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Involute
In mathematics, an involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the locus of a point on a piece of taut string as the string is either unwrapped from or wrapped around the curve. It is a class of curves coming under the roulette family of curves. The evolute of an involute is the original curve. The notions of the involute and evolute of a curve were introduced by Christiaan Huygens in his work titled '' Horologium oscillatorium sive de motu pendulorum ad horologia aptato demonstrationes geometricae'' (1673). Involute of a parameterized curve Let \vec c(t),\; t\in _1,t_2 be a regular curve in the plane with its curvature nowhere 0 and a\in (t_1,t_2), then the curve with the parametric representation \vec C_a(t)=\vec c(t) -\frac\; \int_a^t, \vec c'(w), \; dw is an ''involute'' of the given curve. Adding an arbitrary but fixed number l_0 to the integral \Bigl(\int_a^t, \ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Involute(in Red) Of Parabola(dark Blue)
In mathematics, an involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the Locus (mathematics), locus of a point on a piece of taut string as the string is either unwrapped from or wrapped around the curve. It is a class of curves coming under the Roulette (curve), roulette family of curves. The evolute of an involute is the original curve. The notions of the involute and evolute of a curve were introduced by Christiaan Huygens in his work titled ''Horologium Oscillatorium, Horologium oscillatorium sive de motu pendulorum ad horologia aptato demonstrationes geometricae'' (1673). Involute of a parameterized curve Let \vec c(t),\; t\in [t_1,t_2] be a regular curve in the plane with its Curvature (mathematics), curvature nowhere 0 and a\in (t_1,t_2), then the curve with the parametric representation \vec C_a(t)=\vec c(t) -\frac\; \int_a^t, \vec c'(w), \; dw is an ''involute'' of the g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Involute
In mathematics, an involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the locus of a point on a piece of taut string as the string is either unwrapped from or wrapped around the curve. It is a class of curves coming under the roulette family of curves. The evolute of an involute is the original curve. The notions of the involute and evolute of a curve were introduced by Christiaan Huygens in his work titled '' Horologium oscillatorium sive de motu pendulorum ad horologia aptato demonstrationes geometricae'' (1673). Involute of a parameterized curve Let \vec c(t),\; t\in _1,t_2 be a regular curve in the plane with its curvature nowhere 0 and a\in (t_1,t_2), then the curve with the parametric representation \vec C_a(t)=\vec c(t) -\frac\; \int_a^t, \vec c'(w), \; dw is an ''involute'' of the given curve. Adding an arbitrary but fixed number l_0 to the integral \Bigl(\int_a^t, \ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gear
A gear is a rotating circular machine part having cut teeth or, in the case of a cogwheel or gearwheel, inserted teeth (called ''cogs''), which mesh with another (compatible) toothed part to transmit (convert) torque and speed. The basic principle behind the operation of gears is analogous to the basic principle of levers. A gear may also be known informally as a cog. Geared devices can change the speed, torque, and direction of a power source. Gears of different sizes produce a change in torque, creating a mechanical advantage, through their ''gear ratio'', and thus may be considered a simple machine. The rotational speeds, and the torques, of two meshing gears differ in proportion to their diameters. The teeth on the two meshing gears all have the same shape. Two or more meshing gears, working in a sequence, are called a gear train or a '' transmission''. The gears in a transmission are analogous to the wheels in a crossed, belt pulley system. An advantage of gears is tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Involute Gear
The involute gear profile is the most commonly used system for gearing today, with cycloid gearing still used for some specialties such as clocks. In an involute gear, the profiles of the teeth are ''involutes of a circle.'' The involute of a circle is the spiraling curve traced by the end of an imaginary taut string unwinding itself from that stationary circle called the base circle, or (equivalently) a triangle wave projected on the circumference of a circle. The involute gear profile was a fundamental advance in machine design, since unlike with other gear systems, the tooth profile of an involute gear depends only on the number of teeth on the gear, pressure angle, and pitch. That is, a gear's profile does not depend on the gear it mates with. Thus, n and m tooth involute spur gears with a given pressure angle and pitch will mate correctly, independently of n and m. This dramatically reduces the number of shapes of gears that need to be manufactured and kept in inventory. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Evolute
In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. That is to say that when the center of curvature of each point on a curve is drawn, the resultant shape will be the evolute of that curve. The evolute of a circle is therefore a single point at its center. Equivalently, an evolute is the envelope of the normals to a curve. The evolute of a curve, a surface, or more generally a submanifold, is the caustic of the normal map. Let be a smooth, regular submanifold in . For each point in and each vector , based at and normal to , we associate the point . This defines a Lagrangian map, called the normal map. The caustic of the normal map is the evolute of . Evolutes are closely connected to involutes: A curve is the evolute of any of its involutes. History Apollonius ( 200 BC) discussed evolutes in Book V of his ''Conics''. However, Huygens is sometimes credited with being the first to study them (1673). Huygens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycloid
In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve. The cycloid, with the cusps pointing upward, is the curve of fastest descent under uniform gravity (the brachistochrone curve). It is also the form of a curve for which the period of an object in simple harmonic motion (rolling up and down repetitively) along the curve does not depend on the object's starting position (the tautochrone curve). History The cycloid has been called "The Helen of Geometers" as it caused frequent quarrels among 17th-century mathematicians. Historians of mathematics have proposed several candidates for the discoverer of the cycloid. Mathematical historian Paul Tannery cited similar work by the Syrian philosopher Iamblichus as evidence that the curve was known in antiquity. English mathematician John Wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallel Curve
