the minor axis since a > b; the semi-major axis is the segment between the origin and {a, 0}.
There is a well-known result that there are two focal points situated on the major axis at the same distance f on either side of the origin, the sum of whose distances to any point on the ellipse is constant (see Fig. 2.21 where f = 2). Writing d1 and d2 for the distances of the foci on the left and on the right of the origin from the point on the ellipse with parametric representation {a cos θ, b sin θ}, we find that
d12 = (acosθ + f )2 + b2 sin2θ
d22 = (acosθ − f )2 + b2 sin2θ .
When θ = 0, d1 = a + f , d2 = a − f , so that d1 + d2 = 2a. When θ = π/2, d1 = d2 = f 2 + b2 .
Equating the sum of these distances to 2a and solving for f gives f = a2 − b2 . The quantity
ε = f/a = 1− b2/a2 is called the eccentricity of the ellipse. It can vary from 0 to 1 and measures how far the ellipse departs from being circular (ε = 0). It remains to show that
d1 + d2 = 2a for all values of θ. A little algebra shows that
d12 = a2(1+ ε cosθ )2
d22 = a2(1− ε cosθ )2
from which the result follows. Gardeners use this result to mark out an elliptical flowerbed. Put two pegs in the soil
with a distance 2f between them, place a piece of rope with length 2f + 2a around them and trace out the ellipse with the rope held taut.
An ellipse can also be defined by the focus-directrix definition that is useful for constructing a parabola or a hyperbola. This definition unifies the treatment of these curves. Define a point called the focus, a line called the directrix and a quantity ε called the eccentricity, and write d1 for the distance between a point and the focus and d2 for the distance between this point and the directrix. With appropriate choice of these quantities, the curve comprises all points such that d1 = εd2 . It turns out that 0 < ε < 1 for the ellipse, ε = 1 for the parabola and ε > 1 for the hyperbola.
For the ellipse with parameters a and b (0 < b < a) define the focus as {f, 0}, the eccentricity as f/a and the directrix as the vertical line x = a/ε parallel to the y-axis. For any point {x, y}
d12 − ε 2d2
2 = (x2 − a2)(1− ε 2)+ y2.
Chapter 2
34
Writing x = acosθ , y = bsinθ and b2 = a2(1− ε 2) we find that this is zero. The same result
holds for the other focus {−f, 0} and its associated directrix x = −a/ε. Thus the ellipse meets the criterion of the focus-directrix definition that it comprises the points such that d1 = εd2 .
The ellipse has important scientific uses, in particular to the orbits of the planets around the sun. The most popular explanation of the movements of the sun and the planets during the middle ages was the geocentric Ptolemaic system in which the earth was at the center of the universe and other bodies in the heavens moved around it in circular orbits since the circle was considered the perfect shape. To make this model fit the facts it was developed into a complex interaction of circles (epicycles and deferents). Copernicus proposed a heliocentric theory in 1543 in which the planets moved in circular orbits with centers near the sun, though he could not dispense with epicycles.
At the beginning of the seventeenth century Kepler improved on the Copernican system when he enunciated his three laws of planetary motion, based on Tycho Brahe’s observations on Mars, that are valid today:
1. The planets move in ellipses, with the sun at one focus. 2. The radius vector sweeps out equal areas in equal times. 3. The square of the time of revolution is proportional to the cube of the mean
distance. These laws do not predict either the size or the eccentricity of the elliptical orbits.
In accordance with the first law, the earth orbits round the sun in an ellipse with semi-major axis (the distance from its center to the ellipse along the major axis) a = 93.0 million miles and the sun is at one of its foci 1.5 million miles from the center along the major axis (f = 1.5 million miles, see Fig. 2.21), so that its eccentricity is ε = 0.017. The earth is nearest the sun at 91.5 million miles when it is on the major axis with the sun between it and the center (perihelion) about January 3 in recent times; it is furthest from the sun six months later about July 4 when it is on the major axis on the opposite side of the ellipse (aphelion). Thus winter is slightly warmer and summer slightly colder in the northern hemisphere than they would be if the orbit were circular. Furthermore, the earth travels faster in its orbit when it is closer to the sun at perihelion that when it is farther from it at aphelion by Kepler’s second law. Hence the winter is slightly shorter than the summer in the northern hemisphere. There are 179 days between the autumnal equinox on September 22 or 23 and the vernal equinox on March 20, while there are 186 days between the vernal equinox and the autumnal equinox. Mercury is the closest planet to the sun with semi-major axis a = 36.0 million miles, a large eccentricity ε = 0.206 and an orbital period of 88 earth days compared with 365.25 days for the earth. The ratio of the cube of the semi-major axis to the square of the orbital period, divided by 1018 , is 6.029 for the earth and 6.025 for Mercury, in good agreement with Kepler’s third law. It will be seen in Chapter 7 how Isaac Newton in the Principia (1687) justified Kepler’s empirical laws theoretically.
