For a while, your friends and family may put these digital traces out of their minds.
|
|
Tessa Hulls traces out the racist foundations that still haunt us in a timeline stretching back to the 1500s.
|
|
The word "empty" is unexpectedly joined to a low F-major triad, and a solo double bass traces out a plaintive micro-melody in that key area.
|
|
Olmi traces out the child's successive captivities and introduces us to the fellow slaves she befriends and loses while being marched in chains from Darfur to Khartoum.
|
|
The trajectory even traces out the shape of a heart during one close pass, while the tight spins around the comet leading up to the fatal collision are like stencil drawings etched in orbit.
|
|
Frazier traces out a web of related concerns: the difficulty of family life in such a place, the imperishability of love, the injustice of a hospital closure, the exclusion of black history, the bonds among generations of women.
|
|
The nimble narrative of Patricia Vigderman's The Real Life of the Parthenon (Mad Creek Books, 2018) traces out this history in a very winning personal account, describing her travels to Athens, London, and the Getty Villa in Los Angeles, and also to Naples and Southern Italy.
|
|
It's full of action and gore, but it's essentially a quiet, thoughtful show that posits a supernatural situation with political relevance — a small band led by a scientist, an exterminator and a Holocaust survivor battle parasitic bloodsuckers ruled by an all-powerful hive mind — and traces out its logical effects.
|
|
It traces out the intersections, historical and personal, between a gay man, Prior Walter, who is developing AIDS; Louis, the lover who abandons him; Louis's new, right-wing Mormon lover, Joe Pitt (married to a Valium-popping woman, Harper); Joe's mother; and Louis's friend and caretaker (and the play's only character of color), Belize.
|
|
On the news site Splinter, the writer Alex Pareene has characterized much of modern conservatism as a grift gone wrong — pulling from the historian Rick Perlstein's 2012 Baffler article "The Long Con," which traces out just how much of the movement's far-right fringe was born and nurtured in self-enriching direct-mail and media operations.
|
|
For all regular polygons, each mouse traces out a pursuit curve in the shape of a logarithmic spiral. These curves meet in the center of the polygon.
|
|
The polarization can be thought of as an orientation perpendicular to the momentum. As the fiber traces out its path, the momentum vector of the light traces out a path on the sphere in momentum space. The path is closed since initial and final directions of the light coincide, and the polarization is a vector tangent to the sphere. Going to momentum space is equivalent to taking the Gauss map.
|
|
The classic Spirograph toy traces out epitrochoid and hypotrochoid curves. The orbits of planets in the once popular geocentric Ptolemaic system are epitrochoids. The combustion chamber of the Wankel engine is an epitrochoid.
|
|
The gnomonic projection is said to be the oldest map projection, developed by Thales in the 6th century BC. The path of the shadow-tip or light-spot in a nodus-based sundial traces out the same hyperbolae formed by parallels on a gnomonic map.
|
|
The world sheet of an open string (with loose ends) is a strip; that of a closed string (a loop) resembles a tube. Once the object is not approximated as a mere point but has extended volume, it traces out not a world line but rather a world tube.
|
|
In mathematics, a prime geodesic on a hyperbolic surface is a primitive closed geodesic, i.e. a geodesic which is a closed curve that traces out its image exactly once. Such geodesics are called prime geodesics because, among other things, they obey an asymptotic distribution law similar to the prime number theorem.
|
|
A skew apeirogon in two dimensions forms a zig-zag line in the plane. If the zig-zag is even and symmetrical, then the apeirogon is regular. Skew apeirogons can be constructed in any number of dimensions. In three dimensions, a regular skew apeirogon traces out a helical spiral and may be either left- or right-handed.
|
|
The terms parametric continuity and geometric continuity (Gn) were introduced by Brian Barsky, to show that the smoothness of a curve could be measured by removing restrictions on the speed, with which the parameter traces out the curve. Parametric continuity is a concept applied to parametric curves, which describes the smoothness of the parameter's value with distance along the curve.
|
|
With the help of his colleague Income Tax Inspector Harichandra (Vivek), he tries to trace Mythili out. A chance look at the video of their marriage throws light on Vishwa’s hatred towards Ragunandan. Ragunandan is convinced that Mythili did not go on her own will. He confronts Rajasekhar, but he is of no help. Finally, he traces out the location of Vishwa’s hideout.
