DIRECTION FOR KABAH (Mathematical) - FROM ANYWHERE
Dr.Mohibullah N.Durrani,
National Coordinator for
Astronomical Information,
Islamic Society of North America (ISNA).
Copyright (c) 1994
[Permission for free distribution is given to all.]
The calculation for the Mathematical Direction for Kabah from any point
on the surface of the world is a comparatively simple calculation.
(NOTE: I am NOT presenting a Proof by Fiqh. I am only presenting the
information, based on Spherical Tigonometry.)
The calculations are based on SPHERICAL TRIGONOMETRY.
(The actual shape of the earth can be better approximated as an
oblate ellipsoid of revolution, or spheroid. For the present,
only a spherical shape has been considered. The variation is
not much, for the purposes of determining the direction for the Kabah.
In future, the other shapes will be included, inshallah.)
There are some references given at the end of this document.
The information is presented in the following sections:
A. Overview.
B. True Directions (and Magnetic) for the Kabah
from various cities around the world.
C. Spherical Trigonometry (Fundamentals).
D. Examples: 1. Points around the North Pole (NP),
all on the same Latitude, NEAR THE NP.
2. Points around the North Pole,
on constant Latitudes, upto Equator.
3. Points on the same Latitude as Kabah.
4. Points all around Kabah (0 Deg to 360 Deg).
5. Points on the same Longitude as Kabah.
6. Points at approximately the center of
U.S.A., France, Australia, Japan.
E. References.
A. OVERVIEW
***********
SPHERICAL TRIGONOMETRY IS USED. This is because the Latitudes, Longitudes,
and the Cardinal Directions (North, N; South, S; East, E; and West, W)
are all defined on an equivalent sphere representing the Earth.
The Equivalent Earth redius is 6,378.14 Kilometers.
THE SHORTEST DISTANCE BETWEEN ANY TWO POINTS IS ALONG A GREAT CIRCLE.
*********************************************************************
Consider two places "A" and "B".
If "B" is North-East (N-E) of "A", then IT IS POSSIBLE
that "A" is North-West (N-W) of "B" (Note: NOT South-West, S-W) !!!
This is because of the definitions of N, S, E, and W directions and
the changing North direction (when viewed globally) from point to point.
Please see the examples presented in Section D.
IN NORTH AMERICA, THE QIBLA IS MOSTLY NORTH-OF-EAST.
****************************************************
Please see Section B for actual values of the directions
for the Qibla at places all around the globe.
COUNTRY Latitude Longitude QIBLA DIRECTION (From TRUE NORTH)
==================================================================
Australia: 30 Deg S 130 Deg E 289 Deg E-of-N
(19 Deg N of W)
France: 43 Deg N 3 Deg E 113 Deg E-of-N
(23 Deg S of E)
Japan: 35 Deg N 140 Deg E 293 Deg E-of-N
(23 Deg N of W; NOT S of W !!)
U.S.A.: 40 Deg N 100 Deg W 39 Deg E-of-N
(51 Deg N of E; NOT S of E !!)
The difference between the TRUE NORTH direction and the MAGNETIC NORTH
direction varies from place to place and also varies slightly every year.
The "Variation" between the True and Magnetic North, at places around
the world, can vary upto nearly 25 Degrees East and 25 Degrees West,
for the list of places mentioned in Section B, and the actual value of
the variation can be easily obtained from Reference 2.
