Area of Regions on the Lunar Sphere - Revision history

Strangelv: Strangelv moved page Area of regions on the lunar sphere to Area of Regions on the Lunar Sphere: Capitalization

2019-04-10T21:47:58Z

Strangelv moved page Area of regions on the lunar sphere to Area of Regions on the Lunar Sphere: Capitalization

Laertes: Undo revision 14729 by 209.237.227.133 (Talk)

2008-12-17T00:06:14Z

Undo revision 14729 by 209.237.227.133 (Talk)

209.237.227.133: FIELD_OTHER

2008-12-16T23:33:56Z

FIELD_OTHER

59.95.209.132 at 23:31, 23 August 2008

2008-08-23T23:31:31Z

Cfrjlr: Reverted edits by 74.208.11.169 (Talk); changed back to last version by 80.38.37.26

2007-11-11T17:34:34Z

Reverted edits by 74.208.11.169 (Talk); changed back to last version by 80.38.37.26

74.208.11.169 at 14:52, 11 November 2007

2007-11-11T14:52:42Z

80.38.37.26: /* Area of regions on the lunar sphere */

2007-10-09T08:47:09Z

Area of regions on the lunar sphere

80.38.37.26: /* Area of regions on the lunar sphere */

2007-10-09T08:45:33Z

Area of regions on the lunar sphere

80.38.37.26: /* Area of regions on the lunar sphere */

2007-10-09T08:44:46Z

Area of regions on the lunar sphere

130.207.218.196: /* Area of regions on the lunar sphere */ Added math notation

2007-10-06T02:10:46Z

Area of regions on the lunar sphere: Added math notation

← Older revision		Revision as of 00:06, 17 December 2008
Line 1:		Line 1:
−	~~FIELD_MESSAGE_accnavidele~~	+	==Area of regions on the lunar sphere==
		+
		+	To calculate area on the lunar sphere bounded by pair of latitudes and a pair of longitudes:
		+
		+	The general calculation for the area on the Moon (in square kilometers) bounded by a pair of longitudes and a pair of latitudes is as follows:
		+
		+	<math>{2\pi r^2 [(1 - sin(Latitude_1)) - (1 - sin(Latitude_2))]} [(Longitude_1 - Longitude_2)/360] \,\!</math>
		+
		+	where <math>Longitude_1 \,\!</math> and <math>Longitude_2 \,\!</math> are in degrees
		+
		+	Here is the proof:
		+
		+	First, calculate the area of the strip between two parallels of latitude.
		+
		+	That is the same as the difference in the area of two spherical caps about the same apex.
		+
		+	Those areas are most easily calculated as solid angles with units of steradians.
		+
		+	The steradian is defined as "the solid angle subtended at the center of a sphere of radius r by a portion of the surface of the sphere having an area <math>r^2 \,\!</math>."
		+
		+	The area of a spherical cap (<math>A = 2\pi r h \,\!</math>), where h = distance between the apex of the cap and the center of the base plane of the cap.
		+
		+	To calculate h:
		+
		+	<math>h = r - [r sin(Latitude)] \,\!</math>
		+
		+	So:
		+
		+	Where <math>Latitude_2 > Latitude_1 \,\!</math> (i.e. <math>A1 > A2 \,\!</math>):
		+
		+	Then, for <math>Latitude_1 \,\!</math>: <math>A1 = 2\pi r [r-{ r sin(Latitude_1)}]
		+	= 2\pi r x r [1-sin(Latitude_1)]
		+	= 2\pi r ^2 [1-sin(Latitude_1)] \,\!</math>
		+
		+	And for <math>Latitude_2 \,\!</math>: <math>A2 = 2\pi r ^2 [1-sin(Latitude_2)] \,\!</math>
		+
		+	So area of strip between two latitudes is
		+
		+	<math>A(strip) = A1 - A2 = [2\pi r^2 [1-sin(Latitude_1)]]-[2 \pi r^2 [1-sin(Latitude_2)]]
		+	= 2 \pi r^2 [(1-sin(Latitude_1)) - (1-sin(Latitude_2) )]
		+	\,\!</math>
		+	That gives the area in steradians.
		+
		+	The area of a sphere is <math>4\pi*steradians = 4\pi r^2 \,\!</math>
		+
		+	The mean radius of the Moon = 1738 km<BR>
		+
		+	Surface area of the entire Moon = 37,958,532 Square kilometers<BR>
		+	<BR>
		+	Once we have calculated the area of the strip, then it is a simple matter to pro-rate for the longitude, as follows:
		+
		+	The area of the curved rectangle subtended by the pairs of latitude and longitude can be expressed as follows:
		+
		+	Area of the strip multiplied by the decimal fraction of the longitudinal circumference.
		+
		+	The decimal fraction of the longitudinal circumference (in degrees) is calculated as follows:
		+
		+	<math>(Longitude_1 - Longitude_2)/360 \,\!</math> degrees
		+
		+	Hence the general calculation for the area on the Moon (in square kilometers) bounded by a pair of longitudes and a pair of latitudes is as follows:
		+
		+	<math>{2\pi r^2 [(1-sin(Latitude_1))-(1-sin(Latitude_2))]}x[(Longitude_1-Longitude_2)/360 degrees] \,\!</math>

