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Final Answers 
© 2000-2014   Gérard P. Michon, Ph.D.

Geography 

 The Planet Earth

Geography, the best subject on Earth.     Got it?

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Related Links (Outside this Site)

Why is the Ocean Salty? by Herbert Swenson.
Coordinate Systems Overview  by  Peter H. Dana.
Cartographie  in "Archéologie Aérienne"  by  Jacques Dassié.

Wikipedia :  Figure of the Earth

 
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Geodesy, Cartography, Geography


(2010-10-14)   Geodetic Reference System  (GRS-1980)
Defining the geodetic latitude (j), longitude (q) and elevation (h) of  M.

The so-called  Geoid  is essentially the  mean sea level  surface of the Earth.  Irregularities in the gravitational field of the Earth  (due to irregularities in its mass distribution)  make the actual Geoid a very complicated surface...

 Geodetic latitude and elevation To a very good approximation, that irregular  Geoid  matches the shape of a perfect spheroid,  standardized by the IUGG in 1980 (and known as the  Reference Ellipsoid):
  • Equatorial radius  a = 6378137 m
  • Polar radius  b = 6356752.3141 m
The meridian of that surface at any  longitude  q  is a half-ellipse where a point is identified by its [geodetic]  latitude  j, which is defined as the angle from the equator to the  upward geodetic vertical  (itself  defined  to be perpendicular to the surface of the spheroid, it is virtually indistinguishable from the direction of a plumb line).

At zero longitude  (y = 0)  the cartesian equation of the meridian is:

(x/a) 2  +  (z/b) 2   =   1

Differentiating that equation or writing that  (cos j, sin j)  is perpenticular to  (dx,dz)  we obtain two separate relations, which must be proportional:

(x / a 2 ) dx  +  (z / b 2 ) dz   =   0     and     cos j  dx  +  sin j  dz   =   0

Calling  Rj  the coefficient of proportionality between them,  we have:

x   =   (a2 / Rj ) cos j     and     z   =   (b2 / Rj ) sin j

Plugging those two values into the above equation of the ellipse, we obtain:

Rj2   =   a2 cos2 j  +  b2 sin2 j

The coordinates of a point of nonzero elevation h  (along the geodetic vertical)  are obtained by adding the following corrections to the above:

Dx   =   h cos j     and     Dz   =   h sin j

So, the conversion formulas from geodetic to 3D cartesian coordinates are:

Cartesian coordinates of a point of elevation  h
at 
 [ geodetic ]  latitude  j  and longitude  q :
  x   =   ( a 2/ Rj + h )  cos j  cos q  
y   =   ( a 2/ Rj + h )  cos j  sin q
z   =   ( b 2/ Rj + h )  sin j
where   Rj2   =   a2 cos2 j  +  b2 sin2 j

As shown below, the quantity  Rj  can be given the following  differential  characterization, in terms of the distance  r  to the center of the Earth:

Rj + h   =   r ( ¶r / ¶h )j

Straight-line distance (through the Earth) of two Cities  (Grand Rapids & Melbourne)
Mean Sea Level, GPS and the Geoid  by  Witold Fraczek.
Derived geometrical constants of GRS-1980  by  Miloš Cimbálnik & M. Pick  (1987)
"Geodetic Reference System" 1980 (GRS-1980)   by H. Moritz  [ 1 | 2 | 3 ]
Ellipsoidal density models and hydrostatic equilibrium  by  Marussi, Moritz, Rapp & Vicente  (1974)


(2011-06-28)   Geocentric Coordinates
Geocentric latitude  (j0 )  is almost never used...

Only the above  geodetic coordinates  are used in geography.  Geocentric coordinates  are all but useless, except as  approximations  in physics.

The  geocentric latitude  of a point  (at elevation  h = 0)  on the surface of the Earth is the angle between the plane of the equator and the  radius vector  originating from the center of the Earth.  That unused flavor of latitude is denoted by  j0.  At  q = 0, its tangent is  z/x, which yields:

  tg j0   =   ( b 2/ a 2 )  tg j  

Numerically:   tg j0  =  0.99330562  tg j


(2011-06-28)   Radius of the Earth at a given latitude  j.
Distance to the center of a point located at the surface of the Earth.

The actual distance  r  of a point to the center of the Earth is given by:

r2   =   x2 + y2 + z2   =   ( a 2/ Rj + h ) 2 cos2 j  +  ( b 2/ Rj + h ) 2 sin2 j
=   (Rj4 - 2 a2 b2 sin2 j cos2 j ) / Rj2  +  2 h Rj  +  h2
=   (Rj + h) 2 - 2 (a b sin j cos j / Rj ) 2

Although Mount Everest is the peak of highest elevation  (h = 8848 m)  its distance to the center of the Earth  (r = 6382.3 km)  is about  2 km  less than the record  (r = 6384.4 km)  achieved at the top of the  Chimborazo volcano in Ecuador  (h = 6268 m).  Nevado Huascarán in Yungay  (Peru)  is a  close  second  (by only 18 m).

