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In geometry, a centered trochoid is the roulette formed by a circle rolling along another circle. That is, it is the path traced by a point attached to a circle as the circle rolls without slipping along a fixed circle. The term encompasses both epitrochoid and hypotrochoid. The center of this curve is defined to be the center of the fixed circle. Alternatively, a centered trochoid can be defined as the path traced by the sum of two vectors, each moving at a uniform speed in a circle. Specifically, a centered trochoid is a curve that can be parameterized in the complex plane by : or in the Cartesian plane by : : where : If is rational then the curve is closed and algebraic. Otherwise the curve winds around the origin an infinite number of times, and is dense in the annulus with outer radius and inner radius . ==Terminology== Most authors use ''epitrochoid'' to mean a roulette of a circle rolling around the outside of another circle, ''hypotrochoid'' to mean a roulette of a circle rolling around the inside of another circle, and ''trochoid'' to mean a roulette of a circle rolling along a line. However, some authors (for example () following F. Morley) use "trochoid" to mean a roulette of a circle rolling along another circle, though this is inconsistent with the more common terminology. The term ''Centered trochoid'' as adopted by () combines ''epitrochoid'' and ''hypotrochoid'' into a single concept to streamline mathematical exposition and remains consistent with the existing standard. The term ''Trochoidal curve'' describes epitrochoids, hypotrochoids, and trochoids (see ()). A trochoidal curve can be defined as the path traced by the sum of two vectors, each moving at a uniform speed in a circle or in a straight line (but not both moving in a line). In the parametric equations given above, the curve is an epitrochoid if and have the same sign, and a hypotrochoid if they have opposite signs. 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「centered trochoid」の詳細全文を読む スポンサード リンク
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