Simpler local approximations are possible, e.g., osculating sphere and local tangent plane.
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Vertices are points where the curve has 4-point contact with the osculating circle at that point., p. 126., p. 142. In contrast, generic points on a curve typically only have 3-point contact with their osculating circle.
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That circle is the osculating circle of the road curve at that point.
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A circle with 1st-order contact (tangent) A circle with 2nd-order contact (osculating) A circle with 3rd-order contact at a vertex of a curve For each point S(t) on a smooth plane curve S, there is exactly one osculating circle, whose radius is the reciprocal of κ(t), the curvature of S at t. Where curvature is zero (at an inflection point on the curve), the osculating circle is a straight line. The locus of the centers of all the osculating circles (also called "centers of curvature") is the evolute of the curve. If the derivative of curvature κ'(t) is zero, then the osculating circle will have 3rd-order contact and the curve is said to have a vertex.
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He is also one of the namesakes of the Tait-Kneser theorem on osculating circles.
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Frenet–Serret frame, and the osculating plane (spanned by T and N). In mathematics, particularly in differential geometry, an osculating plane is a plane in a Euclidean space or affine space which meets a submanifold at a point in such a way as to have a second order of contact at the point. The word osculate is from the Latin osculatus which is a past participle of osculari, meaning to kiss. An osculating plane is thus a plane which "kisses" a submanifold. The osculating plane in the geometry of Euclidean space curves can be described in terms of the Frenet-Serret formulas as the linear span of the tangent and normal vectors.
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An osculating orbit and the object's position upon it can be fully described by the six standard Kepler orbital elements (osculating elements), which are easy to calculate as long as one knows the object's position and velocity relative to the central body. The osculating elements would remain constant in the absence of perturbations. Real astronomical orbits experience perturbations that cause the osculating elements to evolve, sometimes very quickly. In cases where general celestial mechanical analyses of the motion have been carried out (as they have been for the major planets, the Moon, and other planetary satellites), the orbit can be described by a set of mean elements with secular and periodic terms.
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In most situations, it is convenient to set each of these curves tangent to the trajectory at the point of intersection. Curves that obey this condition (and also the further condition that they have the same curvature at the point of tangency as would be produced by the object's gravity towards the central body in the absence of perturbing forces) are called osculating, while the variables parameterising these curves are called osculating elements. In some situations, description of orbital motion can be simplified and approximated by choosing orbital elements that are not osculating. Also, in some situations, the standard (Lagrange-type or Delaunay-type) equations furnish orbital elements that turn out to be non-osculating.
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One speaks also of curves and geometric objects having k-th order contact at a point: this is also called osculation (i.e. kissing), generalising the property of being tangent. (Here the derivatives are considered with respect to arc length.) An osculating curve from a given family of curves is a curve that has the highest possible order of contact with a given curve at a given point; for instance a tangent line is an osculating curve from the family of lines, and has first-order contact with the given curve; an osculating circle is an osculating curve from the family of circles, and has second-order contact (same tangent angle and curvature), etc..
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250x250px Historically, the curvature of a differentiable curve was defined through the osculating circle, which is the circle that best approximates the curve at a point. More precisely, given a point on a curve, every other point of the curve defines a circle (or sometimes a line) passing through and tangent to the curve at . The osculating circle is the limit, if it exists, of this circle when tends to . Then the center and the radius of curvature of the curve at are the center and the radius of the osculating circle.
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The Deep Ecliptic Survey (DES) defines centaurs using a dynamical classification scheme, based on the behavior of orbital integrations over 10 million years. The DES defines centaurs as nonresonant objects whose osculating perihelia are less than the osculating semimajor axis of Neptune at any time during the integration. Using the dynamical definition of a centaur, is a centaur.
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An osculating circle Osculating circles of the Archimedean spiral, nested by the Tait–Kneser theorem. "The spiral itself is not drawn: we see it as the locus of points where the circles are especially close to each other." In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point p on the curve has been traditionally defined as the circle passing through p and a pair of additional points on the curve infinitesimally close to p. Its center lies on the inner normal line, and its curvature defines the curvature of the given curve at that point.
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Osculating orbit (inner, black) and perturbed orbit (red) In astronomy, and in particular in astrodynamics, the osculating orbit of an object in space at a given moment in time is the gravitational Kepler orbit (i.e. an elliptic or other conic one) that it would have around its central body if perturbations were absent. That is, it is the orbit that coincides with the current orbital state vectors (position and velocity).
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Before 2014, Rees has been a near-Earth object of the Amor group, as its perihelion was slightly less than 1.3 AU due to the body's osculating orbit.
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The nested osculating circles of an Archimedean spiral. The spiral itself is not shown, but is visible where the circles are more dense. In differential geometry, the Tait–Kneser theorem states that, if a smooth plane curve has monotonic curvature, then the osculating circles of the curve are disjoint and nested within each other. The logarithmic spiral or the pictured Archimedean spiral provide examples of curves whose curvature is monotonic for the entire curve.
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Furthermore, by considering the limit independently on either side of , this definition of the curvature can sometimes accommodate a singularity at . The formula follows by verifying it for the osculating circle.
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Kneser's theorem on differential equations is named after him, and provides criteria to decide whether a differential equation is oscillating. He is also one of the namesakes of the Tait–Kneser theorem on osculating circles.
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The four-vertex theorem was first proved for convex curves (i.e. curves with strictly positive curvature) in 1909 by Syamadas Mukhopadhyaya. His proof utilizes the fact that a point on the curve is an extremum of the curvature function if and only if the osculating circle at that point has 4th-order contact with the curve (in general the osculating circle has only 3rd-order contact with the curve). The four-vertex theorem was proved in general by Adolf Kneser in 1912 using a projective argument.