Distances to the center of the Earth of some famous peaks :
# SummitElevation hLatitude j Radius r
1 Volcán Chimborazo, Ecuador 6268 m-1.469167°    6384391 m  
2 Nevado Huascarán, Peru 6769 m-9.117078°  6384373 m
  Kilimanjaro, Tanzania 5895 m-3.076475°  6383971 m
  Mount Everest, Nepal/Tibet 8848 m+27.988257°  6382306 m
  Mont Blanc, France/Italy 4810 m+45.832866°  6371989 m
  Mount McKinley, Alaska 6194 m+63.069042°  6367362 m
  Vinson Massif, Antartica 4897 m-78.525483°  6362502 m

The top of  Mont Blanc  is actually at the same distance from the center of the Earth as a point  6148 m  below  sea-level at the equator.

The radius of a sphere having the same volume as the reference ellipsoid is:

(a2b) 1/3   =   6371000.79 m

The rounded value  R = 6371 km  is the  conventional radius of the Earth.  It's the distance from the center of the Earth to the surface of the ocean at a latitude of  35.402807°.  The top of  Mont Blanc  is just  989 m  further out.


Jennifer Bishop (Wake Forest, NC.  2004-05-21; e-mail)
What is the volume of the Grand Canyon?

Only a rough number can be given, since the object itself is not even precisely defined geometrically.  Here's the data on which we may base an estimate:

  • Length (from Lees Ferry to the Grand Wash) :  277 miles
  • Average width (rim-to-rim) :  10 miles.
  • Average height of rims above the river:  1 mile.

Taking these numbers at face value, we obtain a volume of 2770 cubic miles.  Considering that this is probably an overestimate (the walls are not vertical, etc.) and that three significant digits would be a misleading representation of the precision thus obtained, we should probably state that the volume of the Grand Canyon is about 2500 cubic miles or around:

1013 m3

This corresponds to about 80% of the volume of Lake Superior or 40% the volume of Lake Baikal (Siberia) and would represent a thickness of around 3/4"  (2 cm)  if spread over the entire surface of the Earth.


(2003-11-01)
What is the oldest city in the World?

Among the contenders below, the title should probably go to Jericho...

Jericho, "West Bank"

A green oasis in the Jordan Valley, 20 km east of Jerusalem (7 km west of the Jordan River and 10 km north of the Dead Sea) Jericho lies 250 m below sea level.  It's the lowest major city in the world and one of the oldest continuously inhabited places, with vestiges dating back to about 8000 BC.

Damascus, capital of Syria

Also known as Al-Sham or Dimashq, Damascus is on record as the capital of an Aramaean kingdom conquered by the Assyrians in 732 BC.  According to ancient Egyptian manuscripts, it was already the capital of a small Aramaean principality as early as the 15th century BC.  The area may have been settled as early as 6000 BC, or even 8000 BC, like Jericho.

Kerma (or Karmah) capital of ancient Nubia

Karmah an Nuzul  is a town of modern Sudan, built on the ruins of the capital of a kingdom which once rivaled the contemporary achievements of ancient Egypt.  Nubia is a region of Black Africa surrounding the Nile valley, between present-day Karthoum (Sudan) and Aswan (Egypt).  Egyptian and Nubian civilizations may have borrowed heavily from each other, but they were once clearly distinct.

Ceramics were found in Nubia dating back to 8000 BC which may predate equivalent "Egyptian" achievements  ("Egypt" was politically unified by the first pharaohs, around 3100 BC).  The urban ruins at Karmah seem to be the most ancient on the African continent.

The region was known as "Kush" to the ancient Egyptians, but this term is now best reserved to a more recent kingdom (capital: Napata) which regained its independence from Egypt around 850 BC and went on to conquer and reunite all Egyptian principalities, around 750 BC.  Egypt's "25th Dynasty" thus consisted of Kushite pharaohs who were perceived as foreigners.  They did not lose control of Egypt until Assyrians invaded the country, almost a century later (674-663 BC).

The earliest settlers of Kerma seem unrelated to the  Badarians  who are apparently, the direct ancestors of the ancient Egyptians.

The Predynastic Period of "Egypt" is often divided into 3 parts, named after archaeological sites:  Badarian (Tasian), Amratian (Naqada I), and Gerzean (Naqada II).  The stone-age Badarian culture existed as early as 4400 BC  or, possibly, 5000 BC.

Rome, capital of Italy

According to its own legend, the "Eternal City" was only founded in 754 BC.

Teotihuacán, pre-Columbian central Mexico

This is the great archeological site which contains the ruins of the oldest city in the Americas.  Teotihuacan  means "City of the Gods" in Nahuatl.  It was probably founded around 300 BC and became prominent before the beginning of the Christian era.  At its peak, around AD 500, the city was home to about 200 000 people, surpassing what would be the size of Shakespeare's London, a millenium later.  By AD 700, Teotihuacan had been sacked and burned (most probably by Toltec invaders), but its regime had lasted longer than the Roman Empire.  Teotihuacan's principal axis, the Avenue of the Dead, is dominated by one of the largest monuments of the ancient world, the 65 m Pyramid of the Sun  (completed before 150 BC).