"AND ALLAH (swt) KNOWS BEST, AND HE IS MOST WISE"
************************************************
B. KIBLA DIRECTION
******************
[======================================================================]
| |
| QIBLA DIRECTIONS FROM AROUND THE WORLD |
| ************************************** |
| 1 Muharram 1415 |
| Dr.Mohibullah N.Durrani 11 June 1994 |
| Copyright (c) 1994 |
| |
| [Permission for free distribution is given to all.] |
| |
+======================================================================+
| D I R E C T I O N O F Q-I-B-L-A, EAST-of-NORTH (AZ) |
|======================================================================|
| CITY - Country Lat Long | From | From | 1992 Annual 1995 |
| (N +) (E +) | TRUE | MAGNT | Var Drift Total |
| (S -) (W -) | NORTH | NORTH | Deg Min Var-Min |
+===============================|=======|=======|======================|
| | | | |
| MAKKAH-S.Arab 21.4 39.8 | - | - | - - - |
| DAR SALAM-Tnz -6.8 39.2 | 1.2 | 3.1 | 2.0W 1.2E 1.9W |
| MOSCOW - USSR 55.8 37.6 | 176.4 | 168.0 | 8.3E 1.3E 8.4E |
| ISTANBUL-Turk 41.0 28.9 | 151.5 | 148.1 | 3.4E 0.8E 3.4E |
| CAIRO - Egypt 30.1 31.3 | 136.7 | 34.3 | 2.3E 1.2E 2.4E |
| KHARTOUM-Sudn 15.5 32.6 | 48.1 | 46.7 | 1.3E 1.7E 1.4E |
| CAPETOWN-S.Af -33.8 18.6 | 23.2 | 46.2 | 22.8W 3.0W 23.0W |
| LAGOS-Nigeria 6.5 3.4 | 63.3 | 68.2 | 5.2W 6.8E 4.9W |
| | | | |
| ALGIERS-Algra 36.8 3.0 | 105.5 | 107.1 | 1.9W 6.2E 1.6W |
| GREENWICH-Engl 51.5 0.0 | 119.2 | 123.7 | 4.8W 6.8E 4.5W |
| DAKAR-Senegal 14.7 -17.5 | 74.0 | 84.7 | 11.0W 7.1E 10.7W |
| | | | |
| RIO DeJENEIRO -22.9 -43.2 | 67.6 | 88.3 | 20.4W 6.2W 20.7W |
| PARAMARIBO-Suri 5.9 -55.2 | 68.2 | 83.9 | 15.4W 6.3W 15.7W |
| BUNOS ARESarg -34.7 -58.4 | 76.3 | 81.6 | 4.9W 7.8W 5.3W |
| LIMA-Peru S.A -12.4 -77.0 | 72.2 | 70.7 | 2.0E 10.6W 1.5E |
| | | | |
| HALIFAX -CNDA 44.6 -63.6 | 65.6 | 85.9 | 20.4W 1.3E 20.3W |
| NEW YORK -USA 40.8 -74.0 | 58.5 | 73.3 | 14.6W 3.7W 14.8W |
| WASH DC -USA 38.9 -77.0 | 56.6 | 66.9 | 10.0W 5.2W 10.3W |
| MIAMI - USA 25.8 -80.2 | 56.6 | 60.7 | 3.7W 8.2W 4.1W |
| CHICAGO - USA 41.2 -87.6 | 48.8 | 50.7 | 1.6W 5.4W 1.9W |
| DALLAS - USA 32.8 -96.8 | 43.5 | 37.9 | 5.8E 4.0W 5.6E |
| DENVER - USA 39.7 -105.0 | 35.0 | 23.9 | 11.3E 3.3W 11.1E |
| SAN DIEGO-USA 32.7 -117.1 | 25.4 | 12.2 | 13.3E 1.7W 13.2E |
| SAN FRANCISCO 37.7 -122.4 | 18.9 | 3.3 | 15.8E 3.3W 15.6E |
| VANCOUVER-CND 49.3 -123.1 | 16.7 | 356.2 | 20.8E 6.7W 20.5E |
| | | | |
| ANCHORAGE-Als 61.1 -150.0 | 350.8 | 326.9 | 24.2E 5.5W 23.9E |
| HONOLULU -Hwi 21.3 -157.5 | 337.3 | 326.6 | 10.8E 2.8W 10.7E |
| | | | |
| SYDNEY-Austra -33.9 151.2 | 277.5 | 265.1 | 12.4E 0.9E 12.5E |
| TOKYO - Japan 35.7 139.7 | 293.0 | 299.8 | 6.7W 1.2W 6.8W |
| BEIJING-China 39.9 116.4 | 278.9 | 284.8 | 5.9W 0.2W 5.9W |
| JAKARTA-Indon -6.3 106.9 | 295.1 | 295.1 | 0.0W 0.7W 0.0W |
| DHAKA -B.Desh 23.8 90.3 | 277.5 | 278.2 | 0.7W 1.0E 0.7W |
| AGRA - India 27.2 77.9 | 269.0 | 268.9 | 0.0E 1.1E 0.1E |
| PESHAWAR -Pak 33.6 71.4 | 254.5 | 252.5 | 2.0E 0.7E 2.0E |
| BUKHARA -Uzbk 39.6 64.6 | 236.6 | 232.6 | 4.0E 0.3E 4.0E |
| TEHRAN - Iran 35.7 51.4 | 218.4 | 214.8 | 3.6E 0.0E 3.6E |
| | | | |
[======================================================================]
C. SPHERICAL TRIGONOMETRY
*************************
I will present, inshallah, some basics of Spherical Astronomy and
then obtain the direction for Kabah from some points on the globe.
