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Lituus (mathematics)
300px, Branch for positive In mathematics, a lituus is a spiral with polar equation :r^2\theta = k where is any non-zero constant. Thus, the angle is inversely proportional to the square of the radius . This spiral, which has two branches depending on the sign of , is asymptotic to the -axis. Its points of inflexion are at :(\theta, r) = \left(\tfrac12, \pm\sqrt\right). The curve was named for the ancient Roman lituus by Roger Cotes Roger Cotes (10 July 1682 – 5 June 1716) was an English mathematician, known for working closely with Isaac Newton by proofreading the second edition of his famous book, the '' Principia'', before publication. He also invented the quadratur ... in a collection of papers entitled ''Harmonia Mensurarum'' (1722), which was published six years after his death. External links * * Interactive example using JSXGraph* * https://hsm.stackexchange.com/a/3181 on the history of the lituus curve. Spirals Plane curves {{geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lituus
The word ''lituus'' originally meant a curved augural staff, or a curved war-trumpet in the ancient Latin language. This Latin word continued in use through the 18th century as an alternative to the vernacular names of various musical instruments. Roman ritual wand The ''lituus'' was a crooked wand (similar in shape to the top part of some Western European crosiers) used as a cult instrument in ancient Roman religion by augurs to mark out a ritual space in the sky (a ''templum''). The passage of birds through this ''templum'' indicated divine favor or disfavor for a given undertaking. The ''lituus'' was also used as a symbol of office for the college of the augurs to mark them out as a priestly group. Music instrument Antiquity The ancient ''lituus'' was an Etruscan high-pitched brass instrument, which was straight but bent at the end, in the shape of a letter J, similar to the Gallic carnyx. It was later used by the Romans, especially for processional music and as a signalli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spiral
In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the American Heritage Dictionary are:Spiral ''American Heritage Dictionary of the English Language'', Houghton Mifflin Company, Fourth Edition, 2009. # a curve on a plane that winds around a fixed center point at a continuously increasing or decreasing distance from the point. # a three-dimensional curve that turns around an axis at a constant or continuously varying distance while moving parallel to the axis; a . The first definition describes a |
Polar Coordinate System
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the origin of a Cartesian coordinate system) is called the ''pole'', and the ray from the pole in the reference direction is the ''polar axis''. 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''. Angles in polar notation are generally expressed in either degrees or radians (2 rad being equal to 360°). Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-17th century, though the actual term "polar coordinates" has been attributed to Gregorio Fontana in the 18th century. The initial motivation for the introduction of the polar system was the study of circula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angle
In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two rays lie in the plane (geometry), plane that contains the rays. Angles are also formed by the intersection of two planes. These are called dihedral angles. Two intersecting curves may also define an angle, which is the angle of the rays lying tangent to the respective curves at their point of intersection. ''Angle'' is also used to designate the measurement, measure of an angle or of a Rotation (mathematics), rotation. This measure is the ratio of the length of a arc (geometry), circular arc to its radius. In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of a rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inversely Proportional
In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constant. Two sequences are inversely proportional if corresponding elements have a constant product, also called the coefficient of proportionality. This definition is commonly extended to related varying quantities, which are often called ''variables''. This meaning of ''variable'' is not the common meaning of the term in mathematics (see variable (mathematics)); these two different concepts share the same name for historical reasons. Two functions f(x) and g(x) are ''proportional'' if their ratio \frac is a constant function. If several pairs of variables share the same direct proportionality constant, the equation expressing the equality of these ratios is called a proportion, e.g., (for details see Ratio). Proportionality is closely r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the spoke of a chariot wheel. as a function of axial position ../nowiki>" Spherical coordinates In a spherical coordinate system, the radius describes the distance of a point from a fixed origin. Its position if further defined by the polar angle measured between the radial direction and a fixed zenith direction, and the azimuth angle, the angle between the orthogonal projection of the radial direction on a reference plane that passes through the origin and is orthogonal to the zenith, and a fixed reference direction in that plane. See also *Bend radius *Filling radius in Riemannian geometry *Radius of convergence * Radius of convexity *Radius of curvature *Radius of gyration ''Radius of gyration'' or gyradius of a body about the axis of r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptote
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity. The word asymptote is derived from the Greek ἀσύμπτωτος (''asumptōtos'') which means "not falling together", from ἀ priv. + σύν "together" + πτωτ-ός "fallen". The term was introduced by Apollonius of Perga in his work on conic sections, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve. There are three kinds of asymptotes: ''horizontal'', ''vertical'' and ''oblique''. For curves given by the graph of a function , horizontal asymptotes are horizontal lines that the graph of the function approaches as ''x'' tends to Vertical asymptotes are vertical lines near which the fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Points Of Inflexion
In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa. For the graph of a function of differentiability class (''f'', its first derivative ''f, and its second derivative ''f'''', exist and are continuous), the condition ''f'' = 0'' can also be used to find an inflection point since a point of ''f'' = 0'' must be passed to change ''f'''' from a positive value (concave upward) to a negative value (concave downward) or vice versa as ''f'''' is continuous; an inflection point of the curve is where ''f'' = 0'' and changes its sign at the point (from positive to negative or from negative to positive). A point where the second derivative vanishes but doe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roger Cotes
Roger Cotes (10 July 1682 – 5 June 1716) was an English mathematician, known for working closely with Isaac Newton by proofreading the second edition of his famous book, the '' Principia'', before publication. He also invented the quadrature formulas known as Newton–Cotes formulas, and made a geometric argument that can be interpreted as a logarithmic version of Euler's formula. He was the first Plumian Professor at Cambridge University from 1707 until his death. Early life Cotes was born in Burbage, Leicestershire. His parents were Robert, the rector of Burbage, and his wife, Grace, ''née'' Farmer. Roger had an elder brother, Anthony (born 1681), and a younger sister, Susanna (born 1683), both of whom died young. At first Roger attended Leicester School, where his mathematical talent was recognised. His aunt Hannah had married Rev. John Smith, and Smith took on the role of tutor to encourage Roger's talent. The Smiths' son, Robert Smith, became a close associate of Rog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spirals
In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the American Heritage Dictionary are:Spiral ''American Heritage Dictionary of the English Language'', Houghton Mifflin Company, Fourth Edition, 2009. # a curve on a plane that winds around a fixed center point at a continuously increasing or decreasing distance from the point. # a three-dimensional curve that turns around an axis at a constant or continuously varying distance while moving parallel to the axis; a . The first definition describes a |