	==External Links==		==External Links==

← Older revision		Revision as of 23:33, 16 December 2008
Line 1:		Line 1:
−	~~==Area of regions on the lunar sphere==~~	+	FIELD_MESSAGE_accnavidele
−
−	~~To calculate area on the lunar sphere bounded by pair of latitudes and a pair of longitudes:~~
−
−	~~The general calculation for the area on the Moon (in square kilometers) bounded by a pair of longitudes and a pair of latitudes is as follows:~~
−
−	~~<math>{2\pi r^2 [(1 - sin(Latitude_1)) - (1 - sin(Latitude_2))]} [(Longitude_1 - Longitude_2)/360] \,\!</math>~~
−
−	~~where <math>Longitude_1 \,\!</math> and <math>Longitude_2 \,\!</math> are in degrees~~
−
−	~~Here is the proof:~~
−
−	~~First, calculate the area of the strip between two parallels of latitude.~~
−
−	~~That is the same as the difference in the area of two spherical caps about the same apex.~~
−
−	~~Those areas are most easily calculated as solid angles with units of steradians.~~
−
−	~~The steradian is defined as "the solid angle subtended at the center of a sphere of radius r by a portion of the surface of the sphere having an area <math>r^2 \,\!</math>."~~
−
−	~~The area of a spherical cap (<math>A = 2\pi r h \,\!</math>), where h = distance between the apex of the cap and the center of the base plane of the cap.~~
−
−	~~To calculate h:~~
−
−	~~<math>h = r - [r sin(Latitude)] \,\!</math>~~
−
−	~~So:~~
−
−	~~Where <math>Latitude_2 > Latitude_1 \,\!</math> (i.e. <math>A1 > A2 \,\!</math>):~~
−
−	~~Then, for <math>Latitude_1 \,\!</math>: <math>A1 = 2\pi r [r-{ r sin(Latitude_1)}]~~
−	~~= 2\pi r x r [1-sin(Latitude_1)]~~
−	~~= 2\pi r ^2 [1-sin(Latitude_1)] \,\!</math>~~
−
−	~~And for <math>Latitude_2 \,\!</math>: <math>A2 = 2\pi r ^2 [1-sin(Latitude_2)] \,\!</math>~~
−
−	~~So area of strip between two latitudes is~~
−
−	~~<math>A(strip) = A1 - A2 = [2\pi r^2 [1-sin(Latitude_1)]]-[2 \pi r^2 [1-sin(Latitude_2)]]~~
−	~~= 2 \pi r^2 [(1-sin(Latitude_1)) - (1-sin(Latitude_2) )]~~
−	~~\,\!</math>~~
−	~~That gives the area in steradians.~~
−
−	~~The area of a sphere is <math>4\pi*steradians = 4\pi r^2 \,\!</math>~~
−
−	~~The mean radius of the Moon = 1738 km<BR>~~
−
−	~~Surface area of the entire Moon = 37,958,532 Square kilometers<BR>~~
−	~~<BR>~~
−	~~Once we have calculated the area of the strip, then it is a simple matter to pro-rate for the longitude, as follows:~~
−
−	~~The area of the curved rectangle subtended by the pairs of latitude and longitude can be expressed as follows:~~
−
−	~~Area of the strip multiplied by the decimal fraction of the longitudinal circumference.~~
−
−	~~The decimal fraction of the longitudinal circumference (in degrees) is calculated as follows:~~
−
−	~~<math>(Longitude_1 - Longitude_2)/360 \,\!</math> degrees~~
−
−	~~Hence the general calculation for the area on the Moon (in square kilometers) bounded by a pair of longitudes and a pair of latitudes is as follows:~~
−
−	~~<math>{2\pi r^2 [(1-sin(Latitude_1))-(1-sin(Latitude_2))]}x[(Longitude_1-Longitude_2)/360 degrees] \,\!</math>~~

