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Spherical Geometry

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Spherical Geometry 987023 2473 < % 52873 52 59 3 1273 ¼4 14 578 2 49873 3 4 7∆��≌ ≠ 3 δ � 2 1 2 7 5 8 4 3 23 5 1 ⅓β ⅝Υ � VCAA-AMSI maths modules 5 1 85 34 5 45 83 8 8 3 12 5 = 2 38 95 123 7 3 7 8 x 3 309 �Υβ⅓⅝ 2 8 13 2 3 3 4 1 8 2 0 9 756 73 < 1 348 2 5 1 0 7 4 9 9 38 5 8 7 1 5 812 2 8 8 ХФУШ 7 δ 4 x ≠ 9 3 ЩЫ � Ѕ � Ђ 4 � ≌ = ⅙ Ё ∆ ⊚ ϟϝ 7 0 Ϛ ό ¼ 2 ύҖ 3 3 Ҧδ 2 4 Ѽ 3 5 ᴂԅ 3 3 Ӡ ⍰1 ⋓ ӓ 2 7 ӕ 7 ⟼ ⍬ 9 3 8 Ӂ 0 ⤍ 8 9 Ҷ 5 2 ⋟ ҵ 4 8 ⤮ 9 ӛ 5 ₴ 9 4 5 € 4 ₦ € 9 7 3 ⅘ 3 ⅔ 8 2 2 0 3 1 ℨ 3 ℶ 3 0 ℜ 9 2 ⅈ 9 ↄ 7 ⅞ 7 8 5 2 ∭ 8 1 8 2 4 5 6 3 3 9 8 3 8 1 3 2 4 1 85 5 4 3 Ӡ 7 5 ԅ 5 1 ᴂ - 4 �Υβ⅓⅝ 0 Ѽ 8 3 7 Ҧ 6 2 3 Җ 3 ύ 4 4 0 Ω 2 = - Å 8 € ℭ 6 9 x ℗ 0 2 1 7 ℋ 8 ℤ 0 ∬ - √ 1 ∜ ∱ 5 ≄ 7 A guide for teachers – Years 11 and 12 ∾ 3 ⋞ 4 ⋃ 8 6 ⑦ 9 ∭ ⋟ 5 ⤍ ⅞ Geometry ↄ ⟼ ⅈ ⤮ ℜ ℶ ⍬ ₦ ⍰ ℨ Spherical Geometry ₴ € ⋓ ⊚ ⅙ ӛ ⋟ ⑦ ⋃ Ϛ ό ⋞ 5 ∾ 8 ≄ ∱ 3 ∜ 8 7 √ 3 6 ∬ ℤ ҵ 4 Ҷ Ӂ ύ ӕ Җ ӓ Ӡ Ҧ ԅ Ѽ ᴂ 8 7 5 6 Years DRAFT 11&12 DRAFT Spherical Geometry – A guide for teachers (Years 11–12) Andrew Stewart, Presbyterian Ladies’ College, Melbourne Illustrations and web design: Catherine Tan © VCAA and The University of Melbourne on behalf of AMSI 2015. Authorised and co-published by the Victorian Curriculum and Assessment Authority Level 1, 2 Lonsdale Street Melbourne VIC 3000 This publication may be used in accordance with the VCAA Copyright Policy: www.vcaa.vic.edu.au/Pages/aboutus/policies/policy-copyright.aspx Copyright in materials appearing at any sites linked to this document rests with the copyright owner/s of those materials. The VCAA recommends you refer to copyright statements at linked sites before using such materials. Australian Mathematical Sciences Institute Building 161 The University of Melbourne VIC 3010 Email: [email protected] Website: www.amsi.org.au Assumed knowledge ..................................... 4 Motivation ........................................... 4 Content ............................................. 4 Circular Geometry ..................................... 5 Calculation Notes ...................................... 6 The Earth as a sphere ................................... 11 Latitude and Longitude .................................. 11 Distances along meridians ................................. 14 Distances between places with the same longitude .................. 17 Distance between places on the Equator ........................ 20 Distances between places with the same latitude ................... 21 Great circle distance between places with the same latitude ............ 24 Time zones and travel times ............................... 26 Using Pythagoras’ theorem in spheres .......................... 29 Extension Activity ....................................... 29 Distance between any two places on the earth’s surface ............... 30 The calculation ....................................... 30 Comparing great circle distance with small circle distance .............. 33 Spherical Geometry Assumed knowledge • Familiarity with circle measurement formulae, including circumference and area • Familiarity with trigonometric calculations, including basic SOHCAHTOA, sine and cosine rules • Familiarity with area of triangles using sine area rule • Familiarity with circle geometry. Motivation • What have circles to do with measuring distances on the surface of the Earth ? • Do I have to use radians for measuring any angles ? • Will it make much difference to the distance calculated if I use an incorrect method ? • Why is the straight-line distance between two points on a sphere not necessarily the shortest distance between those two points ? • How does a spherical model of the Earth help me to understand the different time zones that have been established ? Content A circle is formed by a point moving around a fixed point (the centre) at a constant dis- tance. A sphere is a three-dimensional form of a circle, where all points on the surface of the sphere are at a constant distance from the centre of the sphere. The Earth is not exactly a sphere. A grapefruit is a better analogy, as it is flatter at the top and bottom (an oblate sphere). The diameter as measured through the North and South Poles is about 40 km less than the diameter using two points on the Equator. The radius of the Earth is usually given as 6 370 kilometres. A guide for teachers – Years 11 and 12 • {5} For the purposes of modelling key aspects of travel and