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The curve has a unique vertex at the point of tangency with its defining circle. That is, this point is the only point where the curvature reaches a local minimum or local maximum. The defining circle of the witch is also its osculating circle at the vertex, the unique circle that "kisses" the curve at that point by sharing the same orientation and curvature. Because this is an osculating circle at the vertex of the curve, it has third-order contact with the curve.
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The curvature of a differentiable curve was originally defined through osculating circles. In this setting, Augustin-Louis Cauchy showed that the center of curvature is the intersection point of two infinitely close normal lines to the curve.
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The best local spherical approximation to the ellipsoid in the vicinity of a given point is the osculating sphere. Its radius equals the Gaussian radius of curvature as above, and its radial direction coincides with the ellipsoid normal direction. The center of the osculating sphere is offset from the center of the ellipsoid, but is at the center of curvature for the given point on the ellipsoid surface. This concept aids the interpretation of terrestrial and planetary radio occultation refraction measurements and in some navigation and surveillance applications.
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Diagram of a body's direct orbit around the Sun with its nearest (perihelion) and farthest (aphelion) points. The perihelion (q) and aphelion (Q) are the nearest and farthest points respectively of a body's direct orbit around the Sun. Comparing osculating elements at a specific epoch to effectively those at a different epoch will generate differences. The time-of-perihelion-passage as one of six osculating elements is not an exact prediction (other than for a generic 2-body model) of the actual minimum distance to the Sun using the full dynamical model.
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In the particular set of coordinates exampled above, much of the elements has been omitted as unknown or undetermined; for example, the element n allows an approximate time-dependence of the element M to be calculated, but the other elements and n itself are treated as constant, which represents a temporary approximation (see Osculating elements). Thus a particular coordinate system (equinox and equator/ecliptic of a particular date, such as J2000.0) could be used forever, but a set of osculating elements for a particular epoch may only be (approximately) valid for a rather limited time, because osculating elements such as those exampled above do not show the effect of future perturbations which will change the values of the elements. Nevertheless, the period of validity is a different matter in principle and not the result of the use of an epoch to express the data. In other cases, e.g.
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C/2000 W1 (Utsunomiya-Jones) is a long-period comet discovered on November 18, 2000, by Syogo Utsunomiya and Albert F. A. L. Jones. The comet has an observation arc of 58 days allowing a reasonable estimate of the orbit. But the near-parabolic trajectory with an osculating perihelion eccentricity of 0.9999996 generates an extreme unperturbed aphelion distance of 2,034,048 AU (32 light-years). The orbit of a long-period comet is properly obtained when the osculating orbit is computed at an epoch after leaving the planetary region and is calculated with respect to the center of mass of the solar system.
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In more familiar terms, if λ is the contact lift of a curve γ in the plane, then the preferred cycle at each point is the osculating circle. In other words, after taking contact lifts, much of the basic theory of curves in the plane is Lie invariant.
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If, for such a curve, o is any point of the convex hull of a smooth curve on the sphere that is not a vertex of the curve, then at least four points of the curve have osculating planes passing through o. In particular, for a curve not contained in a hemisphere, this theorem can be applied with o at the center of the sphere. Every inflection point of a spherical curve has an osculating plane that passes through the center of the sphere, but this might also be true of some other points. This theorem is analogous to the four-vertex theorem, that every smooth simple closed curve in the plane has four vertices (extreme points of curvature).
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The characterization of the curvature in terms of the derivative of the unit tangent vector is probably less intuitive than the definition in terms of the osculating circle, but formulas for computing the curvature are easier to deduce. Therefore, and also because of its use in kinematics, this characterization is often given as a definition of the curvature.
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This arc length must be greater than the straight-line distance between the same two centers, so the two circles have centers closer together than the difference of their radii, from which the theorem follows. Analogous disjointness theorems can be proved for the family of Taylor polynomials of a given smooth function, and for the osculating conics to a given smooth curve.
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For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature at a point of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point. The curvature of a straight line is zero.
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In differential geometry, Meusnier's theorem states that all curves on a surface passing through a given point p and having the same tangent line at p also have the same normal curvature at p and their osculating circles form a sphere. The theorem was first announced by Jean Baptiste Meusnier in 1776, but not published until 1785.Jean Meusnier: Mém. prés. par div. Etrangers. Acad. Sci.
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When describing general motion, the actual forces acting on a particle are often referred to the instantaneous osculating circle tangent to the path of motion, and this circle in the general case is not centered at a fixed location, and so the decomposition into centrifugal and Coriolis components is constantly changing. This is true regardless of whether the motion is described in terms of stationary or rotating coordinates.
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Methone's orbit is visibly affected by a perturbing 14:15 mean-longitude resonance with the much larger Mimas. This causes its osculating orbital elements to vary with an amplitude of about in semi-major axis, and 5° in longitude of its periapsis on a timescale of about 450 days. Its eccentricity also varies on different timescales between 0.0011 and 0.0037, and its inclination between about 0.003° and 0.020°.
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If there is an intersection point of multiplicity at least 3, the two curves are said to be osculating. If there is only one intersection point, which has multiplicity 4, the two curves are said to be superosculating.. Furthermore, each straight line intersects each conic section twice. If the intersection point is double, the line is a tangent line. Intersecting with the line at infinity, each conic section has two points at infinity.
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Given the trifocal tensor of three views and a pair of matched points in two views, it is possible to determine the location of the point in the third view without any further information. This is known as point transfer and a similar result holds for lines and conics. For general curves, the transfer can be realized through a local differential curve model of osculating circles (i.e., curvature), which can then be transferred as conics.