	==External Links==		==External Links==

@@ Line 5: / Line 5: @@
 The general calculation for the area on the Moon (in square kilometers) bounded by a pair of longitudes and a pair of latitudes is as follows:
-<math>{2\pi r^2 [{1 - sin(Latitude_1)} - {1 - sin(Latitude_2)}]} [(Longitude_1 - Longitude_2)/360] \,\!</math>
+<math>{2\pi r^2 [(1 - sin(Latitude_1)) - (1 - sin(Latitude_2))]} [(Longitude_1 - Longitude_2)/360] \,\!</math>
 where <math>Longitude_1 \,\!</math> and <math>Longitude_2 \,\!</math> are in degrees
@@ Line 38: / Line 38: @@
 <math>A(strip) =  A1 - A2   = [2\pi r^2 [1-sin(Latitude_1)]]-[2 \pi r^2 [1-sin(Latitude_2)]]
-  = 2 \pi r^2  [{1-sin(Latitude_1)} - {1-sin(Latitude_2) }]
+  = 2 \pi r^2  [(1-sin(Latitude_1)) - (1-sin(Latitude_2) )]
   \,\!</math>
 That gives the area in steradians.
@@ Line 60: / Line 60: @@
 Hence the general calculation for the area on the Moon (in square kilometers) bounded by a pair of longitudes and a pair of latitudes is as follows:
-<math>{2\pi r^2 [{1-sin(Latitude_1)}-{1-sin(Latitude_2)}]}x[(Longitude_1-Longitude_2)/360 degrees] \,\!</math>
+<math>{2\pi r^2 [(1-sin(Latitude_1))-(1-sin(Latitude_2))]}x[(Longitude_1-Longitude_2)/360 degrees] \,\!</math>
 ==External Links==

← Older revision		Revision as of 17:34, 11 November 2007
Line 1:		Line 1:
−	~~trocvidro~~
	==Area of regions on the lunar sphere==		==Area of regions on the lunar sphere==

← Older revision		Revision as of 14:52, 11 November 2007
Line 1:		Line 1:
		+	trocvidro
	==Area of regions on the lunar sphere==		==Area of regions on the lunar sphere==

@@ Line 29: / Line 29: @@
 Where <math>Latitude_2 > Latitude_1 \,\!</math> (i.e. <math>A1 > A2 \,\!</math>):
-Then, for <math>Latitude_1</math>:   <math>A1 = 2\pi r [r-{ r sin(Latitude_1)}]
+Then, for <math>Latitude_1 \,\!</math>:   <math>A1 = 2\pi r [r-{ r sin(Latitude_1)}]
                         = 2\pi r x  r [1-sin(Latitude_1)]
                         = 2\pi r ^2 [1-sin(Latitude_1)] \,\!</math>
-And for <math>Latitude_2</math>:     <math>A2  = 2\pi r ^2 [1-sin(Latitude_2)] \,\!</math>
+And for <math>Latitude_2 \,\!</math>:     <math>A2  = 2\pi r ^2 [1-sin(Latitude_2)] \,\!</math>
 So area of strip between two latitudes is