time on the Earth, for a course such as Further Mathematics in the Victorian Certificate of Education, treating the Earth as a true sphere with a radius of 6 400 kilometres will provide reasonable answers from a conceptually simple framework. Circular Geometry Radii and chords We begin by revisiting the definition of a circle and some of the language used for de- scribing the geometry of circles. • A circle is the set of all points in the plane that are a fixed distance (the ra- dius) from a fixed point (the centre). chord • Any interval joining a point on the cir- cle to the centre is called a radius. By diameter the definition of a circle, any two radii have the same length. • An interval joining two points on the circle is called a chord. radiusradius • A chord that passes through the centre is called a diameter. Since a diameter consists of two radii joined at their endpoints, every diameter has length equal to twice the radius. Let AB be a chord of a circle not passing through its centre O. The chord and the two equal radii OA and BO form an isosceles triangle whose base is the chord. The \AOB is called the angle at the centre subtended by the chord. A O B {6} • Spherical Geometry AMSI module Let A and B be two different points on a circle with centre O. These two points divide the circle into two opposite arcs. A If the chord AB is a diameter, then the minor arc two arcs are called semicircles. Other- wise, one arc is longer than the other. The longer arc is called the major arc AB and O B the shorter arc is called the minor arc AB. Now join the radii OA and OB. The reflex angle AOB is called the angle subtended major arc by the major arc AB. The non-reflex angle AOB is called the angle subtended by the minor arc AB, and is also the angle subtended by the chord AB. The two radii divide the circle into two sectors, called correspondingly the major sector OAB and the minor sector OAB. Segments minor segment A A chord AB of a circle divides the circle into two segments. The two segments are called a major segment (the larger one) and a mi- B nor segment (the smaller one). major segment Subtend The word ’subtend’ literally means ’holds under’, and is often used in geometry to de- scribe an angle. The phrase ’opposite to’ could also be used, since, in the case above, the angle subtended by an arc is located opposite to that arc in the diagram. Calculation Notes Unless it is specifically requested, the ¼ value obtained by accessing the appropriate function in a scientific or graphic calculator should be used in any calculation involv- ing ¼. ALL angles in this unit will be specified in degrees. A guide for teachers – Years 11 and 12 • {7} Measuring arc length In any circle, the length of an arc (l) is proportional to the angle (θ) it creates (subtends) at the centre. For a circle of radius r , the circumference will be 2¼ units. Thus, the length of the arc θ , circumference Æ 360± l θ so . 2¼r Æ 360± θ ) l 2¼r Æ 360± £ θ ¼r Æ 180± £ ¼ Exampler θ (Alternate formulations) Æ £ 180± £ A circle has a radius of 30 cm. Find the length of an arc subtending an angle of 75± at the centre, correct to one decimal place. Solution r ¼ Length of arc θ Æ 180± £ 30¼ 75± Æ 180± £ 39.2699... Æ 39.3 cm ¼ Example An arc of a circle has a length of 30 cm. If the radius of the circle is 25 cm, what angle, correct to the nearest whole degree, does the arc subtend at the centre ? Solution r ¼ 30 θ Æ 180± £ 25¼ θ Æ 180± £ 30 180± θ £ Æ 25 ¼ £ 68.754... 69± Æ ¼ {8} • Spherical Geometry AMSI module Sector The area of a sector depends not only on the radius of the circle r involved, but also the angle be- tween the two straight edges. r 휃 We can see this in the following table : Angle Fraction of circle area Area rule 180± 1 1 2 180± A ¼r 360± Æ 2 Æ 2 £ 90± 1 1 2 90± A ¼r 360± Æ 4 Æ 4 £ 45± 1 1 2 45± A ¼r 360± Æ 8 Æ 8 £ 30± 1 1 2 30± A ¼r 360± Æ 12 Æ 12 £ θ θ θ A ¼r 2 360± Æ 360 £ Example Find the area of a sector with a radius of 40 mm, that contains an angle of 60±, correct to the nearest square millimetre. Solution θ 60± A ¼r 2 ¼402 923.628... 924 mm2 Æ 360± £ Æ 360± £ Æ ¼ A guide for teachers – Years 11 and 12 • {9} Areas of segments Where a chord subtends an angle θ at the r centre of a circle of radius r, the area of the minor segment is given by : 1 µ 2πθ ¶ r Area r 2 sinθ 휃 Æ 2 £ 360± ¡ θ The area of the sector that includes the segment is given by A ¼r 2 Æ 360± £ or rewritten as 1 2πθ A r 2 .
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