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Pallene is visibly affected by a perturbing mean-longitude resonance with the much larger Enceladus, although this effect is not as large as Mimas's perturbations on Methone. The perturbations cause Pallene's osculating orbital elements to vary with an amplitude of about 4 km in semi-major axis, and 0.02° in longitude (corresponding to about 75 km). Eccentricity also changes on various timescales between 0.002 and 0.006, and inclination between about 0.178° and 0.184°.
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The principle of ellipsographs were known to Greek mathematicians such as Archimedes and Proklos. If there is no ellipsograph available, one can draw an ellipse using an approximation by the four osculating circles at the vertices. For any method described below, knowledge of the axes and the semi-axes is necessary (or equivalently: the foci and the semi-major axis). If this presumption is not fulfilled one has to know at least two conjugate diameters.
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A tacnode at the origin of the curve defined by (x2+y2 −3x)2−4x2(2−x)=0 In classical algebraic geometry, a tacnode (also called a point of osculation or double cusp). is a kind of singular point of a curve. It is defined as a point where two (or more) osculating circles to the curve at that point are tangent. This means that two branches of the curve have ordinary tangency at the double point.
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Comet Swift–Tuttle (formally designated 109P/Swift–Tuttle) is a large periodic comet with a 1995 (osculating) orbital period of 133 years that is in a 1:11 orbital resonance with Jupiter. It fits the classical definition of a Halley- type comet with a period between 20 and 200 years. It was independently discovered by Lewis Swift on July 16, 1862 and by Horace Parnell Tuttle on July 19, 1862. It has a comet nucleus 26 km in diameter.
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The red Euler spiral is an example of an easement curve between a blue straight line and a circular arc, shown in green. Animation depicting evolution of a Cornu spiral with the tangential circle with the same radius of curvature as at its tip, also known as an osculating circle (click on thumbnail to observe). This sign aside a railroad (between Ghent and Bruges) indicates the start of the transition curve. A parabolic curve (POB) is used.
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C/1999 F1 (Catalina) is one of the longest known long-period comets. It was discovered on March 23, 1999, by the Catalina Sky Survey. The comet has an observation arc of 2,360 days allowing a good estimate of the orbit. The orbit of a long-period comet is properly obtained when the osculating orbit is computed at an epoch after leaving the planetary region and is calculated with respect to the center of mass of the Solar System.
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Contact forms are particular differential forms of degree 1 on odd-dimensional manifolds; see contact geometry. Contact transformations are related changes of coordinates, of importance in classical mechanics. See also Legendre transformation. Contact between manifolds is often studied in singularity theory, where the type of contact are classified, these include the A series (A0: crossing, A1: tangent, A2: osculating, ...) and the umbilic or D-series where there is a high degree of contact with the sphere.
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In, Mukhin, Tarasov, and Varchenko categorified this fact and showed that the Bethe algebra of the Gaudin model on such a space of invariants is isomorphic to the algebra of functions on the intersection of the corresponding Schubert varieties. As an application, they showed that if the Schubert varieties are defined with respect to distinct real osculating flags, then the varieties intersect transversally and all intersection points are real. This property is called the reality of Schubert calculus.
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Comet C/2006 M4 is in a hyperbolic trajectory (with an osculating eccentricity larger than 1) during its passage through the inner Solar System. After leaving the influence of the planets, the eccentricity will drop below 1 and it will remain bound to the Solar System as an Oort cloud comet. Given the extreme orbital eccentricity of this object, different epochs can generate quite different heliocentric unperturbed two-body best-fit solutions to the aphelion distance (maximum distance) of this object.
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Cybele asteroids (also known as the "Cybeles") are a dynamical group of asteroids, named after the asteroid 65 Cybele. Considered by some as the last outpost of an extended asteroid belt, the group consists of nearly 2000 members and a few collisional families. The Cybeles are in a 7:4 orbital resonance with Jupiter. Their orbit is defined by an osculating semi-major axis of 3.28 to 3.70 AU, with an eccentricity of less than 0.3, and an inclination less than 25°.
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ANSI C79.1-2002, IS 14897:2000, and JIS C 7710:1988 define the "A shape" as "a bulb shape having a spherical end section that is joined to the neck by a radius", where the radius is greater than that of the sphere, corresponds to an osculating circle outside the light bulb, and is tangent to both the neck and the sphere. The Energy Star certification only requires omnidirectional light bulbs to fit the overall dimensions of the corresponding ANSI bulb type.
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This song schedule and patchwork for the film kicked off on 13 January 2013. By second last week of the same month the patch work and re-recording of the background score was completed. Post release of theatrical trailer, it was revealed that the filming of ten second osculating scene between lead actors took four hours to complete after several re-takes. The director had devoided the technicians and crew from the location to ease the situation for the actors.
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This causes its osculating orbital elements to vary with an amplitude of about 20 km in semi-major axis on a timescale of about 2 Earth years. The close proximity to the orbits of Pallene and Methone suggests that these moons may form a dynamical family. Material blasted off Anthe by micrometeoroid impacts is thought to be the source of the Anthe Ring Arc, a faint partial ring about Saturn co-orbital with the moon first detected in June 2007.
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In celestial mechanics, the eccentricity vector of a Kepler orbit is the dimensionless vector with direction pointing from apoapsis to periapsis and with magnitude equal to the orbit's scalar eccentricity. For Kepler orbits the eccentricity vector is a constant of motion. Its main use is in the analysis of almost circular orbits, as perturbing (non-Keplerian) forces on an actual orbit will cause the osculating eccentricity vector to change continuously. For the eccentricity and argument of periapsis parameters, eccentricity zero (circular orbit) corresponds to a singularity.
