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Construction of the limaçon r = 2 + cos(π – θ) with polar coordinates' origin at (x, y) = ( 1 / 2 , 0). In geometry, a limaçon or limacon / ˈ l ɪ m ə s ɒ n /, also known as a limaçon of Pascal or Pascal's Snail, is defined as a roulette curve formed by the path of a point fixed to a circle when that circle rolls around the outside of a circle of equal radius.
The inner loop is defined when + on the polar angle interval / /, and is symmetric about the polar axis. The point furthest from the pole on the inner loop has the coordinates ( a , 0 ) {\displaystyle (a,0)} , and on the polar axis, is one-third of the distance from the pole compared to the furthest point of the outer loop.
In geometry, a trisectrix is a curve which can be used to trisect an arbitrary angle with ruler and compass and this curve as an additional tool. Such a method falls outside those allowed by compass and straightedge constructions, so they do not contradict the well known theorem which states that an arbitrary angle cannot be trisected with that type of construction.
One such method uses compass and straightedge; another method uses a modified string compass. The common traditional construction uses compass and straightedge to approximate the arithmetic spiral. First, a large circle is constructed and its circumference is subdivided by 12 diameters into 12 arcs (of 30 degrees each; see regular dodecagon ).
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Perl provides three loop control keywords that all accept an optional loop label as an argument. If no label is specified, the keywords act on the innermost loop. Within nested loops, the use of labels enables control to move from an inner loop to an outer one, or out of the outer loop altogether.
Notational abuse to be found below includes e X for the exponential map exp given an argument, writing g for the element (g, e H) in a direct product G × H (e H is the identity in H), and analogously for Lie algebra direct sums (where also g + h and (g, h) are used interchangeably).
In some cases, a while loop with an inner while loop performed slower than a while loop without an inner loop. The two examples below, written in Python, present a while loop with an inner for loop and a while loop without an inner loop. Although both have the same terminating condition for their while loops, the first example will finish ...