A parallel of a curve is the envelope of a family of congruent circles centered on the curve. It generalises the concept of '' parallel (straight) lines''. It can also be defined as a curve whose points are at a constant ''normal distance'' from a given curve. These two definitions are not entirely equivalent as the latter assumes smoothness, whereas the former does not. In computer-aided design the preferred term for a parallel curve is offset curve. (In other geometric contexts, the term offset can also refer to translation.) Offset curves are important for example in numerically controlled machining, where they describe for example the shape of the cut made by a round cutting tool of a two-axis machine. The shape of the cut is offset from the trajectory of the cutter by a constant distance in the direction normal to the cutter trajectory at every point. In the area of 2D computer graphics known as vector graphics, the (approximate) computation of parallel curves is invo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallel Curves
A parallel of a curve is the envelope of a family of congruent circles centered on the curve. It generalises the concept of '' parallel (straight) lines''. It can also be defined as a curve whose points are at a constant ''normal distance'' from a given curve. These two definitions are not entirely equivalent as the latter assumes smoothness, whereas the former does not. In computer-aided design the preferred term for a parallel curve is offset curve. (In other geometric contexts, the term offset can also refer to translation.) Offset curves are important for example in numerically controlled machining, where they describe for example the shape of the cut made by a round cutting tool of a two-axis machine. The shape of the cut is offset from the trajectory of the cutter by a constant distance in the direction normal to the cutter trajectory at every point. In the area of 2D computer graphics known as vector graphics, the (approximate) computation of parallel curves is invol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tractrix
In geometry, a tractrix (; plural: tractrices) is the curve along which an object moves, under the influence of friction, when pulled on a horizontal plane by a line segment attached to a pulling point (the ''tractor'') that moves at a right angle to the initial line between the object and the puller at an infinitesimal speed. It is therefore a curve of pursuit. It was first introduced by Claude Perrault in 1670, and later studied by Isaac Newton (1676) and Christiaan Huygens (1693). Mathematical derivation Suppose the object is placed at (or in the example shown at right), and the puller at the origin (mathematics), origin, so is the length of the pulling thread (4 in the example at right). Then the puller starts to move along the axis in the positive direction. At every moment, the thread will be tangent to the curve described by the object, so that it becomes completely determined by the movement of the puller. Mathematically, if the coordinates of the object are , the o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedean Spiral
The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line that rotates with constant angular velocity. Equivalently, in polar coordinates it can be described by the equation r = a + b\cdot\theta with real numbers and . Changing the parameter moves the centerpoint of the spiral outward from the origin (positive toward and negative toward ) essentially through a rotation of the spiral, while controls the distance between loops. From the above equation, it can thus be stated: position of particle from point of start is proportional to angle as time elapses. Archimedes described such a spiral in his book '' On Spirals''. Conon of Samos was a friend of his and Pappus states that this spiral was discovered by Conon. Derivation of general equation of spiral A p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Horologium Oscillatorium
(English: ''The Pendulum Clock: or Geometrical Demonstrations Concerning the Motion of Pendula as Applied to Clocks'') is a book published by Dutch physicist Christiaan Huygens in 1673 and his major work on pendulums and horology. It is regarded as one of the three most important works on mechanics in the 17th century, the other two being Galileo’s '' Discourses and Mathematical Demonstrations Relating to Two New Sciences'' (1638) and Newton’s (1687). Much more than a mere description of clocks, Huygens's is the first modern treatise in which a physical problem (the accelerated motion of a falling body) is idealized by a set of parameters then analyzed mathematically and constitutes one of the seminal works of applied mathematics.Bruce, I. (2007). Christian Huygens: Horologium Oscillatorium'. Translated and annotated by Ian Bruce. The book is also known for its strangely worded dedication to Louis XIV.. The appearance of the book in 1673 was a political issue, since at t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roulette (curve)
In the differential geometry of curves, a roulette is a kind of curve, generalizing cycloids, epicycloids, hypocycloids, trochoids, epitrochoids, hypotrochoids, and involutes. Definition Informal definition Roughly speaking, a roulette is the curve described by a point (called the ''generator'' or ''pole'') attached to a given curve as that curve rolls without slipping, along a second given curve that is fixed. More precisely, given a curve attached to a plane which is moving so that the curve rolls, without slipping, along a given curve attached to a fixed plane occupying the same space, then a point attached to the moving plane describes a curve, in the fixed plane called a roulette. Special cases and related concepts In the case where the rolling curve is a line and the generator is a point on the line, the roulette is called an involute of the fixed curve. If the rolling curve is a circle and the fixed curve is a line then the roulette is a trochoid. If, in this case, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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