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This geometry can be difficult to visualize because Lie transformations do not preserve points in general: points can be transformed into circles (or spheres). To handle this, curves in the plane and surfaces in space are studied using their contact lifts, which are determined by their tangent spaces. This provides a natural realisation of the osculating circle to a curve, and the curvature spheres of a surface. It also allows for a natural treatment of Dupin cyclides and a conceptual solution of the problem of Apollonius.
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Dence is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the stony Phocaea family (). It orbits the Sun in the inner asteroid belt at a distance of 1.8–2.9 AU once every 3 years and 8 months (1,353 days; semi-major axis of 2.39 AU). Its orbit has an eccentricity of 0.23 and an inclination of 25° with respect to the ecliptic.
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Based on recent HCM-analyses, Auricula is a non-family asteroid that belongs to the main belt's background population. On its osculating Keplerian orbital elements, it is located in the Eunomia region (), where the prominent family of stony asteroids is located. It orbits the Sun in the central main-belt at a distance of 2.4–2.9 AU once every 4 years and 4 months (1,593 days; semi-major axis of 2.67 AU). Its orbit has an eccentricity of 0.09 and an inclination of 11° with respect to the ecliptic.
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Graphic rapresentation of JPL Horizons On-Line Ephemeris System output values JPL Horizons On-Line Ephemeris System provides easy access to key Solar System data and flexible production of highly accurate ephemerides for Solar System objects. Osculating elements at a given epoch are always an approximation to an object's orbit (i.e. an unperturbed conic orbit or a "two-body" orbit). The real orbit (or the best approximation to such) considers perturbations by all planets, a few of the larger asteroids, a few other usually small physical forces, and requires numerical integration.
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Kig can handle any classical object of the dynamic geometry, but also: # The center of curvature and osculating circle of a curve; # The dilation, generic affinity, inversion, projective application, homography and harmonic homology; # The hyperbola with given asymptotes; # The Bézier curves (2nd and 3rd degree); # The polar line of a point and pole of a line with respect to a conic section; # The asymptotes of a hyperbola; # The cubic curve through 9 points; # The cubic curve with a double point through 6 points; # The cubic curve with a cusp through 4 points.
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This is known as the theorema egregium, and was a major discovery of Carl Friedrich Gauss. It is particularly striking when one recalls the geometric definition of the Gaussian curvature of as being defined by the maximum and minimum radii of osculating circles; they seem to be fundamentally defined by the geometry of how bends within . Nevertheless, the theorem shows that their product can be determined from the "intrinsic" geometry of , having only to do with the lengths of curves along and the angles formed at their intersections. As said by Marcel Berger:Berger.
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Klopstock is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Vesta family (), one of the largest asteroid families of bright asteroids in the main-belt. It orbits the Sun in the inner main-belt at a distance of 2.2–2.6 AU once every 3 years and 8 months (1,328 days; semi- major axis of 2.36 AU). Its orbit has an eccentricity of 0.09 and an inclination of 5° with respect to the ecliptic.
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Magion is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been considered a member of the Phocaea family (), a large family with two thousand members, named after 25 Phocaea. It orbits the Sun in the inner asteroid belt at a distance of 2.2–2.7 AU once every 3 years and 10 months (1,401 days; semi-major axis of 2.45 AU). Its orbit has an eccentricity of 0.11 and an inclination of 25° with respect to the ecliptic.
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Maximiliana is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Erigone family (), a large asteroid family named after 163 Erigone. It orbits the Sun in the inner asteroid belt at a distance of 2.0–2.7 AU once every 3 years and 7 months (1,318 days; semi- major axis of 2.35 AU). Its orbit has an eccentricity of 0.15 and an inclination of 5° with respect to the ecliptic.
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A Hirayama family of asteroids is a group of minor planets that share similar orbital elements, such as semimajor axis, eccentricity, and orbital inclination. The members of the families are thought to be fragments of past asteroid collisions. Strictly speaking, families and their membership are identified by analysing the so-called proper orbital elements rather than the current osculating orbital elements, which regularly fluctuate on timescales of tens of thousands of years. The proper elements are related constants of motion that are thought to remain almost constant for times of at least tens of millions of years.
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The concept of osculation can be generalized to higher-dimensional spaces, and to objects that are not curves within those spaces. For instance an osculating plane to a space curve is a plane that has second-order contact with the curve. This is as high an order as is possible in the general case.. In one dimension, analytic curves are said to osculate at a point if they share the first three terms of their Taylor expansion about that point. This concept can be generalized to superosculation, in which two curves share more than the first three terms of their Taylor expansion.
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Paeonia is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Themis family (), a very large family of carbonaceous asteroids, named after 24 Themis. It orbits the Sun in the outer asteroid belt at a distance of 2.5–3.8 AU once every 5 years and 6 months (2,017 days; semi-major axis of 3.12 AU). Its orbit has an eccentricity of 0.22 and an inclination of 2° with respect to the ecliptic.
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But for objects at such high eccentricity, the Sun's barycentric coordinates are more stable than heliocentric coordinates. The orbit of a long-period comet is properly obtained when the osculating orbit is computed at an epoch after leaving the planetary region and is calculated with respect to the center of mass of the Solar System. Using JPL Horizons, the barycentric orbital elements for epoch 1 January 2050 generate a hyperbolic solution. On its closest approach, Comet ISON passed about from Mars on 1 October 2013, and the remnants of Comet ISON passed about from Earth on 26 December 2013.
