Introduction to Spheres, and Some Applications

1. Introduction to Spheres,and Some Applications Doug Marquis Weston High School 3 June 2013

2. Outline • Basics of 3-D Circles – Spheres • Definition and Review • Some 2-D geometry interacting with Spheres • Some 3-D geometry interacting with Spheres • Spherical Coordinates • Applications to Satellites (Orbitology)

3. Review: Introduction to Spheres • What makes an object a sphere? • Sphere: A 3-D object wherein every point on the surface is equidistant from a single point, and every direction from that point has a point on the surface • Variants include hemisphere, quartersphere, etc

4. Review: Simple Sphere Properties • Spheres can be solid, or hollow (note that the definition only defines the surface of the sphere) • A sphere has the smallest surface area among all 3-D surfaces enclosing a given volume • Spheres are rotationally symmetric until some point on the sphere becomes special (e.g. north and south pole) • There are 11 properties of spheres – you can find them on wikipedia.org

5. Review: Equation for a Sphere • x, y, and z are the coordinate axes • R is the radius of the sphere • is the origin of the sphere • Spherical work allows a wholedifferent coordinate system – will be briefly discussed R

6. Review: Volume of a Sphere • They tell me you’ve already seen this • Enough said

7. Review: Surface Area of a Sphere • They said you knew this one too • Let’s run some numbers

8. Ex 1: Calculate V & SA of Earth • R ≈ 6438 km • Volume = 4 /3 * π * 64383= ~ 1 T km3 • Surface Area = 4 * π * 64382= ~ 1/2 B km2 • Possibly of interest – Surface Area of USA? • Just under 10 M km2 • Just under 2% of Earth’s Surface

9. Example 2: Relative Volumes • Problem: Find the ratio of the surface areas of a cube and a sphere, where both have a volume of one cubic meter. • Answer: • Vsphere = 1 m3 = 4/3 π rsphere3 • rsphere = 0.62m • Vcube = 1 m3 = side3cube • Sidecube= 1 m • SurfaceAreasphere= 4 π rsphere2 = 4.84 m2 • SurfaceAreacube= 6 side2cube = 6 m2 • Ratio = SurfaceAreacube / SurfaceAreasphere ≈1.24

10. Circles Inscribed on Spheres • Circle at equator is exactly like circles you’ve studied • X2 + Y2 = R2 • Other circles on axes are easy to understand • X2 + Z2 = R2 • Z2+ Y2 = R2 • Off-axis equations get more complicated • Not all circles are full radius

11. Circles Inscribed on Spheres Longitude • For earth, one set of circles inscribed are called “latitude”, others are called “longitude” • Lat Radius at equator = R • Lat Radius at north/south pole = 0 • In between we can use a lookup table or trigonometry to find radius • Note that on the sphere drawn, only latitude lines have radius <r Latitude

12. Line Segments on Spheres - Arc • What if the circle doesn’t go all the way around? • Called an “Arc” • Can calculate “Great Circle Distance” – shortestdistance on sphere • Very useful if you are a airplane pilot or sea captain • Even useful later today • When flying from New York to Beijing, the great circle route passes close to?

13. Example: Fly BostonBejing • Is this what you’d do? • Projections distort

14. Google Earth Plot of BostonBejing • If you flew west: d = ~0.75 * 172.4ᶛ * 69 sm≈ 8928miles • If you fly greatcircle: 6737 miles • Very close to Santa! • ≈ 1/3 farther to go west Long Distance Arcs can be Counter-intuitive!

15. Intersection of Line with Sphere • There are only three ways for a line to ‘intersect’ with the surface of a sphere • No intersection (a.k.a. Swing and a miss!) • Tangent • Two points • Linesegmentcould intersecta fourth way – what is it? • Three intersectinglines inside a sphere form a ???