The EARTH can be considered as an APPROXIMATE SPHERE.
The variation between the actual shape of the earth (oblate ellipsoid, or
spheroid) and the approximating sphere is not more than a few degrees in
azimuth and is not very significant for the purpose of determination of
the direction of the Kabah.
The earth can be sliced "vertically" by LONGITUDES, from 0 Deg Longitude,
through Greenwich, England, to 180 Deg East (E) Longitude, and
from 0 Deg Longitude to 180 Deg West (W) Longitude.
The earth can also be sliced "horizontally" by LATITUDES, from 0 Deg Latitude
(the Equator) to 90 Deg North (N) Latitude (the North Pole), and
from 0 Deg Latitude (Equator) to 90 Deg South (S) Latitude (the South Pole).
The Latitude is the angle at the center of the earth, from the Equator to
the point on the surface of the earth, on the same Longitude.
The intersection of the surface of a sphere with a plane through the
center of the sphere is a circle called the GREAT CIRCLE.
Longitudes are all Great Circles. The Equator is also a Great Circle.
When two Great Circles intersect, the angle between the tangents of the
two Great Circles, at the point of intersection, is the SPHERICAL ANGLE.
***************************************************************************
* THE SHORTEST DISTANCE BETWEEN ANY TWO POINTS ON THE SURFACE OF A SPHERE *
* IS ALONG A GREAT CIRCLE *
* BASED ON THE PRINCIPLES OF SHPERICAL TRIGONOMETRY *
***************************************************************************
>From any point on the earth, the North (N) direction is defined as
the direction pointing to the North Pole. The AZIMUTH ( [AZ], angular
direction, in degrees, of any one point from any other point) is the
angular rotation FROM TRUE NORTH (not magnetic North) in a ClockWise (CW)
direction.
North (N) is 0 Deg CW from N [AZ = 0]
East (E) is 90 Deg CW from N [AZ = 90]
South (S) is 180 Deg CW from N [AZ = 180]
West (W) is 270 Deg CW from N [AZ = 270]
North-East (N-E) is 45 Deg CW from N [AZ = 45]
South-East (S-E) is 135 Deg CW from N [AZ = 135]
South-West (S-W) is 225 Deg CW from N [AZ = 225]
North-West (N-W) is 315 Deg CW from N [AZ = 315]
(S is also 180 Deg CCW from N) [AZ = 180]
(W is also 90 Deg CCW from N) [AZ = 270]
(CCW = CounterClockWise)
***************************************************************
* THE NORTH DIRECTIONS FROM POINTS ON THE SAME LATITUDE *
* ARE NOT PARALLEL TO ONE ANOTHER. *
* (Only exception are points on 0 Deg latitude, the Equator.) *
***************************************************************
In a Spherical Triangle, three points on the surface of a sphere are joined
by three lines on the surface of the sphere. Each of the lines (shortest
distance) joining any two of the three points are on a Great Circle.
The vertex A can represent the North Pole, P,
the vertex B can represent the location of Kabah, K, and
the vertex C can represent any City,
from where the direction of Kabah needs to be located.
A, North Pole (P)
^
/ \
/ ------------- \
/ A, Angle \
/ \ c, Side
/ \
/ \
b, Side / \
/ B, Angle / \
/ / |
/ ------/----v B, Kabah (K)
|\ C, Angle -----------
| \ -----------
v---\----- a, Side
C, City
The SIDES are curves on the spherical surface of the sphere,
ON GREAT CIRCLES (it is difficult to represent curves in ASCII file,
as shown above, but please use your imagination, hopefully just a little !!!)
