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In mathematics, a rose or rhodonea curve is a sinusoid plotted in polar coordinates. == General overview == Up to similarity, these curves can all be expressed by a polar equation of the form :〔''Mathematical Models'' by H. Martyn Cundy and A.P. Rollett, second edition, 1961 (Oxford University Press), p. 73.〕 or, alternatively, as a pair of Cartesian parametric equations of the form : : If ''k'' is an integer, the curve will be rose-shaped with *2''k'' petals if ''k'' is even, and *''k'' petals if ''k'' is odd. When ''k'' is even, the entire graph of the rose will be traced out exactly once when the value of θ changes from 0 to 2π. When ''k'' is odd, this will happen on the interval between 0 and π. (More generally, this will happen on any interval of length 2π for ''k'' even, and π for ''k'' odd.) If ''k'' is a half-integer (''e.g.'' 1/2, 3/2, 5/2), the curve will be rose-shaped with 4''k'' petals. If ''k'' can be expressed as ''n''±1/6, where ''n'' is a nonzero integer, the curve will be rose-shaped with 12''k'' petals. If ''k'' can be expressed as ''n''/3, where ''n'' is an integer not divisible by 3, the curve will be rose-shaped with ''n'' petals if ''n'' is odd and 2''n'' petals if ''n'' is even. If ''k'' is rational, then the curve is closed and has finite length. If ''k'' is irrational, then it is not closed and has infinite length. Furthermore, the graph of the rose in this case forms a dense set (i.e., it comes arbitrarily close to every point in the unit disk). Since : for all , the curves given by the polar equations : and are identical except for a rotation of π/2''k'' radians. Rhodonea curves were named by the Italian mathematician Guido Grandi between the year 1723 and 1728. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Rose (mathematics)」の詳細全文を読む スポンサード リンク
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