16. A Quick Aside, Useful as we Go On • An engineer’s to avoid trigonometry! • For very small angles (10ᶛ or less), you can avoid trigonometry in many cases • sin Θ≈ tan Θ ≈ Θ (radians) • cosΘ ≈ 1 • Radians = degrees * 2 π / 360ᶛ • D/L ≈ Θ (radians) D Θ L

17. Triangles Inside Spheres • Q: Compute Angle at Earth’score between Weston, MA andProvidence, Rhode Island • Answer: • Earth Radius ≈ 6378km • Distance Weston to Providence: 73 km • Θ≈D/L ≈73/6378 = 0.0114 radians • 0.0144 radians * 360 / 2 π = 0.66 degrees

18. Intersection of Plane with Sphere • Similar to a line, plane cutting through has 3 possibilities • Misses • Tangent • Segments • In the segment case, a plane cutting through a sphere creates a circle at the intersection of the two bodies • If that plane intersects the center-of-sphere, the circle will have radius r

19. Intersection of Plane w/ Sphere • Given Sphere: (x-2)2 + (y-2)2 + (z-2)2 = 1 • Given Plane: x=0 • Question: Which – Miss, tangent, segment? • Answer: • If they touch, for some x=0, Sphere equation <=1 • (0-2) 2 + (y-2)2+ (z-2)2<= 1 • (y-2)2 + (z-2)2<= -3 • But (y-2)2 >=0, and (z-2)2>=0: 0 can’t be -3 • X=0 plane isn’t inside sphere • What about x=2?

20. Matlab graph of plane and sphere • Easy when you can visualize it, isn’t it? • The message: Make sure you visualize 3-D geometry problems before getting lost in calculation

21. Triangle Around Sphere • As we’ll discover, satellites may communicate with one-another • What shape does one need to ensure earth doesn’t block communication? • Line? • Triangle? • Square?

22. Example: Minimum Triangle • Q: Minimum equilateral triangle size around sphere of radius r • Answer: • Intersection of bisectors is the center of the incircle (sphere!) • Area of triangle =√ ¾ β2 • r = 2 triangle area / perimeter length • r =2 √ ¾ β2/ (3 β) • r= √ 3 β / 6 • For earth, we’ll see some satellites get away with a triangle, some need more β β β

23. Outline • Basics of 3-D Circles – Spheres • Definition and Review • Some 2-D geometry interacting with Spheres • Some 3-D geometry interacting with Spheres • Spherical Coordinates • Applications to Satellites (Orbitology)

24. Spheres within Cylinders 1 of 2 • When a Sphere ‘exactly fits’ inside a cylinder (i.e. a cylinder of height 2r and radius r), the ratio of their volumes form is surprisingly simple h

25. Spheres within Cylinders 2 of 2 • Sphere volume: • Cylinder volume: • But h = 2r • / h

26. “Touch and Go” on Spherical Triangles • Three joined arcs on the surface of a sphere • Planar triangle, angles sum to 180ᶛ. • Spherical? • Excess E = α + β + γ - 180ᶛ • Area = R2 ∙ E (in radians) • Useful to pilots flying three legged routes α γ β

27. Spherical Coordinates • Angles in 2-D geometry are often measured w.r.t. an assumed axis or another segment • Angles in 3-D geometry are often measured w.r.t. an axis • Ө w.r.t x-axis • Ф w.r.t z-axis • (ρ, Ө, Ф) instead of (x,y,z) • Sphere equation in Spherical Coordinates:r = ρ

28. Spherical Coordinate Example • Q: Rotate P=(1/√3,1/√3,1/√3) aboutxy plane by 80ᶛand find point P’ • Answer in spherical coordinates: • ρ‘ = ρ = 1 • φ‘ = φ = 45ᶛ • Θ’ = Θ + 80ᶛ= 45ᶛ + 80ᶛ = 125ᶛ • Answer in cartesian coordinates:Lots of trigonometry! Ouch! • x = 1 sin (Ө+80) cosϕ • y = 1 sin (Ө+80) sin ϕ • z = 1 cos (Ө+80) • Sometimes (e.g. rotations),spherical coordinates area loteasier than xyz (and vice versa)

29. Outline • Basics of 3-D Circles – Spheres • Applications to Satellites (Orbitology) • Circles around Spheres • Basics of Orbits

30. Circles Around Spheres • The earth is (almost) a sphere • Circles around a sphere describe the path of an orbit

31. Different Types Orbits • Orbits have altitude (low, medium, high) and inclination (low, medium, high) • All relate to the relative size of sphere and circle radius (there are NASA definitions), or the relation of the sphere axis to imaginary circle axis 1

32. Low Earth Orbit Example: Iridium • 66 satellites, 11 satellites in each of 6-7 orbital planes (circles) • Think of earth (sphere) as rotating under fixed constellation • Very low altitude (notice distance circle-to-sphere); ~500 mi

33. Medium Earth Orbit Example: GPS • Global Positioning System • Note higher altitude (about 12,600 miles) • One of the ways your iPhones and Droids know where they are • 24 satellites (including 3 spare) • Need to “hear” 3 satellites to solve for your position on the sphere in 2 dimensions • Need 4 satellites to solve for your position in 3 dimensions – another geometry problem!