The parabola
A typical parabola is the graph of the function y = ax2 , shown in the top row of Fig. 2.22 for
a = 1 and a = 2. These two parabolas are similar because, starting from y = x2 and making
Geometry and Trigonometry
35
the isotropic scaling ′x = 2x, ′y = 2y, we obtain ′y = 2 ′x 2. All other parabolas may be obtained from this typical parabola by rotation and/or translation. For a counterclockwise rotation by φ radians the point {x, y} becomes
{ ′x , ′y }= Rφ i{x, y} where the rotation matrix
Rφ is defined on p. 30. To find the relationship between ′x and ′y in the rotated parabola,
make the clockwise rotation {x, y}= R−φ i{ ′x , ′y } so that
x = (cosφ) ′x + (sinφ) ′yy = (−sinφ) ′x + (cosφ) ′y .
Since y = ax2 before rotation, we obtain
(−sinφ) ′x + (cosφ) ′y = a (cosφ) ′x + (sinφ) ′y⎡⎣ ⎤⎦2
.
Dropping the dashes, which refer to the rotated parabola with the original axes, we obtain
(−sinφ)x + (cosφ)y = a (cosφ)x + (sinφ)y⎡⎣ ⎤⎦2
.
Figure 2.22. Vertical parabolas (top row); horizontal and inverted parabolas (bottom row).
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Chapter 2
36
For φ = π /2, this gives (bottom left in Fig. 2.22); for φ = π , it gives y = −ax2
(bottom right in Fig. 2.22); and for φ = π/4 it gives If it is desired to translate the parabola from the origin to {c, d}, replace x by (x − c) and y by (y − d) in its equation.
All parabolas are similar and they are all congruent to the vertical parabola y = ax2, illustrated at the top of Fig. 2.22, for some value of a, so that it may be taken as a model. It has a vertex at the origin and an axis of symmetry that is the upper half of the y-axis starting at the vertex. It has two other important features, illustrated in Fig. 2.23: the focus is a point {0, 1/4a} situated on the axis of symmetry at a distance 1/4a above the vertex and the directrix is a horizontal line y = −1/4a at the same distance below the vertex. An important property of the parabola is that any point on it is at the same distance from the focus and the directrix. Suppose that {x, y} is a point on the parabola. Its squared distance from the directrix is ( y +1/ 4a)2 and its squared distance from the focus is x
2 + ( y −1/ 4a)2. These
quantities are easily shown to be equal after substituting y/a for x2 . Under the focus-directrix definition the parabola can be regarded as a curve with eccentricity 1 since the distance of a point from the focus is equal to its distance from the directrix. (There is only one focus with its associated directrix for a parabola.)
Figure 2.23. The parabola y = x2/ 6 showing the focus {0, 1.5} and the directrix y = −1.5.
The point on the parabola at {−3.5, 2.04} is equidistant from each of them. The reflective property of the parabola states that, if a parabola can reflect light or
similar electromagnetic radiation, then light that enters it parallel to the axis of symmetry is reflected to the focus. Fig. 2.24 shows a vertical parabola with focus at F = {0, 1/4a} and with a point P = {X, Y} on the parabola. The perpendicular from P to the directrix meets the
x = −ay2
2( y − x) = a(x + y)2.
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Geometry and Trigonometry
37
latter at A = {X, −1/4a}. The slope of the curve y = ax2 is dy/dx = 2ax so that the tangent to the point P on the parabola is the line y = Y + 2aX (x − X ), which meets the x-axis at B =
{X/2, 0} given that Y = aX 2 . Thus B is halfway along the straight line between A and F. Hence the triangles APB and FPB are congruent since they have three equal sides; in
fact they are mirror images reflected about PB. The figure also shows a ray of light parallel to the axis hitting the parabola at P and making an angle α with the tangent at this point. Now
(vertical angles). Since the law of the reflection of light states that the vertical incident light will be reflected towards the focus along PF. This will hold whatever point on the parabola the vertical incident light hits.
Figure 2.24. The reflective property of the parabola.