The length of SIDES are represented by "degrees" at the center of the sphere.
The three Spherical SIDES are denoted by LOWER CASE LETTERS, ex. a,b,c.
The three Spherical ANGLES are denoted by UPPER CASE LETTERS, ex. A,B,C.
The Angles are at the intersection points (vertices) of the sides, and
the Sides are the ones opposite to the intersection points (angles).
Spherical Trigonometric formulae exist which relate the sides and angles
of Spherical Triangles. We shall use one such formula.
THE FOUR PART FORMULA
=====================
cos B COS A = sin b cot c - SIN A COT C
rearranging the terms, we obtain
__ __
-1 | SIN A |
C = TAN | ----------------------------------- |
|__ sin b cot c - cos b COS A __|
Note: UPPER CASE Letters are ANGLES,
Lower Case Letters are Sides.
Computed answers could be supplements, compliments, etc.
of actual values depending on your computing machine's
evaluation of trigonometric functions.
Let us now consider a numerical example, "near" the North Pole.
The formula is valid for any choice of points B and C.
(For directions for the Qibla, please see later section of this document.)
Consider two points "B" and "C" on the SAME Latitude of 80 Degrees North.
Let the Longitude of "B" be 0 Degrees and that of "C" be 90 Degrees East.
Then, the spherical triangle would have:
Angle "A" = 90 Degrees (the difference in the Longitudes of "B" and "C")
Side "b" = 10 Degrees (90 - Latitude of Point "C" = 90 - 80 = 10 Deg)
Side "c" = 10 Degrees (90 - Latitude of Point "B" = 90 - 80 = 10 Deg)
__ __
-1 | SIN A |
C = TAN | ----------------------------------- |
|__ sin b cot c - cos b COS A __|
__ __
-1 | SIN 90 |
C = TAN | --------------------------------------- |
|__ sin 10 cot 10 - cos 10 COS 90 __|
__ __
-1 | 1.000 |
C = TAN | --------------------------------------- |
|__ sin 10 cot 10 - cos 10 COS 90 __|
C = 45.44 Degrees
Note: Even though B and C are on the SAME LATITUDE,
B is approximately North-East (45 Deg E of N) of C !!!
D. EXAMPLES
***********
All the examples in this section do NOT include the variation of
the Magnetic North Pole from the (True) North Pole. The values
given in Section B includes, in addition, the Qibla direction as read
from a magnetic compass, and hence from the Magnetic North Pole.
Examples: 1. Points around the North Pole (NP),
all on the same Latitude, NEAR THE NP.
2. Points around the North Pole,
on constant Latitudes, upto Equator.
3. Points on the same Latitude as Kabah.
4. Points all around Kabah (0 Deg to 360 Deg).
5. Points on the same Longitude as Kabah.
6. Points at approximately the center of
U.S.A., France, Australia, Japan.
EXAMPLE D-1: Points around the North Pole (NP), all on the same Latitude,
=========================================================================
NEAR THE NP.
============
This is an INTERESTING EXAMPLE. It seems contrary to "intuition".
Many CONCEPTS OF SPHERICAL TRIGONOMETRY can be understood by this
"extreme" case of pints on a constant Latitude circle around and
NEAR THE NORTH POLE.
Consider a Circle around the North Pole with a small radius
of only ten (10) feet. Choose "N" as the center of the circle.
We choose four points around the circumference, "A" at 0 Deg,
"B" at 90 Deg, "C" at 180 Deg, and "D" at 270 Deg. Choose the
positive direction of the "X" axis from the center of the circle,
"N", through the Point "A" on the circumference. Choose the positive
direction of the "Y" axis from the center, "N", through the Point "B" on
the circumference. On this small scale for our circle, ALL POINTS "A", "B",
"C", "D", and "N" ARE DIRECTLY VISIBLE FROM EACH OF THE OTHER POINTS.
Join DA and drop a perpendicular from N to DA to intersect DA at "E".
Exprapolate NE to meet the circumference of the circle at "F".
