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Spirals
In mathematics, a spiral is a curve which emanates from a point, moving further away as it revolves around the point. It is a subtype of whorled patterns, a broad group that also includes concentric objects. Two-dimensional A two-dimensional, or plane, spiral may be easily described using polar coordinates, where the radius r is a monotonic continuous function of angle \varphi: * r=r(\varphi)\; . The circle would be regarded as a degenerate case (the function not being strictly monotonic, but rather constant). In ''x-y-coordinates'' the curve has the parametric representation: * x=r(\varphi)\cos\varphi \ ,\qquad y=r(\varphi)\sin\varphi\; . Examples Some of the most important sorts of two-dimensional spirals include: * The Archimedean spiral: r=a \varphi * The hyperbolic spiral: r = a/ \varphi * Fermat's spiral: r= a\varphi^ * The lituus: r = a\varphi^ * The logarithmic spiral: r=ae^ * The Cornu spiral or ''clothoid'' * The Fibonacci spiral and golden spiral * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golden Spiral
In geometry, a golden spiral is a logarithmic spiral whose growth factor is , the golden ratio. That is, a golden spiral gets wider (or further from its origin) by a factor of for every quarter Turn (angle), turn it makes. Approximations of the golden spiral There are several comparable spirals that approximate, but do not exactly equal, a golden spiral. For example, a golden spiral can be approximated by first starting with a rectangle for which the ratio between its length and width is the golden ratio. This rectangle can then be partitioned into a square and a similarity (geometry), similar rectangle and this rectangle can then be split in the same way. After continuing this process for an arbitrary number of steps, the result will be an almost complete partitioning of the rectangle into squares. The corners of these squares can be connected by quarter-circles. The result, though not a true logarithmic spiral, closely approximates a golden spiral. Another approximation i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spiral Of Theodorus
In geometry, the spiral of Theodorus (also called the square root spiral, Pythagorean spiral, or Pythagoras's snail) is a spiral composed of right triangles, placed edge-to-edge. It was named after Theodorus of Cyrene. Construction The spiral is started with an isosceles right triangle, with each leg having unit length. Another right triangle (which is the ''only'' automedian right triangle) is formed, with one leg being the hypotenuse of the prior right triangle (with length the square root of 2) and the other leg having length of 1; the length of the hypotenuse of this second right triangle is the square root of 3. The process then repeats; the nth triangle in the sequence is a right triangle with the side lengths \sqrt and 1, and with hypotenuse \sqrt. For example, the 16th triangle has sides measuring 4=\sqrt, 1 and hypotenuse of \sqrt. History and uses Although all of Theodorus' work has been lost, Plato put Theodorus into his dialogue '' Theaetetus'', which tells of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Spiral
A hyperbolic spiral is a type of spiral with a Pitch angle of a spiral, pitch angle that increases with distance from its center, unlike the constant angles of logarithmic spirals or decreasing angles of Archimedean spirals. As this curve widens, it approaches an asymptotic line. It can be found in the view up a spiral staircase and the starting arrangement of certain footraces, and is used to model spiral galaxy, spiral galaxies and Volute, architectural volutes. As a plane curve, a hyperbolic spiral can be described in polar coordinates (r,\varphi) by the equation r=\frac, for an arbitrary choice of the scale factor a. Because of the Multiplicative inverse, reciprocal relation between r and \varphi it is also called a reciprocal spiral. The same relation between Cartesian coordinates would describe a hyperbola, and the hyperbolic spiral was first discovered by applying the equation of a hyperbola to polar coordinates. Hyperbolic spirals can also be generated as the inverse cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