Cities of the Ancient World by George Modelski

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United States of America (& Territories)
50 State Quarters ® Program (Statehood Coins)

Name, Postal Code CapitalStatehood License PlatesArea Code(s)
Alabama ALMontgomery1819 (22) Heart of Dixie205 251 256 334
Alaska AKJuneau1959 (49) The Last Frontier907
American
Samoa
ASPago Pago1900 ( T ) Motu o Fiafiaga684
Arizona AZPhoenix1912 (48) Grand Canyon State480 520 602 623 928
Arkansas ARLittle Rock1836 (25) The Natural State501 870
California CASacramento1850 (31) The Golden State209 213 310 323 408
415 510 530 559 562
619 626 650 661 707
714 760 805 818 831
858 909 916 925 949
Colorado CODenver1876 (38) Colorful303 719 720 970
Connecticut CTHartford1788 (05) Constitution State203 860
Delaware DEDover1787 (01) The First State302
District of Columbia DCWashington  Nation's Capital
Celebrate & Discover
202
Florida FLTallahassee1845 (27) Sunshine State 305 321 352 386 407
561 727 754 786 813
850 863 904 941 954
Guam GUHagåtña1898 ( T ) Where America's Day Begins671
Georgia GAAtlanta1788 (04) Peach State229 404 478 678 706
770 912
Hawaii HIHonolulu1959 (50) Aloha State808
Idaho IDBoise1890 (43) Famous Potatoes208
Illinois ILSpringfield1818 (21) Land of Lincoln217 309 312 618 630
708 773 815 847
Indiana INIndianapolis1816 (19) Hoosier State219 260 317 574 765
812
Iowa IADes Moines1846 (29) The Corn State319 515 641 712
Kansas KSTopeka1861 (34) The Wheat State316 620 785 913
Kentucky KYFrankfort1792 (15) Bluegrass State270 502 606 859
Louisiana LABaton Rouge1812 (30) Sportsman's Paradise225 318 337 504 985
Maine MEAugusta1820 (23) Vacationland207
Marshall Islands MHMajuroIndependent
since 1986
 +692 [country code]
Maryland MDAnnapolis1788 (07) Old Line State240 301 410 443
Massachusetts MABoston1788 (06) The Spirit of America339 351 413 508 617
774 781 857 978
Michigan MILansing1837 (26) Great Lakes231 248 313 517 586
616 734 810 906 989
Micronesia FMKolonia   691
Minnesota MNSaint Paul1858 (32) 10,000 Lakes218 320 507 612 651
763 952
Mississippi MSJackson1817 (20) The Hospitality State228 601 662
Missouri MOJefferson City1821 (24) Show Me State314 417 573 636 660
816
Montana MTHelena1889 (41) The Treasure State
Big Sky Country
406
Nebraska NELincoln1867 (37) Cornhusker State308 402
Nevada NVCarson City1864 (36) The Silver State702 775
New Hampshire NHConcord1788 (09) Live Free or Die603
New Jersey NJTrenton1787 (03) Garden State201 609 732 856 908
973
New Mexico NMSanta Fe1912 (47) Land of Enchantment505
New York NYAlbany1788 (11) The Empire State212 315 347 516 518
607 631 646 716 718
845 914 917
North Carolina NCRaleigh1789 (12) Tarheel State
First in Flight
252 336 704 828 910
919 980
North Dakota NDBismark1889 (39) Peace Garden State701
North Marianas MPSaipan1947 ( T )  670
Ohio OHColumbus1803 (17) Buckeye State
Birthplace of Aviation
216 234 330 419 440
513 614 740 937
Oklahoma OKOklahoma City1907 (46) Sooner State405 580 918
Oregon ORSalem1859 (33) Beaver State503 541 971
Pennsylvania PAHarrisburg1787 (02) Keystone State215 267 412 484 570
610 717 724 814 878
Puerto Rico PRSan Juan1952 ( T ) Isla Del Encanto787 939
Rhode Island RIProvidence1790 (13) Ocean State401
South Carolina SCColumbia1788 (08) Palmetto State803 843 864
South Dakota SDPierre1889 (40) Rushmore Memorial605
Tennessee TNNashville1796 (16) Volunteer State423 615 731 865 901
931
Texas TXAustin1845 (28) The Lone Star State210 214 254 281 361
409 469 512 682 713
806 817 830 832 903
915 936 940 956 972
979
Utah UTSalt Lake City1896 (45) Beehive State435 801
Vermont VTMontpelier1791 (14) Green Mountains802
Virginia VARichmond1788 (10) Old Dominion276 434 540 571 703
757 804
Virgin Islands VICharlotte Amalie1917 ( T ) American Paradise340
Washington WAOlympia1889 (42) Evergreen State206 253 360 425 509
West Virginia WVCharleston1863 (35) Wild, Wonderful304
Wisconsin WIMadison1848 (30) America's Dairyland262 414 608 715 920
Wyoming WYCheyenne1890 (44) The Spirit of '76307
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 (c) Copyright 2000-2014, Gerard P. Michon, Ph.D.