|
|
It thus makes sense to define the hyperbolic angle from P0 to an arbitrary point on the curve as a logarithmic function of the point's value of x.Bjørn Felsager, Through the Looking Glass – A glimpse of Euclid's twin geometry, the Minkowski geometry , ICME-10 Copenhagen 2004; p.14. See also example sheets exploring Minkowskian parallels of some standard Euclidean resultsViktor Prasolov and Yuri Solovyev (1997) Elliptic Functions and Elliptic Integrals, page 1, Translations of Mathematical Monographs volume 170, American Mathematical Society Whereas in Euclidean geometry moving steadily in an orthogonal direction to a ray from the origin traces out a circle, in a pseudo-Euclidean plane steadily moving orthogonally to a ray from the origin traces out a hyperbola. In Euclidean space, the multiple of a given angle traces equal distances around a circle while it traces exponential distances upon the hyperbolic line.
|
|
Meanwhile, at Kavitha's betrothal function, Muthazhagu shows up wearing fake jewellery, for which she is humiliated by Kavitha and Valliyamai. Meanwhile, Chellamma arranges an alliance for Amudha, which she disrupts, giving the same reasons she gave to Vasu. As this goes on, Chellamma does all she can to get justice for Anjali and traces out the other three rapists. Meanwhile, Anjali gets pregnant and is disheartened.
|
|
A second example is linearly polarized light entering a single-mode optical fiber. Suppose the fiber traces out some path in space and the light exits the fiber in the same direction as it entered. Then compare the initial and final polarizations. In semiclassical approximation the fiber functions as a waveguide and the momentum of the light is at all times tangent to the fiber.
|
|
The length of M is called the proper time of the worldline or particle. If the worldline M is a line segment, then the particle is said to be in free fall. A world line traces out the path of a single point in spacetime. A world sheet is the analogous two-dimensional surface traced out by a one-dimensional line (like a string) traveling through spacetime.
|
|
A coordinate grid in spacetime is the set of curves one obtains if three out of four coordinate functions are set to a constant. Sometimes, the term world line is loosely used for any curve in spacetime. This terminology causes confusions. More properly, a world line is a curve in spacetime that traces out the (time) history of a particle, observer or small object.
|
|
On each parallel line we mark the midpoint of the line segment joining these two intersection points. For each parallel line we get a midpoint, and so the locus of midpoints traces out a curve starting at p. The limiting tangent line to the locus of midpoints as we approach p is exactly the affine normal line, i.e. the line containing the affine normal vector to γ(I) at γ(t0).
|
|
He comes through Pisachini's den which leads to the Shergill house and realises that they are a very rich family and decides to rob them. Rakshit traces out the real Shikhar through some detectives using his childhood pictures. The prisoner who has now named himself as Murari, is brought to the Shergill family as the real Shikhar. Shachini kills Patali and puts the blame on Divya, sending her to hell.
|
|
Hyperbolas may be seen in many sundials. On any given day, the sun revolves in a circle on the celestial sphere, and its rays striking the point on a sundial traces out a cone of light. The intersection of this cone with the horizontal plane of the ground forms a conic section. At most populated latitudes and at most times of the year, this conic section is a hyperbola.
|
|
An affine connection defines a notion of development of curves. Intuitively, development captures the notion that if is a curve in , then the affine tangent space at may be rolled along the curve. As it does so, the marked point of contact between the tangent space and the manifold traces out a curve in this affine space: the development of . In formal terms, let be the linear parallel transport map associated to the affine connection.
|
|
The Earth is seen from the lunar surface to rotate, with a period of approximately one Earth day (differing slightly due to the Moon's orbital motion). If the Moon's rotation were purely synchronous, Earth would not have any noticeable movement in the Moon's sky. However, due to the Moon's libration, Earth does perform a slow and complex wobbling movement. Once a month, as seen from the Moon, Earth traces out an approximate oval 18° in diameter.
|
|
This causes the beam to be deflected slightly to the opposite side of the boresight. The antenna is then spun so that it rotates (or nutates) around the boresight axis (pointing in the direction of the target). As it rotates, the beam traces out a cone, with its tip at the antenna and its long axis aligned with the boresight. A target that is centred in the boresight will always return some signal to the radar, creating a strong, constant signal.