Y
^
|
| B (90 Deg)
------+------
/ | \
/ | \
( | )
C | | N | A (0 Deg)
(180 Deg)+-----------+-----------+---------------> X
| | \ / |
( | \E/ )
\ | / \ /
\ | / / F
------+------
D (270 Deg)
Since the center of this circle is the North Pole, "N", each INWARD-DRAWN
RADIUS, from any point on the circumference would be the NORTH DIRECTION
for that particular point on the circumference. Similarly, EN is the North
direction for point E. These North directions are NOT parallel to each other.
The North directions converge to the center of the circle, which we had
chosen as the North Pole.
The North (N) direction is defined as the direction pointing to the
North Pole. The AZIMUTH (angular direction, in degrees, of any one
point from any other point) is the angular rotation FROM NORTH,
in a ClockWise direction.
The following can be obtained, by direct inspection (no need to calculate):
>From "A" the North direction is AN (0 Deg Azimuth).
The direction of "B" from "A" is "North-East" (N-E), ( 45 Deg Azimuth).
The direction of "C" from "A" is "North" (N), ( 0 Deg Azimuth).
The direction of "D" from "A" is "North-West" (N-W), (315 Deg Azimuth).
>From "B" the North direction is BN (0 Deg Azimuth).
The direction of "C" from "B" is "North-East" (N-E), ( 45 Deg Azimuth).
The direction of "D" from "B" is "North" (N), ( 0 Deg Azimuth).
The direction of "A" from "B" is "North-West" (N-W), (315 Deg Azimuth).
>From "C" the North direction is CN (0 Deg Azimuth).
The direction of "D" from "C" is "North-East" (N-E), ( 45 Deg Azimuth).
The direction of "A" from "C" is "North" (N), ( 0 Deg Azimuth).
The direction of "B" from "C" is "North-West" (N-W), (315 Deg Azimuth).
>From "D" the North direction is DN (0 Deg Azimuth).
The direction of "A" from "D" is "North-East" (N-E), ( 45 Deg Azimuth).
The direction of "B" from "D" is "North" (N), ( 0 Deg Azimuth).
The direction of "C" from "D" is "North-West" (N-W), (315 Deg Azimuth).
Since the circle has the North Pole as its center, the
points on the circle can be considered as a having the same latitude,
the latititude being very close to 90 Deg (but smaller than 90 Deg,
since the Latitude of the North Pole is exactly 90 Deg, by definition).
We note some VERY INTERESTING facts:
************************************
All points on the circle have the SAME LATITUDE.
Even though "B", "C", and "D" are on the same latitude as "A",
"B" is North-East (N-E) of "A", C is N of A, and D is N-W of A.
Similarly, C, D, and A are on the same latitude as B,
C is N-E of B, D is N of B, and A is N-W of B; and similarly,
D is N-E of C, A is N of C, and B is N-W of C;
A is N-E of D, B is N of D, and C is N-W of D.
*******************************************************************
* When "B" is North-East (N-E) of "A", for this small circle, *
* "A" is North-West (N-W) of "B" !!! (NOT S-W !!!) *
*******************************************************************
The direction from "A" to other points on the same Latitude circle is
along points that are on the LINE-of-the-SHORTEST-DISTANCE joining that
particular point and the point "A". The Line of the Shortest Distance is
along the GREAT CIRCLE. The direct direction is NOT along points on the
same Latitude. (Only exception is for points on the equator).
The SHORTEST (Direct) DISTANCE between any two points on the surface of
the earth is along the Great Circle.
We also note that at the point "A", due "East" is 90 Degrees ClockWise (CW)
from the North direction (N is along the radius AN), hence is perpendicular
to the radius AN, and hence tangential to the circumference of the circle.
If a person travels due East from point "A", and that person travels in a
"straight line" (along the Great Circle) and does not change direction,
then he would be travelling along the tangent (at"A") to the circle,
and hence moving "away" from the circle. This would cause the latitude
of the person to change while traveling in the "straight line" !!
As the person travels in the "straight line" the direction of East
would no longer be "straight ahead" since the North direction would
change and hence the Ease direction would also change !
Consider again the "small" circle around the North Pole.