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Demeter is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, it has also been classified as a member of the Phocaea family (), a large family of stony asteroids, different to Demeter spectral type (see below). It orbits the Sun in the inner main-belt at a distance of 1.8–3.1 AU once every 3 years and 9 months (1,381 days; semi- major axis of 2.43 AU). Its orbit has an eccentricity of 0.26 and an inclination of 25° with respect to the ecliptic.
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Cabot is located in a 10:3 orbital resonance with Jupiter (10/3J), a mean-motion resonance of moderate order and a location of orbital instability. Asteroids in these resonances are known for their chaotic orbits with a relatively short Lyapunov time. It is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Flora family (), a giant asteroid family and the largest family of stony asteroids in the main-belt.
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Special affine curvature, also known as the equiaffine curvature or affine curvature, is a particular type of curvature that is defined on a plane curve that remains unchanged under a special affine transformation (an affine transformation that preserves area). The curves of constant equiaffine curvature are precisely all non-singular plane conics. Those with are ellipses, those with are parabolae, and those with are hyperbolae. The usual Euclidean curvature of a curve at a point is the curvature of its osculating circle, the unique circle making second order contact (having three point contact) with the curve at the point.
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When applying the hierarchical clustering method to its proper orbital elements, Tycho Brahe is a member of the Maria family (), a large family of stony asteroids. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Eunomia family (), the largest family in the intermediate main belt with more than 5,000 members. It orbits the Sun in the central main- belt at a distance of 2.3–2.8 AU once every 4.03 years (1,472 days; semi-major axis of 2.53 AU). Its orbit has an eccentricity of 0.11 and an inclination of 15° with respect to the ecliptic.
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According to modern HCM-analyses by Nesvorný, as well as by Milani and Knežević, Berna is a non-family asteroid from the main belt's background population. Based on osculating Keplerian orbital elements, it is located in the region of the Eunomia family (), a prominent family of stony asteroids. It orbits the Sun in the central asteroid belt at a distance of 2.1–3.2 AU once every 4 years and 4 months (1,584 days; semi-major axis of 2.66 AU). Its orbit has an eccentricity of 0.21 and an inclination of 13° with respect to the ecliptic.
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Adriángalád is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid is located in the densely populated region of the Flora family (), a giant family or clan of stony bodies in the inner asteroid belt. It orbits the Sun in the inner main-belt at a distance of 1.8–2.6 AU once every 3 years and 3 months (1,185 days; semi-major axis of 2.19 AU). Its orbit has an eccentricity of 0.19 and an inclination of 6° with respect to the ecliptic.
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" This was incorrect: an 1832 letter written by Darwin commented that William Sharp Macleay "never imagined such an inosculating creature". The letter preceded Blyth's publication, and indicates that both Darwin and Blyth had independently taken the term from Macleay whose Quinarian system of classification had been popular for a time after its first publication in 1819–1820. In a mystical scheme this grouped separately created genera in "osculating" (kissing) circles. Both Ernst Mayr and Cyril Darlington interpret Blyth's view of natural selection as maintaining the type: :"Blyth's theory was clearly one of elimination rather than selection.
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Kugel is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been considered a member of the Flora family (), a giant asteroid clan and the largest family of stony asteroids in the main belt. It orbits the Sun in the inner asteroid belt at a distance of 1.8–2.8 AU once every 3 years and 6 months (1,289 days; semi-major axis of 2.32 AU). Its orbit has an eccentricity of 0.22 and an inclination of 3° with respect to the ecliptic.
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Hiromiyuki is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Flora family (), a giant asteroid family and the largest family of stony asteroids in the main-belt. Hiromiyuki orbits the Sun in the inner asteroid belt at a distance of 2.0–2.7 AU once every 3 years and 7 months (1,315 days; semi-major axis of 2.35 AU). Its orbit has an eccentricity of 0.16 and an inclination of 2° with respect to the ecliptic.
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Hal is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Flora family (), a giant asteroid family and the largest family of stony asteroids in the main-belt. Hal orbits the Sun in the inner asteroid belt at a distance of 1.8–2.7 AU once every 3 years and 4 months (1,216 days; semi-major axis of 2.23 AU). Its orbit has an eccentricity of 0.21 and an inclination of 6° with respect to the ecliptic.
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Giordano is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a Themistian asteroid that belongs to the Themis family (), a very large family of carbonaceous asteroids, named after 24 Themis. It orbits the Sun in the outer asteroid belt at a distance of 2.7–3.6 AU once every 5 years and 6 months (2,008 days; semi-major axis of 3.11 AU). Its orbit has an eccentricity of 0.15 and an inclination of 1° with respect to the ecliptic.
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Beethoven is a non-family asteroid from the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements and in previous analysis by Zappalà, the asteroid has also been classified as a member of the Themis family (), a very large family of carbonaceous asteroids, named after 24 Themis. It orbits the Sun in the outer main-belt at a distance of 2.5–3.8 AU once every 5 years and 7 months (2,043 days; semi-major axis of 3.15 AU). Its orbit has an eccentricity of 0.19 and an inclination of 3° with respect to the ecliptic.
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Meeus is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Flora family (), a giant asteroid family and the largest family of stony asteroids in the main-belt. It orbits the Sun in the inner main-belt at a distance of 1.7–2.7 AU once every 3 years and 3 months (1,190 days; semi-major axis of 2.2 AU). Its orbit has an eccentricity of 0.23 and an inclination of 5° with respect to the ecliptic.