|
|
The world line (yellow path) of a photon, which is at location x = 0 at time ct = 0. A spacetime diagram is a graphical illustration of the properties of space and time in the special theory of relativity. Spacetime diagrams allow a qualitative understanding of the corresponding phenomena like time dilation and length contraction without mathematical equations. The history of an object's location throughout all time traces out a line, referred to as the object's world line, in a spacetime diagram.
|
|
Hermeticism is a historiographical term describing the work that attempts to reconstruct the mode of thought held by 17th century scientists. It primarily traces out the connections of Renaissance (16th century) modes of thought with those of the Scientific Revolution (17th century). This type of analysis began with English historians of science in the 1960s. This category of history of science work has largely subsumed earlier academic philosophers' work on the problem of transition from Aristotelianism to 17th century science.
|
|
Geological overview map of the Teltow Today's Teltow plateau in Brandenburg-Berlin was formed around 20,000 years ago during the Brandenburg stage of Weichsel glaciation. The Weichsel ice sheet pushed southwards right over the Teltow before reaching the northern edge of the Baruth Urstromtal, the limit of its expansion to the south. Terminal moraines can be found there, for example, around Dobbrikow in Luckenwalde (Weinberg) and near Sperenberg. However, the line of terminal moraines is very patchy and is traces out an ice front.
|
|
It is said to remain ordinary wine and is used only to facilitate swallowing the bread and so that the people can receive Communion in their customary way. This view is a subject of some controversy. The already consecrated bread used in this Liturgy has been united, at the time it is reserved, with the consecrated wine by placing some of the consecrated wine on the bread with the spoon. In the Russian tradition the wine is placed so that it traces out a cross.
|
|
In the middle figure, the two orthogonal components have the same amplitudes and are 90° out of phase. In this case, one component is zero when the other component is at maximum or minimum amplitude. There are two possible phase relationships that satisfy this requirement: the x component can be 90° ahead of the y component or it can be 90° behind the y component. In this special case, the electric vector traces out a circle in the plane, so this polarization is called circular polarization.
|
|
Satellites in geostationary orbit. A geosynchronous satellite is a satellite in geosynchronous orbit, with an orbital period the same as the Earth's rotation period. Such a satellite returns to the same position in the sky after each sidereal day, and over the course of a day traces out a path in the sky that is typically some form of analemma. A special case of geosynchronous satellite is the geostationary satellite, which has a geostationary orbit – a circular geosynchronous orbit directly above the Earth's equator.
|
|
One of them holds the cockerel. The man closes the blind and returns to the table, where he finds the map and, using his own map of the city, traces out the location marked. The next day, he goes to the spot indicated and enters a dilapidated building just as a man rushes from it in fear. He continues into the interior and descends to a dressing room, where he finds a charred script, a robe embroidered with sigils, greasepaint, a wig with a beard and a cap.
|
|
A conic in the projective plane is a curve C that has the following property: If P is a point not on C, and if a variable line through P meets C at points A and B, then the variable harmonic conjugate of P with respect to A and B traces out a line. The point P is called the pole of that line of harmonic conjugates, and this line is called the polar line of P with respect to the conic. See the article Pole and polar for more details.
|
|
As the tangent plane is rolled on , the point of contact traces out a curve on . Conversely, given a curve on , the tangent plane can be rolled along that curve. This provides a way to identify the tangent planes at different points along the curve: in particular, a tangent vector in the tangent space at one point on the curve is identified with a unique tangent vector at any other point on the curve. These identifications are always given by affine transformations from one tangent plane to another.
|
|
John Durham Peters (born 1958) is the María Rosa Menocal Professor of English and of Film & Media Studies at Yale University. A media historian and social theorist, he has authored a number of noted scholarly works. His first book, Speaking into the Air: A History of the Idea of Communication, traces out broad historical, philosophical, religious, cultural, legal, and technological contexts for the study of communication. His second book Courting the Abyss: Free Speech and the Liberal Tradition updates the philosophy of free expression with a history of liberal thought since Paul of Tarsus.
|
|
It is possible to modify the Bricard polyhedra by adding more faces, in order to move the self- crossing parts of the polyhedron away from each other while still allowing it to flex. The simplest of these modifications is a polyhedron discovered by Klaus Steffen with nine vertices and 14 triangular faces. Steffen's polyhedron is the simplest possible flexible polyhedron without self-crossings. By connecting together multiple shapes derived from the Bricard octahedron, it is possible to construct horn-shaped rigid origami forms whose shape traces out complicated space curves..
