Actually, we here in Aus know that the ancient Egyptians had it right, aligning the top of their map with the source of the Nile. This link<\/a> will show you why this correctly reveals the true arrangement of a world map.<\/p>\n Anyway, a while back this got me wondering about just how imprecise using the sunrise as the basis for orientation could be, so I set out estimating just how far the sun ranges from sunrise in mid winter to sunrise in mid summer around Jerusalem. For those of you who’d rather not wade through the maths, skip to the end. For the rest of you, here goes (and if you spot any glaring mathematical errors, please let me know).<\/p>\n Before we begin, some housekeeping. I’m using standard spherical polar coordinates here, as described at MathWorld<\/a>.<\/p>\n For simplicity, we shall make a couple of assumptions about our system, which are:<\/p>\n Given these constraints, we may proceed to calculate the extreme points of sunrise. To do so we require the following data:<\/p>\n Jerusalem is thus defined as the point P on the globe, and since we are assuming that the sun is infinitely distant, we may (for simplicity) simply define this as a unit vector:<\/p>\n Where \u03b8 ranges from 0\u00b0 to 360\u00b0 as the earth rotates during the day. For our purposes we shall determine what value of \u03b8 is appropriate for sunrise in mid-summer and mid-winter.<\/p>\n The position of the sun will remain constant. For mid-summer in Jerusalem, we shall position the sun at: The local coordinate system at Jerusalem may thus be defined by the spanning vectors:<\/p>\n (Here k is the vector (0, 0, 1) in the main coordinate system.) This system defines the local directions up = i<\/strong>J<\/sub><\/em>, east = j<\/strong>J<\/sub><\/em>, and north = k<\/strong>J<\/sub><\/em>. Consequently, the local vector to the sun may be defined as:<\/p>\n At last we are in a position to define sunrise (or sunset) as occuring when the direction to the sun has no vertical component in the local coordinate system\u2014that is when<\/p>\n Converting these values from spherical to rectangular coordinates gives the following:<\/p>\n i<\/strong>J<\/sub><\/em> = ( sin(58\u00b014′) cos \u03b8, sin(58\u00b014′) sin \u03b8, cos(58\u00b014′) ) Substituting into the previous equation and solving for \u03b8 gives:<\/p>\n Calculating the result gives: \u03b8 = \u00b1105\u00b0 34′ 52″<\/p>\n Given this value, we may now determine the position on the horizon where the sun will rise mid-summer in Jerusalem by converting the vector s to local coordinates. We now know that at sunrise, the position of Jerusalem is defined as:<\/p>\n This allows us to determine the entire set of local coordinates (according to the set of equations set out above) and gives the rectangular values:<\/p>\n \ni<\/strong>J<\/sub><\/em> = ( \u20130.2284, \u20130.8190, 0.5265 )\n
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\nFor mid-winter the position of the sun will be:
\n![\"spanning](\"http:\/\/blog.shields-online.net\/wp-content\/uploads\/2008\/12\/picture-1.png\" "\\"spanning")
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\ns<\/strong> = ( sin(66.55\u00b0), 0, cos(66.55\u00b0) )<\/p>\n![\"\"](\"http:\/\/blog.shields-online.net\/wp-content\/uploads\/2008\/12\/picture-2.png\" "\\"sunrise")
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\nj<\/strong>J<\/sub><\/em> = ( 0.9633, \u20130.2686, 0 )
\nk<\/strong>J<\/sub><\/em> = ( 0.1414, 0.5071, 0.8502 )\n<\/p><\/blockquote>\n
![\"\"](\"http:\/\/blog.shields-online.net\/wp-content\/uploads\/2008\/12\/picture-3.png\" "\\"angle")
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