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Phystech is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Flora family (), a giant asteroid family and the largest family of stony asteroids in the main-belt. It orbits the Sun in the inner asteroid belt at a distance of 2.0–2.4 AU once every 3 years and 4 months (1,206 days; semi-major axis of 2.22 AU). Its orbit has an eccentricity of 0.10 and an inclination of 2° with respect to the ecliptic.
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Okamoto is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Flora family (), a giant asteroid family and the largest family of stony asteroids in the main-belt. It orbits the Sun in the inner asteroid belt at a distance of 1.8–2.5 AU once every 3 years and 2 months (1,160 days; semi- major axis of 2.16 AU). Its orbit has an eccentricity of 0.15 and an inclination of 5° with respect to the ecliptic.
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Owa is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Flora family (), a giant asteroid family and the largest family of stony asteroids in the main-belt. It orbits the Sun in the inner main-belt at a distance of 1.9–2.6 AU once every 3 years and 5 months (1,233 days; semi-major axis of 2.25 AU). Its orbit has an eccentricity of 0.17 and an inclination of 5° with respect to the ecliptic.
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Lapponica is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Flora family (), a giant asteroid family and the largest family of stony asteroids in the main-belt. The asteroid orbits the Sun in the inner main-belt at a distance of 1.9–2.5 AU once every 3 years and 3 months (1,184 days; semi-major axis of 2.19 AU). Its orbit has an eccentricity of 0.14 and an inclination of 5° with respect to the ecliptic.
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Rika is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Flora family (), a giant asteroid family and the largest family of stony asteroids in the main-belt. It orbits the Sun in the inner asteroid belt at a distance of 1.7–2.7 AU once every 3 years and 3 months (1,193 days; semi-major axis of 2.2 AU). Its orbit has an eccentricity of 0.23 and an inclination of 7° with respect to the ecliptic.
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Based on its osculating Keplerian orbital elements, Williams qualifies as a member of the Flora family (), a giant asteroid family and the largest family of stony asteroids in the main- belt (according to Zappalà but not Nesvorý). However, analysis using proper orbital elements in a hierarchical clustering method showed that Williams is a background asteroid, not belonging to any known family (Nesvorý, Milani and Knežević). The asteroid orbits the Sun in the inner main-belt at a distance of 1.7–2.6 AU once every 3 years and 3 months (1,183 days). Its orbit has an eccentricity of 0.20 and an inclination of 4° with respect to the ecliptic.
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Beryl is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Flora family (), a giant asteroid family and the largest family of stony asteroids in the main-belt. It orbits the Sun in the inner asteroid belt at a distance of 2.0–2.5 AU once every 3 years and 4 months (1,216 days; semi-major axis of 2.23 AU). Its orbit has an eccentricity of 0.10 and an inclination of 2° with respect to the ecliptic.
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Augustesen is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Vesta family (), one of the main belt's largest asteroid families named after 4 Vesta, the family's parent body. It orbits the Sun in the inner main-belt at a distance of 2.1–2.7 AU once every 3 years and 9 months (1,379 days; semi-major axis of 2.42 AU). Its orbit has an eccentricity of 0.13 and an inclination of 7° with respect to the ecliptic.
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Given the orbital eccentricity of this object, its orbital period is not a fixed value, because it is frequently perturbed by the gravity of the planets. Near perihelion, using an August 2011 epoch, Kazuo Kinoshita shows C/2010 X1 to have a heliocentric orbital period of 600,000 years, though more perturbations will occur. For objects at such high eccentricity, the Sun's barycentric coordinates are more stable than heliocentric coordinates. The orbit of a long-period comet is properly obtained when the osculating orbit is computed at an epoch after leaving the planetary region and is calculated with respect to the center of mass of the Solar System.
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Britta is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Flora family (), a giant asteroid family and the largest family of stony asteroids in the main-belt. It orbits the Sun in the inner main-belt at a distance of 1.9–2.5 AU once every 3 years and 3 months (1,203 days; semi-major axis of 2.21 AU). Its orbit has an eccentricity of 0.12 and an inclination of 4° with respect to the ecliptic.
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C/1992 J1 (Spacewatch) is a comet that was discovered 1 May 1992 by David Rabinowitz of the Spacewatch Project. This was the first comet to be discovered using an automated system. Using a generic heliocentric (two-body) solution calculated near the time of perihelion (closest approach to the Sun), it is estimated to have an aphelion (Q) (furthest distance from the Sun) of 154,202 AU (more than 2 Light-years). But the orbit of a long-period comet is properly obtained when the osculating orbit is computed at an epoch after leaving the planetary region and is calculated with respect to the center of mass of the solar system.
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Shkodrov is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Flora family (), a giant asteroid family and the largest family of stony asteroids in the asteroid belt. It orbits the Sun in the inner main belt at a distance of 2.0–2.7 AU once every 3 years and 7 months (1,297 days; semi-major axis of 2.33 AU). Its orbit has an eccentricity of 0.14 and an inclination of 2° with respect to the ecliptic.
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Sijthoff is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Flora family (), a giant asteroid family and the largest family of stony asteroids in the main-belt. It orbits the Sun in the inner asteroid belt at a distance of 2.1–2.5 AU once every 3 years and 5 months (1,239 days; semi-major axis of 2.26 AU). Its orbit has an eccentricity of 0.09 and an inclination of 3° with respect to the ecliptic.
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Spahr is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Flora family (), a giant asteroid family and the largest family of stony asteroids in the main-belt. It orbits the Sun in the inner asteroid belt at a distance of 2.0–2.5 AU once every 3 years and 4 months (1,232 days; semi-major axis of 2.25 AU). Its orbit has an eccentricity of 0.09 and an inclination of 7° with respect to the ecliptic.