|
|
As seen from the orbiting Earth, the Sun appears to move with respect to the fixed stars, and the ecliptic is the yearly path the Sun follows on the celestial sphere. This process repeats itself in a cycle lasting a little over 365 days. The ecliptic is the plane of Earth's orbit around the Sun. From the perspective of an observer on Earth, the Sun's movement around the celestial sphere over the course of a year traces out a path along the ecliptic against the background of stars.
|
|
Linear polarization diagram Linear Circular polarization diagram Circular Elliptical polarization diagram Elliptical polarization In the leftmost figure above, the x and y components of the light wave are in phase. In this case, the ratio of their strengths is constant, so the direction of the electric vector (the vector sum of these two components) is constant. Since the tip of the vector traces out a single line in the plane, this special case is called linear polarization. The direction of this line depends on the relative amplitudes of the two components.
|
|
This potential produces a saddle point in the centre of the trap, which traps ions along the axial direction. The electric field causes ions to oscillate (harmonically in the case of an ideal Penning trap) along the trap axis. The magnetic field in combination with the electric field causes charged particles to move in the radial plane with a motion which traces out an epitrochoid. The orbital motion of ions in the radial plane is composed of two modes at frequencies which are called the magnetron \omega_-and the modified cyclotron \omega_+ frequencies.
|
|
In a CD experiment, equal amounts of left and right circularly polarized light of a selected wavelength are alternately radiated into a (chiral) sample. One of the two polarizations is absorbed more than the other one, and this wavelength-dependent difference of absorption is measured, yielding the CD spectrum of the sample. Due to the interaction with the molecule, the electric field vector of the light traces out an elliptical path after passing through the sample. It is important that the chirality of the molecule can be conformational rather than structural.
|
|
GW invariants are of interest in string theory, a branch of physics that attempts to unify general relativity and quantum mechanics. In this theory, everything in the universe, beginning with the elementary particles, is made of tiny strings. As a string travels through spacetime it traces out a surface, called the worldsheet of the string. Unfortunately, the moduli space of such parametrized surfaces, at least a priori, is infinite-dimensional; no appropriate measure on this space is known, and thus the path integrals of the theory lack a rigorous definition.
|
|
In practical terms, the shadow of the tip of a pole traces out a hyperbola on the ground over the course of a day (this path is called the declination line). The shape of this hyperbola varies with the geographical latitude and with the time of the year, since those factors affect the cone of the sun's rays relative to the horizon. The collection of such hyperbolas for a whole year at a given location was called a pelekinon by the Greeks, since it resembles a double-bladed axe.
|
|
In an FCBGA package, the die is mounted upside-down (flipped) and connects to the package balls via a substrate that is similar to a printed-circuit board rather than by wires. FCBGA packages allow an array of input-output signals (called Area-I/O) to be distributed over the entire die rather than being confined to the die periphery. Traces out of the die, through the package, and into the printed circuit board have very different electrical properties, compared to on-chip signals. They require special design techniques and need much more electric power than signals confined to the chip itself.
|
|
Some Iranians accuse Britain of "trying to topple the regime by supporting insurgents and separatists". Other states however are also believed to be involved in the politics of ethnicity in southern Iran. Professor Efraim Karsh traces out the origins of Saddam Hussein's wish to annex Khuzestan using the ethnic card:Efraim Karsh, The Iran–Iraq War 1980–1988, Osprey Publishing, 2002, pg 27. During Iran's 1979 revolution, after sending thousands of Iraqi Shi'ites into exile in Iran and the quick and brutal suppression of Kurdish dissent, During the cold war, the Soviet Union's "tentacles extended into Iranian Kurdistan".
|
|
The 1919 edition of The Encyclopedia Americana: A Library of Universal Knowledge further adjusted the definition to be "Physiography (geomorphology), now generally recognized as a science distinct from geology, deals with the origins and development of land forms, traces out the topographic expression of structure, and embodies a logical history of oceanic basins, and continental elevations; of mountains, plateaus and plains; of hills and valleys. Physical geography is used loosely as a synonym, but the term is more properly applied to the borderland between geography and physiography; dealing, as it does, largely with the human element as influenced by its physiographic surroundings".
