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Every rotation in three dimensions is defined by its axis (a vector along this axis is unchanged by the rotation), and its angle — the amount of rotation about that axis (Euler rotation theorem). There are several methods to compute the axis and angle from a rotation matrix (see also axis–angle representation ).
In this table, parentheses mark letters that stand in for themselves or for another. For instance, a rotated 'b' would be a 'q', and indeed some physical typefaces didn't bother with distinct sorts for lowercase b vs. q, d vs. p, or n vs. u; while a rotated 's' or 'z' would be itself.
Turned characters, those that have been rotated 180 degrees and thus appear upside-down (this is the most common); Sideways characters, those that have been rotated 90 degrees counterclockwise (generally the least supported, and used only for a handful of vowels in the Uralic Phonetic Alphabet system).
A point P has coordinates (x, y) with respect to the original system and coordinates (x′, y′) with respect to the new system. [1] In the new coordinate system, the point P will appear to have been rotated in the opposite direction, that is, clockwise through the angle . A rotation of axes in more than two dimensions is defined similarly.
A strobogrammatic number is a number whose numeral is rotationally symmetric, so that it appears the same when rotated 180 degrees. The numeral looks the same right-side up and upside down (e.g., 69, 96, 1001). [54] [55] [56] Some dates are natural numeral ambigrams. [57]
The angle is typically measured in degrees from the mark of number 12 clockwise. The time is usually based on a 12-hour clock. A method to solve such problems is to consider the rate of change of the angle in degrees per minute. The hour hand of a normal 12-hour analogue clock turns 360° in 12 hours (720 minutes) or 0.5° per minute.
The number 619 is strobogrammatic.. A strobogrammatic number is a number whose numeral is rotationally symmetric, so that it appears the same when rotated 180 degrees. [1] In other words, the numeral looks the same right-side up and upside down (e.g., 69, 96, 1001). [2]
In the active transformation (left), a point P is transformed to point P ′ by rotating clockwise by angle θ about the origin of a fixed coordinate system. In the passive transformation (right), point P stays fixed, while the coordinate system rotates counterclockwise by an angle θ about its origin.