For a person to "FACE" Point "A" from Point "D", that person would have
to face in the North-East (N-E) direction at "D". When that person travels
in a "STRAIGHT LINE" along DA, the person is INITIALLY ORIENTED towards
the NORTH-EAST direction (defined at Point D) while AT POINT D. Continuing
to travel from D, WITHOUT CHANGING THE DIRECTION OF TRAVEL IN THE
"STRAIGHT LINE" to an intermediate point E (note that NE is perpendicular
to DEA), the INTERMEDIATE ORIENTATION is DUE EAST AT POINT E, to continue
from E towards A. Further continuing to travel from E, WITHOUT CHANGING
THE DIRECTION OF TRAVEL IN THE "STRAIGHT LINE" to a final pint A, the
FINAL ORIENTATION is SOUTH-EAST at POINT A.
**********************************************
* TRAVEL EVEN IN A "STRAIGHT LINE" CAUSES *
* THE ORIENTATION TO CHANGE WHILE TRAVELLING *
**********************************************
STRAIGHT LINE TRAVEL FROM POINT "D", THROUGH POINT "E", TO POINT "A":
LOCATION POINT: Point D Point E Point A
ORIENTATION: North-East East South-East
NORTH DIRECTION: DN EN AN
If the person travelled a little distance due East from Point D,
since direction East is no longer in the person's line of travel
(the North direction changes from point to point) the person
re-orients towards due East and again travels a short distance,
and again re-orients towards the new East direction and travels a
short distance, then that person WOULD BE TRAVELLING IN A CIRCLE
(from Point D, to Point F, to Point A) and would NOT be travelling
in a "straight line" from Point D, to Point E, to Point A.
EXAMPLE D-2: Points around the North Pole (NP), on constant Latitudes,
======================================================================
upto the EQUATOR
================
This is the general case of Example D-1 and D-3. We will consider only
four (4) points on each Latitude, at Longitudes 0, 90E, 180, 90W Degrees.
Since both the points are on the same latitude, the equation for the
Azimuth (given in Section C) is reduced to:
__ __
-1 | 1.000 |
C = TAN | ------------- |
|__ sin (PHI) __|
where ANGLE PHI = Latitude
Near NP Near Equator
N Latitude: 89.99 89.9 89 85 80 70 60 50 40 30 21 20 10 00
Direction
"A from D": 45.00 45.0 45 45 45 47 49 53 57 63 70 71 80 90
"B from D": 00.00 00.0 00 00 00 00 00 00 00 00 00 00 00 00
"C from D": 315.00 315.0 315 315 315 313 311 307 303 297 290 289 280 270
(All Directions are Azimuthal, ie E of N.)
180
/------|------\
/ B | \
/ | \
( | )
90 | | N A | 90
West |-----------+-----------| East
| C | |
( | )
\ | /
\ | D /
\------|------/
0 Deg
|
West <-------|-------> East
|
"P L A N"
90N
/------|------\
21N / | \
\ / | \ / 21N
' .( | ). ' KABAH
| ' . | . ' |
Equator 0 |-----------+-----------| 0 Equator
| . ' | |
/ ' ( | )
21S \ | /
\ | /
\------|------/
90S
140 W | 40 E
Long <-------|-------> Long
|
"E L E V"
Places on any particular Longitude are the same distance apart
if they have the same difference in Latitude.
A 10 Deg difference in Latitude has the same distance at any Longitude,
and the distance is approximately 692 miles.
Places on any particular Latitude are the same distance apart
if they have the same difference in Longitude.
A 10 Deg difference in Longitude has a greater distance at
Latitudes closer to the Equator than at latitudes closer to the Poles.
At the Equator (00 Deg Lat) a 1 Deg Long difference is 69 miles,
further north (40 Deg Lat) a 1 Deg Long difference is 53 miles, and
further north (60 Deg Lat) a 1 Deg Long difference is 35 miles.
EXAMPLE D-3: Points on the same Latitude as Kabah
=================================================
All points have the same Latitude as that of Kabah (21 Deg N).
The Longitudes are taken in steps of 30 Degrees (Kabah: Long 40 Deg E).
For ease of presenting the directions, the Lat and Long of Kabah have been
approximated to: Latitude 21 Deg North (21 Degrees 26 Minutes)
Longitude 40 Deg East (39 Degrees 49 Minutes)
Latitude 21N 21N 21N 21N 21N 21N 21N 21N 21N 21N 21N
Longitude 10E 20W 50W 80W 110W 140W 170W 160E 130E 100E 70E
Direction * 85 78 70 58 37 0 323 302 290 282 275
* All Directions are Azimuthal, ie E of N.