|
|
This neutral hydrogen traces out the large scale structures in the universe, and so can be used to map out the large scale Baryon Acoustic Oscillation (BAO) structure of the universe. The BAO are a fixed comoving size, and so they act as a standard ruler, marking the expansion of the universe over time, and therefore giving information about dark energy and dark matter. For example, if dark energy is not a cosmological constant, as the standard ΛCDM theory of cosmology predicts, then the rate of acceleration of the universe may not be constant over time.
|
|
The lens's optical axis sweeps in the plane of the nominal X and Y axes around the nominal optical Z axis, pivoting on the optical convergence point (out along the Z axis), so that it passes through positions having parallax in relation to the optical convergence point. The circular scanning of the lens's optical axis traces out a coaxial cone pattern with the convergence point as its apex. Early tests revealed that the brain will translate parallax scanned information into depth information at scanning frequencies of between 3–6 Hz, and that the ideal frequency is 4.31 Hz.
|
|
The rotation direction in the circle depends on which of the two phase relationships exists and corresponds to right-hand circular polarization and left-hand circular polarization. In all other cases, where the two components either do not have the same amplitudes and/or their phase difference is neither zero nor a multiple of 90°, the polarization is called elliptical polarization because the electric vector traces out an ellipse in the plane (the polarization ellipse). This is shown in the above figure on the right. Detailed mathematics of polarization is done using Jones calculus and is characterised by the Stokes parameters.
|
|
The ICIJ investigation traces out many levels of offshore holdings in multiple countries related to the business dealings of Beny Steinmetz, with many serious findings such as a request that Mossack Fonseca backdate the revocation of a power of attorney. Mossack Fonseca records show that Sierra Leone diamond exporter Octea, based in the British Virgin Islands with the Steinmetz family as its beneficiaries, is wholly owned by Guernsey-based BSGR Resources, linked to a bribery scandal in Guinea. Foundations in Switzerland and Liechtenstein, among them Nysco and Balda, own BSGR. In 2007, one of Nysco's bank accounts contained $27.7 million.
|
|
To understand why f is single-valued in this domain, imagine a circuit around the unit circle, starting with on the first sheet. When we are still on the first sheet. When we have crossed over onto the second sheet, and are obliged to make a second complete circuit around the branch point before returning to our starting point, where is equivalent to , because of the way we glued the two sheets together. In other words, as the variable z makes two complete turns around the branch point, the image of z in the w-plane traces out just one complete circle.
|
|
On larger, more sophisticated audio mixing consoles an oscilloscope may be built-in for this purpose. On an oscilloscope, we suppose is CH1 and is CH2, is the amplitude of CH1 and is the amplitude of CH2, is the frequency of CH1 and is the frequency of CH2, so is the ratio of frequencies of the two channels, and is the phase shift of CH1. A purely mechanical application of a Lissajous curve with , is in the driving mechanism of the Mars Light type of oscillating beam lamps popular with railroads in the mid-1900s. The beam in some versions traces out a lopsided figure-8 pattern on its side.
|
|
Generally, a state function is of the form F(P, V, T, \ldots) = 0, where denotes pressure, denotes temperature, denotes volume, and the ellipsis denotes other possible state variables like particle number and entropy . If the state space is two-dimensional as in the above example, it can be visualized as a three-dimensional graph (a surface in three-dimensional space). However, the labels of the axes are not unique (since there are more than three state variables in this case), and only two independent variables are necessary to define the state. When a system changes state continuously, it traces out a "path" in the state space.
|
|
Watt's Curve with parameters a=2.1, b=2.2, and c=0.6 Watt's Curve with parameters a=3.1, b=1.1, and c=3.0 Watt's Curve with parameters a=1, b=\sqrt2, and c=1 In mathematics, Watt's curve is a tricircular plane algebraic curve of degree six. It is generated by two circles of radius b with centers distance 2a apart (taken to be at (±a, 0). A line segment of length 2c attaches to a point on each of the circles, and the midpoint of the line segment traces out the Watt curve as the circles rotate partially back and forth or completely around. It arose in connection with James Watt's pioneering work on the steam engine.
|
|
In physics, a world tube is the path of an object that occupies a nonzero region of space (nonzero volume) at every moment in time, as it travels through 4-dimensional spacetime. That is, as it propagates in spacetime, a world tube traces out a three-dimensional volume for every moment in time.Malcolm Ludvigsen: General relativity: a geometric approach, Cambridge University Press, 1999, , p. 74 The world tube is analogous to the one- dimensional world line in that it describes the time evolution of an object in space, with the difference that a world line represents the path of a point particle (of nonzero volume), whereas a world tube occupies finite space at all moments in time.