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As the probability density function of the Cauchy distribution, the witch of Agnesi has applications in probability theory. It also gives rise to Runge's phenomenon in the approximation of functions by polynomials, has been used to approximate the energy distribution of spectral lines, and models the shape of hills. The witch is tangent to its defining circle at one of the two defining points, and asymptotic to the tangent line to the circle at the other point. It has a unique vertex (a point of extreme curvature) at the point of tangency with its defining circle, which is also its osculating circle at that point.
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Dientzenhofer is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Flora family (), a giant asteroid family and the largest family of stony asteroids in the main-belt. The asteroid orbits the Sun in the inner main-belt at a distance of 2.0–2.6 AU once every 3 years and 6 months (1,266 days; semi-major axis of 2.29 AU). Its orbit has an eccentricity of 0.13 and an inclination of 3° with respect to the ecliptic.
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' is a non-family asteroid from the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Flora family (), a giant asteroid family and the largest family of stony asteroids in the main-belt. It orbits the Sun in the inner asteroid belt at a distance of 1.7–2.6 AU once every 3 years and 2 months (1,164 days; semi-major axis of 2.17 AU). Its orbit has an eccentricity of 0.19 and an inclination of 4° with respect to the ecliptic.
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This monotonicity cannot happen for a simple closed curve (by the four- vertex theorem, there are at least four vertices where the curvature reaches an extreme point) but for such curves the theorem can be applied to the arcs of the curves between its vertices. The theorem is named after Peter Tait, who published it in 1896, and Adolf Kneser, who rediscovered it and published it in 1912. Tait's proof follows simply from the properties of the evolute, the curve traced out by the centers of osculating circles. For curves with monotone curvature, the arc length along the evolute between two centers equals the difference in radii of the corresponding circles.
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Biyo is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Flora family (), a giant asteroid clan and the largest family of stony asteroids in the main-belt. It orbits the Sun in the inner asteroid belt at a distance of 2.1–2.4 AU once every 3 years and 5 months (1,252 days; semi-major axis of 2.27 AU). Its orbit has an eccentricity of 0.06 and an inclination of 7° with respect to the ecliptic.
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Desforges is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Eunomia family (), a prominent family of stony S-type asteroid and the largest one in the intermediate main belt with more than 5,000 members. It orbits the Sun in the central main-belt at a distance of 2.2–3.1 AU once every 4 years and 4 months (1,585 days; semi-major axis of 2.66 AU). Its orbit has an eccentricity of 0.18 and an inclination of 11° with respect to the ecliptic.
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Billbaum is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Eunomia family (), a prominent family of stony S-type asteroid and the largest one in the intermediate main belt with more than 5,000 members. It orbits the Sun in the central main-belt at a distance of 2.2–3.2 AU once every 4 years and 5 months (1,606 days; semi-major axis of 2.68 AU). Its orbit has an eccentricity of 0.19 and an inclination of 14° with respect to the ecliptic.
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Petrpravec is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Eunomia family (), a prominent family of stony S-type asteroid and the largest one in the intermediate main belt with more than 5,000 members. It orbits the Sun in the central asteroid belt at a distance of 2.4–2.9 AU once every 4 years and 3 months (1,554 days; semi-major axis of 2.63 AU). Its orbit has an eccentricity of 0.09 and an inclination of 13° with respect to the ecliptic.
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Based on osculating Keplerian orbital elements, Janice is located in the region of the Themis family (), a very large family of carbonaceous asteroids, named after 24 Themis. When applying the hierarchical clustering method to its proper orbital elements, the object is both a non-family asteroid of the main belt's background population (according to Nesvorný), as well as a core member of the Themis family (according to Milani and Knežević). It orbits the Sun in the outer main-belt at a distance of 2.5–3.6 AU once every 5 years and 5 months (1,977 days; semi-major axis of 3.08 AU). Its orbit has an eccentricity of 0.18 and an inclination of 0° with respect to the ecliptic.
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Maupertuis is a member of the Vesta family (), a giant asteroid family of typically bright V-type asteroids. Vestian asteroids have a composition akin to cumulate eucrites (HED meteorites) and are thought to have originated deep within 4 Vesta's crust, possibly from the Rheasilvia crater, a large impact crater on its southern hemisphere near the South pole, formed as a result of a sub-catastrophic collision. Vesta is the main belt's second-largest and second-most-massive body after . Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Flora family (), a giant asteroid family and the largest family of stony asteroids in the main-belt.
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Burney is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. In the HCM assessment by Zappala and based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Flora family (), a giant asteroid family and the largest family of stony asteroids in the main-belt. It orbits the Sun in the inner asteroid belt at a distance of 1.9–2.6 AU once every 3 years and 4 months (1,226 days; semi-major axis of 2.24 AU). Its orbit has an eccentricity of 0.14 and an inclination of 3° with respect to the ecliptic.
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The evolute of a curve will generically have a cusp when the curve has a vertex; other, more degenerate and non-stable singularities may occur at higher-order vertices, at which the osculating circle has contact of higher order than four. Although a single generic curve will not have any higher-order vertices, they will generically occur within a one-parameter family of curves, at the curve in the family for which two ordinary vertices coalesce to form a higher vertex and then annihilate. The symmetry set of a curve has endpoints at the cusps corresponding to the vertices, and the medial axis, a subset of the symmetry set, also has its endpoints in the cusps.