The reflective property has several useful applications. A satellite dish is a paraboloid
(a surface obtained by revolving a parabola about its axis) with a receiver at its focus. It is aligned to point directly towards the satellite transmitter. The idea also works in reverse. A simple car headlamp is a steel paraboloid with a light bulb at its focus. When the bulb is switched on, the light from it is reflected forward from the paraboloid as parallel rays illuminating the road.
Another application is that the trajectory of a projectile, under the influence of gravity but ignoring air resistance, is a parabola whose parameters depend on the initial velocity of the projectile. This result was obtained by Galileo in 1638 and was clarified by Newton in the Principia in 1687. Suppose that a projectile is fired from a position h meters above a flat plain at an angle θ radians from the horizontal and with initial velocity v = {vcosθ , vsinθ}; thus the initial velocity has components vcosθ horizontally and vsinθ vertically. This velocity is maintained through inertia under Newton’s first law of motion. In addition gravity pulls the projectile downwards with acceleration g = 9.81 m/s2. At time t
∠FPB = ∠APB (congruent triangles) =α ∠FPB =α
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Chapter 2
38
x(t) = vt cosθ
y(t) = h+ vt sinθ − 12
gt2.
Substituting for t from the first equation in the second equation gives
y = h+ (tanθ )x −12 g
v2 cos2θ
⎛
⎝⎜
⎞
⎠⎟ x2 .
This is the quadratic formula for an inverted parabola. Newton’s laws of motion will be discussed in Chapter 7.
The hyperbola
Figure 2.25. Top row: east-west hyperbolas with a = b = 1 and a = 1, b = 2. Bottom row: north-south hyperbola and reciprocal function after rotation by 90° and 45° respectively
A typical hyperbola can be represented by the equation
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Geometry and Trigonometry
39
The top row of Fig. 2.25 shows two hyperbolas with this equation. They have two mirror image branches and each branch is symmetrical about the x-axis, crossing it at the vertex. Each branch asymptotically approaches one of the straight lines y = (b/a)x or y = −(b/a)x (replace 1 by 0 in the equation, which makes little difference for large x). This is called the east-west orientation of the hyperbola and its shape depends only on the ratio b/a, as shown on the top row of Fig. 2.25 for ratios 1 and 2. All east-west hyperbolas with the same value of this ratio are congruent.
Define the eccentricity as ε = 1+ b2/a2 , the focus as the point {aε, 0}, the directrix as the line x = a/ε, and as the distances of a point from the focus and the directrix
respectively. A little algebra will show that for any point on the hyperbola so that the criterion for the focus-directrix definition is satisfied. The other focus at {−aε, 0} with its directrix x = −a/ε might also have been used. Note that ε > 1.
Rotation can be done by the technique described for the parabola on p. 35. The rect-angular east-west parabola with a = b = 1 when rotated by π/2 radians is the north-south parabola illustrated in Fig. 2.25 (bottom left) and has the formula
y2
a2 − x2
b2 = 1.
The rectangular east-west hyperbola with b = a when rotated by π/4 radians has the formula
2xy = a2 so that the reciprocal function y = 1/x, shown in Fig. 2.25 (bottom right), is obtained
by setting a = 2. In discussing Kepler’s laws of planetary motion around the sun Newton showed that
they should be elliptical provided that the planet did not have enough velocity to escape the sun’s gravitational pull. He also showed that an object entering the sun’s gravitational field with more than enough velocity to escape would follow a hyperbola. In the unlikely intermediate case it should follow a parabolic trajectory.
Spherical Geometry and Trigonometry
So far we have considered geometry on a flat two-dimensional plane extending indefinitely in all directions. But the earth is not flat (although this provides a good approximation on a small, local scale) so that we must extend the study to geometry on the surface of a sphere. The other important application of spherical geometry is to the study of the celestial sphere in astronomy but attention will here be limited to the terrestrial sphere. This section comprises only a brief sketch of the subject with detailed proofs omitted.
Because it revolves on its axis the earth is slightly wider at the equator than between the poles and is most accurately represented as an ellipsoid. The eccentricity is so slight that for practical purposes it can be treated as a sphere (the three-dimensional analog of the circle) with a radius (the distance from its center to any point on its surface) of about 3959 statute miles. This sphere will be called the earth. It is convenient in navigational problems to use nautical miles containing 6080 feet rather than statute miles containing 5280 feet, so that the
d1 and d2
d1 = εd2
Chapter 2
40
radius of the earth in this measure is about 3438 nautical miles. The reason for doing this is that 1 nautical mile is approximately 1 minute of arc measured along any great circle since 3438π /(180× 60) ! 1. In this context angles are usually measured in degrees and minutes rather than in radians.