|
|
Axial precession is the movement of the rotational axis of an astronomical body, whereby the axis slowly traces out a cone. In the case of Earth, this type of precession is also known as the precession of the equinoxes, lunisolar precession, or precession of the equator. Earth goes through one such complete precessional cycle in a period of approximately 26,000 years or 1° every 72 years, during which the positions of stars will slowly change in both equatorial coordinates and ecliptic longitude. Over this cycle, Earth's north axial pole moves from where it is now, within 1° of Polaris, in a circle around the ecliptic pole, with an angular radius of about 23.5°.
|
|
Additionally, due to the arrangement of the fields, with the field being stronger at the outside of the volume, the ion orbits will precess around the inner area. This causes the circular path to move its center of rotation. For instance, if the particle is initially fired into the storage area so that it is orbiting around the bottom half of the mirror area, it will slowly move so the orbit is on one side, then the top, the other side, and then the bottom again. If one traces out the path of a single ion over time, it forms a pattern similar to that of a Spirograph, creating a series of circles that fill the volume.
|
|
Parallel transport of a vector around a closed loop on the sphere: The angle by which it twists, , is proportional to the area inside the loop. In a near- inertial frame moving in tandem with Earth, but not sharing the rotation of the earth about its own axis, the suspension point of the pendulum traces out a circular path during one sidereal day. At the latitude of Paris, 48 degrees 51 minutes north, a full precession cycle takes just under 32 hours, so after one sidereal day, when the Earth is back in the same orientation as one sidereal day before, the oscillation plane has turned by just over 270 degrees. If the plane of swing was north–south at the outset, it is east–west one sidereal day later.
|
|
These disparate research areas can be linked by the following theme: the cosmic dust particles evolve cyclically; chemically, physically and dynamically. The evolution of dust traces out paths in which the Universe recycles material, in processes analogous to the daily recycling steps with which many people are familiar: production, storage, processing, collection, consumption, and discarding. Observations and measurements of cosmic dust in different regions provide an important insight into the Universe's recycling processes; in the clouds of the diffuse interstellar medium, in molecular clouds, in the circumstellar dust of young stellar objects, and in planetary systems such as the Solar System, where astronomers consider dust as in its most recycled state. The astronomers accumulate observational ‘snapshots’ of dust at different stages of its life and, over time, form a more complete movie of the Universe's complicated recycling steps.
|
|
This linkage does not generate a true straight line motion, and indeed Watt did not claim it did so. Rather, it traces out Watt's curve, a lemniscate or figure eight shaped curve; when the lengths of its bars and its base are chosen to form a crossed square, it traces the lemniscate of Bernoulli.. In a letter to Boulton on 11 September 1784 Watt describes the linkage as follows. Although the Peaucellier–Lipkin linkage, Hart's inversor, and other straight line mechanisms generate true straight- line motion, Watt's linkage has the advantage of much greater simplicity than these other linkages. It is similar in this respect to the Chebyshev linkage, a different linkage that produces approximate straight-line motion; however, in the case of Watt's linkage, the motion is perpendicular to the line between its two endpoints, whereas in the Chebyshev linkage the motion is parallel to this line.
|
|
An economic input-output life-cycle assessment, or EIO-LCA involves the use of aggregate sector-level data to quantify the amount of environmental impact that can be directly attributed to each sector of the economy and how much each sector purchases from other sectors in producing its output. Combining such data sets can enable accounting for long chains (for example, building an automobile requires energy, but producing energy requires vehicles, and building those vehicles requires energy, etc.), which somewhat alleviates the scoping problem of traditional life-cycle assessments. EIO-LCA analysis traces out the various economic transactions, resource requirements and environmental emissions (including all the various manufacturing, transportation, mining and related requirements) required for producing a particular product or service. EIO-LCA relies on sector-level averages that may or may not be representative of the specific subset of the sector relevant to a particular product.
|
|
Consider a right angle moving rigidly so that one leg remains on the point P and the other leg is tangent to the curve. Then the vertex of this angle is X and traces out the pedal curve. As the angle moves, its direction of motion at P is parallel to PX and its direction of motion at R is parallel to the tangent T = RX. Therefore, the instant center of rotation is the intersection of the line perpendicular to PX at P and perpendicular to RX at R, and this point is Y. If follows that the tangent to the pedal at X is perpendicular to XY. Draw a circle with diameter PR, then it circumscribes rectangle PXRY and XY is another diameter. The circle and the pedal are both perpendicular to XY so they are tangent at X. Hence the pedal is the envelope of the circles with diameters PR where R lies on the curve.