@@ Line 33: / Line 33: @@
                         = 2\pi r ^2 [1-sin(Latitude_1)] \,\!</math>
-And for <math>Latitude_2</math>:     <math>A2  = 2\pi r ^2 [1-sin(Latitude_2)]</math>
+And for <math>Latitude_2</math>:     <math>A2  = 2\pi r ^2 [1-sin(Latitude_2)] \,\!</math>
 So area of strip between two latitudes is
@@ Line 39: / Line 39: @@
 <math>A(strip) =  A1 - A2   = [2\pi r^2 [1-sin(Latitude_1)]]-[2 \pi r^2 [1-sin(Latitude_2)]]
   = 2 \pi r^2  [{1-sin(Latitude_1)} - {1-sin(Latitude_2) }]
-</math>
+ \,\!</math>
 That gives the area in steradians.
-The area of a sphere is  <math>4\pi*steradians =  4\pi r^2</math>
+The area of a sphere is  <math>4\pi*steradians =  4\pi r^2 \,\!</math>
 The mean radius of the Moon = 1738 km<BR>
@@ Line 56: / Line 56: @@
 The decimal fraction of the longitudinal circumference (in degrees) is calculated as follows:
-<math>(Longitude_1 - Longitude_2)/360</math> degrees
+<math>(Longitude_1 - Longitude_2)/360 \,\!</math> degrees
 Hence the general calculation for the area on the Moon (in square kilometers) bounded by a pair of longitudes and a pair of latitudes is as follows:
-<math>{2\pi r^2 [{1-sin(Latitude_1)}-{1-sin(Latitude_2)}]}x[(Longitude_1-Longitude_2)/360 degrees]</math>
+<math>{2\pi r^2 [{1-sin(Latitude_1)}-{1-sin(Latitude_2)}]}x[(Longitude_1-Longitude_2)/360 degrees] \,\!</math>
 ==External Links==

@@ Line 5: / Line 5: @@
 The general calculation for the area on the Moon (in square kilometers) bounded by a pair of longitudes and a pair of latitudes is as follows:
-<math>{2\pi r^2 [{1 - sin(Latitude_1)} - {1 - sin(Latitude_2)}]} [(Longitude_1 - Longitude_2)/360]</math>
+<math>{2\pi r^2 [{1 - sin(Latitude_1)} - {1 - sin(Latitude_2)}]} [(Longitude_1 - Longitude_2)/360] \,\!</math>
-where <math>Longitude_1</math> and <math>Longitude_2</math> are in degrees
+where <math>Longitude_1 \,\!</math> and <math>Longitude_2 \,\!</math> are in degrees
 Here is the proof:
@@ Line 17: / Line 17: @@
 Those areas are most easily calculated as solid angles with units of steradians.
-The steradian is defined as "the solid angle subtended at the center of a sphere of radius r by a portion of the surface of the sphere having an area <math>r^2</math>."
+The steradian is defined as "the solid angle subtended at the center of a sphere of radius r by a portion of the surface of the sphere having an area <math>r^2 \,\!</math>."
-The area of a spherical cap (<math>A = 2\pi r h</math>), where h = distance between the apex of the cap and the center of the base plane of the cap.
+The area of a spherical cap (<math>A = 2\pi r h \,\!</math>), where h = distance between the apex of the cap and the center of the base plane of the cap.
 To calculate h:
-<math>h =  r - [r sin(Latitude)]</math>
+<math>h =  r - [r sin(Latitude)] \,\!</math>
 So:
-Where <math>Latitude_2 > Latitude_1</math> (i.e. <math>A1 > A2</math>):
+Where <math>Latitude_2 > Latitude_1 \,\!</math> (i.e. <math>A1 > A2 \,\!</math>):
 Then, for <math>Latitude_1</math>:   <math>A1 = 2\pi r [r-{ r sin(Latitude_1)}]
                         = 2\pi r x  r [1-sin(Latitude_1)]
-                        = 2\pi r ^2 [1-sin(Latitude_1)]</math>
+                        = 2\pi r ^2 [1-sin(Latitude_1)] \,\!</math>
 And for <math>Latitude_2</math>:     <math>A2  = 2\pi r ^2 [1-sin(Latitude_2)]</math>