34. GEO Satellite Example: DirectTV • Geostationary orbit • Appears “fixed in sky”from any one pointon earth underneath it • Customers must initiallyaim dish towards satellite,afterwards little/noadjustment • Initial pointing angle usually approximated with right triangle

35. Earth Coverage Calculation Ө • Geometry is often approximated as ‘cone on a sphere' • Geosynchronous orbit = 42,164 km • Small angles: Ө≈d/L radians • Question: Calculate spot on earth for0.1 degree beam • Answer: • 0.1 degrees = 0.00175 radians • d ≈ ӨL  0.00175 * 42,164 = 73km wide • Approx like driving Weston MA to Providence RI • Remember triangle around sphere? If gravitywere weaker, earth would block triangle of satellites L d

36. Iridium Flares • Up to 30x brighter than Venus • Geometry of reflected sunlightonto sphere (Earth) • Not predicted beforehand –geometry worked outafter observations • Careful geometry letsyou see a 6’ manmadeobject 500 km away (about Washington DC) • http://www.satobs.org/iridium.html

37. Spheres and Applications Summary • I hope you’ve learned a little about spheres: • You knew what they were, and Volume, and SA formulas • You learned about intersections with lines & planes • You learned about being encompassed by cylinders and triangles • You learned the basics of a new coordinate system: spherical coordinates, and where it might be useful • I hope you learned a little about satellites, and how incredibly important geometry is to their design and operation

38. Conclusion • If you’ve enjoyed today, you owe me something: • Someday, even when it isn’t convenient, do what I’m doing today for the next generation – offer-up a lecture on what you know – that’s my payback

39. References • All materials and ideas herein came from the world wide web, including • Ancillary Description Writer's Guide, 2013.Global Change Master Directory.National Aeronautics and Space Administration.[http://gcmd.nasa.gov/add/ancillaryguide/]. • Ccar.colorado.edu • wikipedia.org • Mathworks.com • Iridium.com • Images.google.com

40. For Two Points of Intersection Case • Looks like quadratic formula! • l is ‘unit vector’ for a the line • o is origin of the line • r is the radius of the sphere • If value under square root < 0 – no intersection • If value under square root = 0 – tangent • If value under square root > 0 – two points

41. Pseudoranges • For simple cases, 3-D euclidean geometry

42. Heron Proof • he triangle is ABC. Draw the inscribed circle, touching the sides at D, E and F, and having its center at O. • Since OD = OE = OF, area ABC = area AOB + area BOC + area COA, • 2.area ABC = p.OD, where p = a+b+c. (a = (length of) BC, etc.) • Extend CB to H, so that BH = AF. Then CH = p/2 = s. (since BD = BF, etc.) • Thus (area ABC)2 = CH2.OD2. • Draw OL at right angles to OC cutting BC in K, and BL at right angles to BC meeting OL in L. Join CL. • Then, since each of the angles COL, CBL is right, COBL is a quadrilateral in a circle. • Therefore, angle COB + angle CLB = 180 degrees. • But angle COB + angle AOF = 180 degrees, because AO bisects angle FOE, etc., so • Angle AOF = angle CLB • Therefore, the right-angled triangles AOF, CLB are similar, and • BC:BL = AF:FO = BH:OD • CB:BH = BL:OD = BK:KD • And from CB/BH = BK/KD, adding one to each side, CH/HB = BD/DK, or • CH:HB = BD:DK • It follows that • CH2:CH.HB = BD.DC:CD.DK = BD.DC:OD2 (since angle COK = 90 degrees) • Therefore (area ABC)2 = CH2.OD2 = CH.HB.BD.DC = s(s-a)(s-b)(s-c).