|
|
Rotation of a Reuleaux triangle within a square, showing also the curve traced by the center of the triangle Any curve of constant width can form a rotor within a square, a shape that can perform a complete rotation while staying within the square and at all times touching all four sides of the square. However, the Reuleaux triangle is the rotor with the minimum possible area. As it rotates, its axis does not stay fixed at a single point, but instead follows a curve formed by the pieces of four ellipses.. Because of its 120° angles, the rotating Reuleaux triangle cannot reach some points near the sharper angles at the square's vertices, but rather covers a shape with slightly rounded corners, also formed by elliptical arcs. At any point during this rotation, two of the corners of the Reuleaux triangle touch two adjacent sides of the square, while the third corner of the triangle traces out a curve near the opposite vertex of the square.
|
|
Between the mirrors a light signal is bouncing, and for the observer resting in the same reference frame as A, the period of clock A is the distance between the mirrors divided by the speed of light. But if the observer looks at clock B, he sees that within that clock the signal traces out a longer, angled path, thus clock B is slower than A. However, for the observer moving alongside with B the situation is completely in reverse: Clock B is faster and A is slower. Also Lorentz (1910–1912) discussed the reciprocity of time dilation and analyzed a clock "paradox", which apparently occurs as a consequence of the reciprocity of time dilation. Lorentz showed that there is no paradox if one considers that in one system only one clock is used, while in the other system two clocks are necessary, and the relativity of simultaneity is fully taken into account.
|
|
The probability of producing the resonance at a given energy is proportional to , so that a plot of the production rate of the unstable particle as a function of energy traces out the shape of the relativistic Breit–Wigner distribution. Note that for values of off the maximum at such that , (hence for ), the distribution has attenuated to half its maximum value, which justifies the name for Γ, width at half-maximum. In the limit of vanishing width, Γ → 0, the particle becomes stable as the Lorentzian distribution sharpens infinitely to . In general, Γ can also be a function of ; this dependence is typically only important when Γ is not small compared to and the phase space-dependence of the width needs to be taken into account. (For example, in the decay of the rho meson into a pair of pions.) The factor of 2 that multiplies Γ2 should also be replaced with 2 (or 4/2, etc.) when the resonance is wide.
|
|
Animated arrowhead construction of Sierpinski gasket Arrowhead construction of the Sierpinski gasket Another construction for the Sierpinski gasket shows that it can be constructed as a curve in the plane. It is formed by a process of repeated modification of simpler curves, analogous to the construction of the Koch snowflake: # Start with a single line segment in the plane # Repeatedly replace each line segment of the curve with three shorter segments, forming 120° angles at each junction between two consecutive segments, with the first and last segments of the curve either parallel to the original line segment or forming a 60° angle with it. At every iteration, this construction gives a continuous curve. In the limit, these approach a curve that traces out the Sierpenski triangle by a single continuous directed (infinitely wiggly) path, which is called the Sierpinski arrowhead.. In fact, the aim of the original article by Sierpinski of 1915, was to show an example of a curve (a Cantorian curve), as the title of the article itself declares.
|
|
If one of the short (uncrossed) edges of an antiparallelogram linkage is fixed in place, and the remaining linkage moves freely, then the crossing point of the antiparallelogram traces out an ellipse that has the fixed edge's endpoints as its foci. The other moving short edge of the antiparallelogram has as its endpoints the foci of another moving ellipse, formed from the first one by reflection across a tangent line through the crossing point.. For both the parallelogram and antiparallelogram linkages, if one of the long (crossed) edges of the linkage is fixed as a base, the free joints move on equal circles, but in a parallelogram they move in the same direction with equal velocities while in the antiparallelogram they move in opposite directions with unequal velocities.. As James Watt discovered, if an antiparallelogram has its long side fixed in this way it forms a variant of Watt's linkage, and the midpoint of the unfixed long edge will trace out a lemniscate or figure eight curve. For the antiparallelogram formed by the sides and diagonals of a square, it is the lemniscate of Bernoulli., pp. 58–59.
|
|