semiz@yalehp.bitnet (01/12/90)
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Greetings to all...
I think the direction of Mecca is the direction of the great circle
passing through your position and Mecca's.
On how to find this direction, first some preliminaries from Math
(Vector Algebra) :
A plane is determined by the direction of one vector.
(The vector perpendicular to it)
A plane is spanned by two (noncolinear) vectors
That is, any point in a plane can be reached by
starting from a special point (origin) in the plane
and putting x numbers of the first vector and y
numbers of the second vector. x and y are called
the coordinates of that point in the plane.
Any vector in the plane is perpendicular to the vector
determining the plane.
The "cross-product" of any two vectors gives another
vector perpendicular to the first two (and length
equal to the product of the lengths of the first
two, if the first two are perpendicular.)
Think of the earth in 3 dimensions. The great circle defines a plane.
The (unit) vector pointing from the center of the earth to Mecca (M) and
the (unit) vector pointing from the center of the earth to your location (Y)
are both in it. Therefore the vector (N) determining the plane will be
proportional to the cross-product of the two. We need the direction of
Mecca (K). This will be the tangent vector (at your location) of the great
circle. Being a tangent vector, it will be perp. to the radius vector, i.e Y.
Being in the plane, it will be perp. to N. Therefore it will be proportional
to the cross-product of the two. Finally, projections on the local east- and
north-vectors need to be taken.
For those who know the math,
N = Y cross M
T = N cross Y (You can use the bac-cab
identity, if you want)
(The east component) e = T dot E ( E is unit vector to
east, phi-hat)
(The south component) s = T dot S ( S is unit vector to
south, theta-hat)
HERE IS THE STEP-BY-STEP RECIPE FOR NON-MATH PEOPLE
Your location - Latitude in (degrees) L = ____
Longitude in (degrees) phi = ____
Colatitude in (degrees) theta = 90 - L = ____
Vector Y - first component Y1 = sin(theta)*cos(phi) = ____
- second component Y2 = sin(theta)*sin(phi) = ____
- third component Y3 = cos(theta) = ____
Location of Mecca - 23.2 deg N, 40.3 deg. E
Colatitude in (degrees) Mtheta = 66.8
Longitude in (degrees) Mphi = 40.3
Vector M - first component M1 = sin(Mtheta) * cos(Mphi) = .7010
- second component M2 = sin(Mtheta) * sin(Mphi) = .5945
- third component M3 = cos(Mtheta) = .3939
Vector N = Y cross M
- first component N1 = Y2*M3 - Y3*M2 = ___
- second component N2 = Y3*M1 - Y1*M3 = ___
- third component N3 = Y1*M2 - Y2*M1 = ___
Vector T = N cross Y
- first component T1 = N2*Y3 - N3*Y2 = ___
- second component T2 = N3*Y1 - N1*Y3 = ___
- third component T3 = N1*Y2 - N2*Y1 = ___
Vector S - first component S1 = cos(theta) * cos(phi) = ___
- second component S2 = cos(theta) * sin(phi) = ___
- third component S3 = - sin(theta) = ___
Vector E - first component E1 = - sin(phi) = ___
- second component E2 = cos(phi) = ___
- third component E3 = 0 (zero)
s = T dot S = T1*S1 + T2*S2 + T3*S3 = ____
e = T dot E = T1*E1 + T2*E2 + T3*E3 = T1*E1 + T2*E2 = ____
Now, take a sheet of paper. Draw directions, north-south-east-west (Of course,
they will be perpendicular. Starting from the origin, measure s (cm,m,feet,
whatever...) to south. Of course, if s is negative, you will be measuring
to north. Starting from that point, measure e (in same units as s) to
east. Of course, if e is negative, you will be measuring to west. Draw a
line from the origin to this point.
This will give you the direction of Mecca.
(This is true up to a sign. If you are to west of Mecca, i.e Middle-
or West Europe, or the Americas, it gives the correct result. If you are to
East of Mecca, i.e Asia, it will show the opposite direction, but this
should be easy to correct by a look at the globe.)
Ibrahim Semiz Yale-Physics "semiz@yalehep"