A plane passing through the center of the earth cuts its surface in a circle with its center at the center of the earth. Such a circle is called a great circle; it has a radius of 3438 nautical miles and a circumference of 2π × 3438 ! 21,600 nautical miles. A plane not passing through the center of the earth cuts its surface in a small circle with a smaller radius. The equator is the great circle defined by a plane perpendicular to the earth’s axis of rotation and passing through it midway between the poles.
Latitude measures the position north or south of the equator of a point on the earth’s surface. Lines of equal latitude run from east to west as small circles parallel to the equator. Longitude measures the position of a point east or west of the prime meridian, which by convention passes through the Royal Observatory, Greenwich. Meridians of longitude are halves of great circles through the north and south poles. (A meridian is only half a great circle because, for example, the meridian 30° east running near St Petersburg, Russia is counted as different from the meridian on the opposite side of the world 150° West running near Anchorage, Alaska.) Parallels of latitude and meridians of longitude are shown diagrammatically in Fig. 2.26.
Figure 2.26. Diagrammatic representation of latitude and longitude.
Equator
Parallelof
Latitude
Meridianof
Longitude
North pole
South pole
Geometry and Trigonometry
41
Fig. 2.27 shows a section from the north to the south pole (N and S) through the center of the earth (C); its circumference is the meridian of longitude containing the point A meeting the equator at E. The latitude of A is defined as the angle δ = ∠ACE north or south of the equator. If r is the radius of the small circle formed by the line of equal latitude (measured by AD) and R is the radius of the earth at the equator (measured by EC = AC), it is clear from the figure that r = Rcosδ since
cosδ = BC
AC= AD
AC= r
R.
Figure 2.27. Definition of the latitude of A. See text for explanation.
Longitude may be defined in a similar way. Suppose that the Greenwich meridian
meets the equator at G* and that the meridian of a point A meets the equator at A*. Consider a section of the earth through the equator passing through its center C and with G* and A* marked on its circumference. The longitude of A is the angle ∠A*CG* east or west of the Greenwich meridian. Latitude and longitude uniquely define the position of a point on the surface of the earth, The interesting question how they can be measured in practice will not be considered here.
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Chapter 2
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A straight line between two points on the surface of the earth is defined as the arc of the great circle through them. This excludes small circles, including parallels of latitude apart from the equator, from being straight lines. The reason for this definition is that it is the shortest distance between the two points. Consider the flight of an airplane from 60° N 30° E (St Petersburg) to 60° N 150° W (just south of Anchorage). The great circle route is over the north pole because they are on opposite meridians, which means flying 30° due north to the pole and then 30° due south to Anchorage, a total distance of 60× 60 = 3600 nautical miles. The rhumb line (loxodrome) going due east (or due west) along the parallel of latitude involves a distance of 180× 60× cos60! = 5400 nautical miles, which is 50% further. A great circle usually changes direction continuously. The great circle course between two points and the distance along it can be found by using spherical trigonometry.
If two great circles meet at a point A, the angle between them at that point is defined as the difference between their directions there. (Strictly speaking it is the angle between the tangents to them at A.) For example, if the first great circle is heading in a direction 48° east of north at A and the second great circle is heading in a direction 23° west of north, the angle between them is 71°.
A spherical triangle is formed by the meeting of three great circle at A, B and C on the surface of the earth. Write a, b and c for the lengths of the arcs opposite these points (in nautical miles or minutes of arc) and α, β and γ for the angles at these points (cf. Fig. 2.17). The equivalent of the law of cosines for plane triangles (p. 26) is
cos a = cosb cosc + sinb sinc cosα with two similar equations after exchanging symbols. There is a similar law for angles:
cosα = sinβ sinγ cos a − cosβ cosγ . The equivalent of the law of sines for plane triangles (p. 26) is
sinαsin a
= sinβsinb
= sinγsinc
.
Given three of the six lengths and angles, these laws and similar ones can be used to calculate the other three. (Remember that the lengths are angles in appropriate units when calculating the trigonometric functions.) As a simple example, consider an equilateral triangle with a = b = c = 2700 nautical miles (45°). The law of cosines shows that α = β = γ = 65.53°. The sum of the angles in the triangle is 196.59°, which is greater than 180°. It can be shown that, for any spherical triangle, the sum of the angles is 180°(1 + 4f) where f is the area of the triangle as a proportion of the surface area of the sphere. The Euclidean result of 180° for plane triangles is only a good approximation for small areas.
