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Epispiral
The epispiral is a plane curve with polar equation :\ r=a \sec. There are ''n'' sections if ''n'' is odd and 2''n'' if ''n'' is even. It is the polar or circle inversive geometry, inversion of the rose (mathematics), rose curve. In astronomy the epispiral is Newton%27s_theorem_of_revolving_orbits#Illustrative_example:_Cotes's_spirals, related to the equations that explain planet, planets' orbits. Alternative definition There is another definition of the epispiral that has to do with tangents to circles: Begin with a circle. Rotate some single point on the circle around the circle by some angle \theta and at the same time by an angle in constant proportion to \theta, say c\theta for some constant c. The intersections of the tangent lines to the circle at these new points rotated from that single point for every \theta would trace out an epispiral. The polar equation can be derived through simple geometry as follows: To determine the polar coordinates (\rho,\phi) of the int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plane Curve
In mathematics, a plane curve is a curve in a plane that may be a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves. Plane curves also include the Jordan curves (curves that enclose a region of the plane but need not be smooth) and the graphs of continuous functions. Symbolic representation A plane curve can often be represented in Cartesian coordinates by an implicit equation of the form f(x,y)=0 for some specific function ''f''. If this equation can be solved explicitly for ''y'' or ''x'' – that is, rewritten as y=g(x) or x=h(y) for specific function ''g'' or ''h'' – then this provides an alternative, explicit, form of the representation. A plane curve can also often be represented in Cartesian coordinates by a parametric equation of the form (x,y)=(x(t), y(t)) for specific functions x(t) and y(t). Plane curves can sometimes also be repr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Equation
In mathematics, the polar coordinate system specifies a given point in a plane by using a distance and an angle as its two coordinates. These are *the point's distance from a reference point called the ''pole'', and *the point's direction from the pole relative to the direction of the ''polar axis'', a ray drawn from the pole. The distance from the pole is called the ''radial coordinate'', ''radial distance'' or simply ''radius'', and the angle is called the ''angular coordinate'', ''polar angle'', or ''azimuth''. The pole is analogous to the origin in a Cartesian coordinate system. Polar coordinates are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point in a plane, such as spirals. Planar physical systems with bodies moving around a central point, or phenomena originating from a central point, are often simpler and more intuitive to model using polar coordinates. The polar coordinate system is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inversive Geometry
In geometry, inversive geometry is the study of ''inversion'', a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves. Many difficult problems in geometry become much more tractable when an inversion is applied. Inversion seems to have been discovered by a number of people contemporaneously, including Steiner (1824), Quetelet (1825), Bellavitis (1836), Stubbs and Ingram (1842–3) and Kelvin (1845). The concept of inversion can be generalized to higher-dimensional spaces. Inversion in a circle Inverse of a point To invert a number in arithmetic usually means to take its reciprocal. A closely related idea in geometry is that of "inverting" a point. In the plane, the inverse of a point ''P'' with respect to a ''reference circle (Ø)'' with center ''O'' and radius ''r'' is a point ''P'', lying on the ray from ''O'' through ''P'' such that :OP \cdot OP^ = r^2. This is calle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rose (mathematics)
In mathematics, a rose or rhodonea curve is a sinusoid specified by either the cosine or sine functions with no phase angle that is plotted in polar coordinates. Rose curves or "rhodonea" were named by the Italian mathematician who studied them, Guido Grandi, between the years 1723 and 1728. General overview Specification A rose is the set of points in polar coordinates specified by the polar equation :r=a\cos(k\theta) or in Cartesian coordinates using the parametric equations :\begin x &= r\cos(\theta) = a\cos(k\theta)\cos(\theta) \\ y &= r\sin(\theta) = a\cos(k\theta)\sin(\theta) \end Roses can also be specified using the sine function. Since :\sin(k \theta) = \cos\left( k \theta - \frac \right) = \cos\left( k \left( \theta-\frac \right) \right). Thus, the rose specified by is identical to that specified by rotated counter-clockwise by radians, which is one-quarter the period of either sinusoid. Since they are specified using the cosine or sine function, rose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Astronomy
Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest include planets, natural satellite, moons, stars, nebulae, galaxy, galaxies, meteoroids, asteroids, and comets. Relevant phenomena include supernova explosions, gamma ray bursts, quasars, blazars, pulsars, and cosmic microwave background radiation. More generally, astronomy studies everything that originates beyond atmosphere of Earth, Earth's atmosphere. Cosmology is a branch of astronomy that studies the universe as a whole. Astronomy is one of the oldest natural sciences. The early civilizations in recorded history made methodical observations of the night sky. These include the Egyptian astronomy, Egyptians, Babylonian astronomy, Babylonians, Greek astronomy, Greeks, Indian astronomy, Indians, Chinese astronomy, Chinese, Maya civilization, M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planet
A planet is a large, Hydrostatic equilibrium, rounded Astronomical object, astronomical body that is generally required to be in orbit around a star, stellar remnant, or brown dwarf, and is not one itself. The Solar System has eight planets by the most restrictive definition of the term: the terrestrial planets Mercury (planet), Mercury, Venus, Earth, and Mars, and the giant planets Jupiter, Saturn, Uranus, and Neptune. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a young protostar orbited by a protoplanetary disk. Planets grow in this disk by the gradual accumulation of material driven by gravity, a process called accretion (astrophysics), accretion. The word ''planet'' comes from the Greek () . In Classical antiquity, antiquity, this word referred to the Sun, Moon, and five points of light visible to the naked eye that moved across the background of the stars—namely, Me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmic Spiral
A logarithmic spiral, equiangular spiral, or growth spiral is a self-similarity, self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). More than a century later, the curve was discussed by René Descartes, Descartes (1638), and later extensively investigated by Jacob Bernoulli, who called it ''Spira mirabilis'', "the marvelous spiral". The logarithmic spiral is distinct from the Archimedean spiral in that the distances between the turnings of a logarithmic spiral increase in a geometric progression, whereas for an Archimedean spiral these distances are constant. Definition In polar coordinates (r, \varphi) the logarithmic spiral can be written as r = ae^,\quad \varphi \in \R, or \varphi = \frac \ln \frac, with e (mathematical constant), e being the base of natural logarithms, and a > 0, k\ne 0 being real constants. In Cartesian coordinates The logarithmi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rose (mathematics)
In mathematics, a rose or rhodonea curve is a sinusoid specified by either the cosine or sine functions with no phase angle that is plotted in polar coordinates. Rose curves or "rhodonea" were named by the Italian mathematician who studied them, Guido Grandi, between the years 1723 and 1728. General overview Specification A rose is the set of points in polar coordinates specified by the polar equation :r=a\cos(k\theta) or in Cartesian coordinates using the parametric equations :\begin x &= r\cos(\theta) = a\cos(k\theta)\cos(\theta) \\ y &= r\sin(\theta) = a\cos(k\theta)\sin(\theta) \end Roses can also be specified using the sine function. Since :\sin(k \theta) = \cos\left( k \theta - \frac \right) = \cos\left( k \left( \theta-\frac \right) \right). Thus, the rose specified by is identical to that specified by rotated counter-clockwise by radians, which is one-quarter the period of either sinusoid. Since they are specified using the cosine or sine function, rose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
