http___en.m.wikipedia.org_wiki_Cardioid_1.xml
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Кардиоида В геометрията кардиоида е равнинна алгебрична крива от четвърта степен, вид епициклоида с единствена рогова точка. Кардиоидата е крива, която се получава като геометричното място на фиксирана точка от окръжност с радиус a, която се търкаля по външната страна на друга окръжност със същия радиус. Кардиоидата също така е специален вид охлюв на Паскал - с една рогова точка, която се получава, когато съотношението на радиусите на двете окръжности е 1. По трети начин тази крива може да се дефинира и като обратна трансформация на парабола. Името на кардиоидата идва от гръцки: kapδιa "сърце" + eιδoς "форма". Откриването на кривата се приписва на холандския математик Кьорсма в края на XVII век, а названието е въведено от италианеца Джовани Франческо Кастилион през 1741 г. в статията му "De curva cardioïde". Кардиоидата е позната и от фракталните изображения. Първият и най-голям елемент в множеството на Манделброт e тъкмо кардиоида. Уравнения[редактиране | редактиране на кода] - Уравнение в декартови координати: - Уравнение в полярни координати: - Дължина на кардиоидата: - Лице на повърхнината на заградената от кардиоидата област: Използвани източници[редактиране | редактиране на кода] - "The Penguin Dictionary of Mathematics", John Daintith, R.D. Nelson, Penguin Books, 1989 - "Лексикон Математика", Георги Симидчиев, Георги Чобанов, Иван Чобанов, ИК Абагар, София, 1995, ISBN 954-584-146-Х - "Математически термини", Н.В. Александрова, ДИ Наука и изкуство, София, 1984 - "Физико-математическа и техническа енциклопедия", Издателство на БАН, София, 1990 - "Математически енциклопедичен речник", В. Гелерт, Х. Кестнер, З. Нойбер, ДИ Наука и изкуство, София, 1983 Външни препратки[редактиране | редактиране на кода] - Страница, посветена на кардиоидата на сайта на Система Mathematica - Hearty Munching on Cardioids at cut-the-knot.org] - Xah Lee, Cardioid (1998) (Сайтът представя множество алтернативни конструкции). - Jan Wassenaar, Cardioid, (2005) в "862 двумерни математически криви".
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A cardioid (from the Greek kapδίa "heart") is a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius. It is therefore a type of limaçon and can also be defined as an epicycloid having a single cusp. It is also a type of sinusoidal spiral, and an inverse curve of the parabola with the focus as the center of inversion.[1] The name was coined by de Castillon in 1741[2] but had been the subject of study decades beforehand.[3] Named for its heart-like form, it is shaped more like the outline of the cross section of a round apple without the stalk. A cardioid microphone exhibits an acoustic pickup pattern that, when graphed in two dimensions, resembles a cardioid, (any 2d plane containing the 3d straight line of the microphone body.) In three dimensions, the cardioid is shaped like an apple centred on the microphone which is the "stalk" of the apple. EquationsEdit Based on the rolling circle description, with the fixed circle having the origin as its center, and both circles having radius a, the cardioid is given by the following parametric equations: In the complex plane this becomes Here a is the radius of the circles which generate the curve, and the fixed circle is centered at the origin. The point generating the curve touches the fixed circle at (a, 0), the cusp. The parameter t can be eliminated giving or, in rectangular coordinates, These equations can be simplified somewhat by shifting the fixed circle to the right a units and choosing the point on the rolling circle so that it touches the fixed circle at the origin; this changes the orientation of the curve so that the cusp is on the left. The parametric equations are then: or, in the complex plane, With the substitution u=tan t/2, giving a rational parameterization: or The parametrization can also be written and in this form it is apparent that the equation for this cardioid may be written in polar coordinates as where θ replaces the parameter t. This can also be written which implies that the curve is a member of the family of sinusoidal spirals. In Cartesian coordinates, the equation for this cardioid is Metrical propertiesEdit The area enclosed by a cardioid can be computed from the polar equation: This is 6 times the area of the circles used in the construction with rolling circles,[3] or 1.5 times the area of the circle used in the construction with circle and tangent lines. The arc length of a cardioid can be computed exactly, a rarity for algebraic curves. The total length is[3] Inverse curveEdit The cardioid is one possible inverse curve for a parabola. Specifically, if a parabola is inverted across any circle whose center lies at the focus of the parabola, the result is a cardioid. The cusp of the resulting cardioid will lie at the center of the circle, and corresponds to the vanishing point of the parabola. In terms of stereographic projection, this says that a parabola in the Euclidean plane is the projection of a cardioid drawn on the sphere whose cusp is at the north pole. Not every inverse curve of a parabola is a cardioid. For example, if a parabola is inverted across a circle whose center lies at the vertex of the parabola, then the result is a cissoid of Diocles. The picture to the right shows the parabola with polar equation In Cartesian coordinates, this is the parabola . When this parabola is inverted across the unit circle, the result is a cardioid with the reciprocal equation Cardioids in complex analysisEdit In complex analysis, the image of any circle through the origin under the map is a cardioid. One application of this result is that the boundary of the central bulb of the Mandelbrot set is a cardioid given by the equation The Mandelbrot set contains an infinite number of slightly distorted copies of itself and the central bulb of any of these smaller copies is an approximate cardioid. CausticsEdit Certain caustics can take the shape of cardioids. The catacaustic of a circle with respect to a point on the circumference is a cardioid. Also, the catacaustic of a cone with respect to rays parallel to a generating line is a surface whose cross section is a cardioid. This can be seen, as in the photograph to the right, in a conical cup partially filled with liquid when a light is shining from a distance and at an angle equal to the angle of the cone.[4] The shape of the curve at the bottom of a cylindrical cup is half of a nephroid, which looks quite similar. See alsoEdit ReferencesEdit - Weisstein, Eric W., "Cardioid", MathWorld. "Cardioid". A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards. pp. 4 ff. Further readingEdit The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. External linksEdit - O'Connor, John J.; Robertson, Edmund F., "Cardioid", MacTutor History of Mathematics archive, University of St Andrews. - Hearty Munching on Cardioids at cut-the-knot - Weisstein, Eric W., "Epicycloid--1-Cusped", MathWorld. - Weisstein, Eric W., "Heart Curve", MathWorld. - "Cardioid" at Encyclopédie des Formes Mathématiques Remarquables - Xah Lee, Cardioid (1998) (This site provides a number of alternative constructions). - Jan Wassenaar, Cardioid, (2005)