Note that the Azimuths to Kabah CHANGES even though all the points are
on the same Latitude of Kabah (21 Deg N). In fact, the Azimuth is 0 Deg,
ie due North, when the Long of a point is 140 Deg West, this direction is
90 Deg CCW from East !!
140 180
West\ /------|------\
/ \ | \
/ \ | \
( \ | )
90 | \| | 90
West |-----------+-----------| East
| | \ |
( | \ )
\ | \ /
\ | \/
\------|------/ \ 40 East
0 Deg KABAH
|
West <-------|-------> East
|
"P L A N"
EXAMPLE D-4: Points around Kabah (0 Deg to 360 Deg).
====================================================
We will consider eight (8) points around the Kabah, with each point at
a constant difference (DIFF) in Latitude (Lat) and/or Longitude (Long)
from Kabah. The direction for the Qibla would be the direction for the
"Shortest Distance" to Kabah.
For ease of presenting the directions, the Lat and Long of Kabah have been
approximated to: Latitude 21 Deg North (21 Degrees 26 Minutes)
Longitude 40 Deg East (39 Degrees 49 Minutes)
^ N D = Difference in
| Lat &/or Long
| O = Point "Oh", Origin
6 | 3 7 0 = Numeral Zero
[--------------+--------------]
(O+D , O-D) (O+D , O+0) (O+D , O+D)
| | |
| | |
| | |
2 | | 0 | 4
+--------------+--------------+--------> E
(O+0 , O-D) (O+0 , O+0) (O+0 , O+D)
| | |
| | |
| | |
5 | | 1 | 8
[--------------+--------------]
(O-D , O-D) (O-D , O+0) (O-D , O+D)
(Latitude, Longitude)
DIRECTIONS TO KABAH FOR POINTS AROUND KABAH
===========================================
***************************************************************************
* Pt|Guess| DIFFerence in Latitude and/or Longitude of points from Kabah *
*-------------------------------------------------------------------------*
* No|Angle| 0.01 0.10 0.5 1 2 5 10 20 40 60 *
***************************************************************************
* | | *
* 1 | 0 | 0.00 0.00 0.0 0.0 0.0 0.0 0.0 0 0 0 *
* 2 | 90 | 89.99 89.98 89.9 89.8 89.6 89.1 88.2 86 83 78 *
* 3 | 180 | 180.00 180.00 180.0 180.0 180.0 180.0 180.0 180 180 180 *
* 4 | 270 | 270.00 270.02 270.1 270.2 270.4 270.9 271.8 274 277 282 *
* | | *
* 5 | 45 | 43.03 43.02 42.99 42.95 42.88 42.7 42.6 43 46 55 *
* 6 | 135 | 136.97 136.96 136.93 136.88 136.79 136.5 135.7 134 127 117 *
* 7 | 225 | 223.03 223.04 223.08 223.12 223.22 223.5 224.3 226 233 243 *
* 8 | 315 | 316.97 316.98 317.01 317.05 317.12 317.3 317.4 317 314 305 *
* | | *
***************************************************************************
Note that our "Guess" at Point No.2, for the direction of the Kabah is
90 Deg (ie Due East). As the difference in Longitude from Kabah increases
(but at the same Latitude), the directions to the Kabah changes upto
78 Deg E of N (NOT CONSTANT AT 90 Deg) for a Longitude difference of 60 Deg.
from Kabah. Similar variations are evident for many of the other points.
Also note that the actual distance for 10 Deg North/South is greater
than the distance for 10 Deg East/West because Kabah is not at the Equator
but is at 21 Deg North.
EXAMPLE D-5: Points on the same Longitude as Kabah
==================================================
All points have the same Longitude as that of Kabah (40 Deg E).
The Latitudes are taken in steps of 30 Degrees, (21N, 51N, 81N,
69N, 39N, 09N, 21S, 51S, 81S, 69S, 39S, and 09S) with intermediate
latitudes at locations of interest.
