Introduction to Spheres, and Some Applications

Introduction to Spheres,and Some Applications

Doug MarquisWeston High School

3 June 2013

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)

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

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

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

Review: Volume of a Sphere

• They tell me you’ve already seen this

• Enough said

Review: Surface Area of a Sphere

• They said you knew this one too

• Let’s run some numbers

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

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 rsphere

3

– rsphere = 0.62m– Vcube = 1 m3 = side3

cube

– Sidecube = 1 m– SurfaceAreasphere = 4 π rsphere

2 = 4.84 m2

– SurfaceAreacube = 6 side2cube = 6 m2

– Ratio = SurfaceAreacube / SurfaceAreasphere ≈1.24

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

Circles Inscribed on Spheres• 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

LatitudeLongitude

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?

Example: Fly BostonBejing

• Is this what you’d do?

• Projections distort

Google Earth Plot of BostonBejing

• If you flew west: d = ~0.75 * 172.4ᶛ * 69 sm≈ 8928 miles

• If you fly greatcircle: 6737 miles

• Very close to Santa!• ≈ 1/3 farther to go

west Long Distance Arcs can be Counter-intuitive!

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 ???

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)– cos Θ ≈ 1– Radians = degrees * 2 π / 360ᶛ

• D/L ≈ Θ (radians)

ΘD

L

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

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

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?

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

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?

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

ββ

β

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)

Spheres within Cylinders 1 of 2

h

• 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

Spheres within Cylinders 2 of 2

h

• Sphere volume: • Cylinder volume: • But h = 2r• /

“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

β

γ

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

Spherical Coordinate Example• Q: Rotate P=(1/√3,1/√3,1/√3) about

xy plane by 80ᶛ and 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) cos ϕ – y = 1 sin (Ө+80) sin ϕ– z = 1 cos (Ө+80)

• Sometimes (e.g. rotations),spherical coordinates area lot easier than xyz (and vice versa)

Outline

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

Circles Around Spheres

• The earth is (almost) a sphere

• Circles around a sphere describe the path of an orbit

1

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

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

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!

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

Earth Coverage Calculation• 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 for

0.1 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 gravity

were weaker, earth would block triangle of satellites

Ө

L

d

Iridium Flares• Up to 30x brighter than Venus• Geometry of reflected sunlight

onto 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

http://www.satobs.org/iridium.html

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

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

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

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

Pseudoranges

• For simple cases, 3-D euclidean geometry

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).

Introduction to Spheres, and Some Applications

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