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Wallis derived this infinite product using interpolation, though his method is not regarded as rigorous. A modern derivation can be found by examining for even and odd values of , and noting that for large , increasing by 1 results in a change that becomes ever smaller as increases.
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For each dimensional k, the boundary maps d k : δD k → RP k−1 /RP k−2 is the map that collapses the equator on S k−1 and then identifies antipodal points. In odd (resp. even) dimensions, this has degree 0 (resp. 2): = + ().
The number is taken to be 'odd' or 'even' according to whether its numerator is odd or even. Then the formula for the map is exactly the same as when the domain is the integers: an 'even' such rational is divided by 2; an 'odd' such rational is multiplied by 3 and then 1 is added.
If the point is on the inside of the polygon then it will intersect the edge an odd number of times. The status of a point on the edge of the polygon depends on the details of the ray intersection algorithm. This algorithm is sometimes also known as the crossing number algorithm or the even–odd rule algorithm, and was known as early as 1962. [3]
Snyder [6] describes generating formulae for the projection, as well as the projection's characteristics. Coordinates from a spherical datum can be transformed into Albers equal-area conic projection coordinates with the following formulas, where is the radius, is the longitude, the reference longitude, the latitude, the reference latitude and and the standard parallels:
Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even—as the last digit of any even number is 0, 2, 4, 6, or 8.
When this definition is used, a standard calendar year is known as a "human year". A standard (365-day) calendar year of a dog's life, whereas a "human year" is the period of a dog's (or other animal's) life that is claimed to be equivalent to a year of a human being's life (or seven calendar years). [72]