For ease of presenting the directions, the Lat and Long of Kabah have been
approximated to: Latitude 21 Deg North (21 Degrees 26 Minutes)
Longitude 40 Deg East (39 Degrees 49 Minutes)
Latitude 20N 21N 22N 51N 81N 89N 90N 89N
Longitude 40E 40E 40E 40E 40E 40E 40E 140W
Direction * 0 - 180 180 180 180 180 0
N - S S S S S N
Latitude 90N 90N 89N 69N 39N 22N 21N 20N 09N 20S 21S 22S
Longitude 40E 140W 140W 140W 140W 140W 140W 140W 140W 140W 140W 140W
Direction * 180 0 0 0 0 0 0 0 0 0 + 180
S N N N N N N N N N + S
Latitude 20S 21S 22S 51S 81S 89S 90S 90S
Longitude 140W 140W 140W 140W 140W 140W 140W 40E
Direction * 0 + 180 180 180 180 180 0
N + S S S S S N
Latitude 90S 90S 89S 69S 39S 09S 20N 21N 22N
Longitude 140W 40E 40E 40E 40E 40E 40E 40E 40E
Direction * 180 0 0 0 0 0 0 - 180
S N N N N N N - S
* All Directions are Azimuthal, ie E of N.
Direction of 0 Deg is due North = N
Direction of 180 Deg is due South = S
+ The Qibla direction can be in ANY DIRECTION since
this point (21S 140W) is directly opposite to the Kabah
on the spherical globe.
Note that even though at the Latitude of 69N (this Latitude is N of Kabah),
and at Longitude of 140W, the direction for the Qibla is NORTH !!
Similarly even though at the Latitude is 51S (this Latitude is S of Kabah),
and at Longitude of 140W, the direction for the Qibla is SOUTH !!
90N
/------|------\
21N / | \
\ / | \ / 21N
' .( | ). ' KABAH
| ' . | . ' |
Equator 0 |-----------+-----------| 0 Equator
| . ' | |
/ ' ( | )
21S \ | /
\ | /
\------|------/
90S
140 W | 40 E
Long <-------|-------> Long
|
"E L E V"
EXAMPLE D-6: Points at approximately the center of various countries
====================================================================
Latitude Longitude Qibla Direction from TRUE North
Australia: 30 Deg S 130 Deg E 289 Deg E-of-N (19 Deg N of W)
France: 43 Deg N 3 Deg E 113 Deg E-of-N (23 Deg S of E)
Japan: 35 Deg N 140 Deg E 293 Deg E-of-N (23 Deg N of W; NOT S of W)
U.S.A.: 40 Deg N 100 Deg W 39 Deg E-of-N (51 Deg N of E; NOT S of E)
E. REFERENCES
*************
1. Textbook of Spherical Astronomy,
W.M.Smart, Cambridge University Press
2. Magnetic Variation (Declination)
(Epoch 1990), Map No.42 (10th Ed, Feb 1992)
World Map, U.S.A Defense Mapping Agency
3. Explanatory Supplement to the Astronomical Almanac, 1992
Ed. P.K.Seidelmann, U.S.Naval Observatory, Wash.DC.
4. Map Projections used by the U.S.Geological Survey
Geological Survey Bulletin 1532, 2nd Ed, 1983
"AND ALLAH (swt) KNOWS BEST, AND HE IS MOST WISE"
************************************************
"YOU (the Muslims) are the BEST NATION that was brought to humanity,
enjoining al-ma'ruf (the right) and forbidding al-munkar (the worng)
and believing in ALLAH"
THE HOLY QURAN, Chap 3, Verse 110.
OF ALL THE DIVINE BOOKS THAT THEMSELVES CLAIM TO BE THE WORD OF GOD
THE HOLY QURAN IS THE MOST AUTHENTIC TO ITS ORIGINAL REVELATION !!!
@@@@@@@@@@@@@@@@@@@@@@@@@@ End of Document @@@@@@@@@@@@@@@@@@@@@@@@@@@@@
Look For The CRESCENT MOON ( HILAL ), --- >>> )
It Is One Of THE MOST BEAUTIFUL OF CREATIONS; ---- >>>> )
Then Offer An INTENSE PRAYER To The ONE CREATOR, ---- >>>> )
All Sincere DEVOTIONS Are Surely ACCEPTED. --- >>> )
___________________________
mnd@columbia.edu