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Sarema is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Nysa family (), the largest asteroid family of the main belt, consisting of stony and carbonaceous subfamilies. The family, named after 44 Nysa, is located in the inner belt near the Kirkwood gap (3:1 orbital resonance with Jupiter), a depleted zone that separates the central main belt. It orbits the Sun in the inner asteroid belt at a distance of 2.1–2.8 AU once every 3 years and 11 months (1,426 days; semi-major axis of 2.48 AU).
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Helvetia is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Flora family (), a giant asteroid family and the largest family of stony asteroids in the main-belt. It orbits the Sun in the inner main-belt at a distance of 1.8–2.8 AU once every 3 years and 6 months (1,276 days; semi- major axis of 2.3 AU). Its orbits the Sun in the inner main-belt at a distance of 1.8–2.8 AU once every 3 years and 6 months (1,277 days; semi-major axis of 2.3 AU).
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Volkonskaya is a member of the Vesta family (), when applying the hierarchical clustering method to its proper orbital elements. Vestian asteroids have a composition akin to cumulate eucrites (HED meteorites) and are thought to have originated deep within 4 Vesta's crust, possibly from the Rheasilvia crater, a large impact crater on its southern hemisphere near the South pole, formed as a result of a subcatastrophic collision. Vesta is the main belt's second-largest and second-most-massive body after . Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Flora family (), a giant asteroid family and the largest family of stony asteroids in the main-belt.
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Lydina is a non-family asteroid from the main belt's background population when applying the hierarchical clustering method to its proper orbital elements. Based on osculating Keplerian orbital elements, the asteroid is considered a member of the dynamical Cybele group, which are asteroid with low orbital inclinations and eccentricities, and with a semi- major axis between 3.3 and 3.5 AU, near the 4:7 orbital resonance with Jupiter. It orbits the Sun in the outermost asteroid belt at a distance of 3.0–3.8 AU once every 6 years and 3 months (2,297 days; semi-major axis of 3.41 AU). Its orbit has an eccentricity of 0.11 and an inclination of 9° with respect to the ecliptic.
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Based on osculating Keplerian orbital elements, Zhuhai has also been classified as a member of the Maria family (), a large family of stony asteroids, named after 170 Maria. When applying the hierarchical clustering method to its proper orbital elements, the asteroid is both a non- family asteroid of the main belt's background population (according to Nesvorný), as well as a core member of the Maria family (according to Milani and Knežević). It orbits the Sun in the central main-belt at a distance of 2.4–2.7 AU once every 4 years and 1 month (1,497 days; semi-major axis of 2.56 AU). Its orbit has an eccentricity of 0.06 and an inclination of 14° with respect to the ecliptic.
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Purple Mountain is a core member of the Vesta family (), a giant asteroid family of typically bright V-type asteroids. Vestian asteroids have a composition akin to cumulate eucrites (HED meteorites) and are thought to have originated deep within 4 Vesta's crust, possibly from the Rheasilvia crater, a large impact crater on its southern hemisphere near the South pole, formed as a result of a subcatastrophic collision. Vesta is the main belt's second-largest and second-most-massive body after . Based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Flora family (), a giant asteroid family and the largest family of stony asteroids in the main-belt.
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Pilcher is a non-family asteroid of the main belt's background population when applying the hierarchical clustering method (HCM) to its proper orbital elements (Nesvorný, Milani and Knežević). In a previous HCM-analysis (Zappalà) and based on osculating Keplerian orbital elements, the asteroid has also been classified as a member of the Flora family (), a giant asteroid family and the largest family of stony asteroids in the main-belt. It orbits the Sun in the Florian region of the inner asteroid belt at a distance of 2.1–2.3 AU once every 3 years and 3 months (1,171 days; semi-major axis of 2.17 AU). Its orbit has an eccentricity of 0.05 and an inclination of 3° with respect to the ecliptic.
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Comet C/2006 P1 follows a hyperbolic trajectory (with an osculating eccentricity larger than 1) during its passage through the inner Solar System, but the eccentricity will drop below 1 after it leaves the influence of the planets and it will remain bound to the Solar System as an Oort cloud comet. Given the orbital eccentricity of this object, different epochs can generate quite different heliocentric unperturbed two-body best-fit solutions to the aphelion distance (maximum distance) of this object. For objects at such high eccentricity, the Sun's barycentric coordinates are more stable than heliocentric coordinates. Using JPL Horizons, the barycentric orbital elements for epoch 2050 generate a semi-major axis of 2050 AU and a period of approximately 92,700 years.
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De Sanctis a member of the Vesta family () when applying the hierarchical clustering method to its proper orbital elements. Vestian asteroids have a composition akin to cumulate eucrites and are thought to have originated deep within 4 Vesta's crust, possibly from the Rheasilvia crater, a large impact crater on its southern hemisphere near the South pole, formed as a result of a subcatastrophic collision. Based on osculating Keplerian orbital elements, De Sanctis has also been classified as a member of the Flora family (), a giant asteroid family and the largest family of stony asteroids in the main-belt. The asteroid orbits the Sun in the inner main-belt at a distance of 2.0–2.6 AU once every 3 years and 7 months (1,313 days; semi-major axis of 2.35 AU).
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The orbit of a long-period comet is properly obtained when the osculating orbit is computed at an epoch after leaving the planetary region and is calculated with respect to the center of mass of the solar system. Using JPL Horizons, the barycentric orbital elements for epoch 2030-Jan-01 generate a semi-major axis of 7,500 AU, an apoapsis distance of 15,000 AU, and a period of approximately 650,000 years. Before entering the planetary region (epoch 1950), C/2007 Q3 had a calculated barycentric orbital period of ~6.4 million years with an apoapsis (aphelion) distance of about 69,000 AU (1.09 light-years). The comet was probably in the outer Oort cloud for millions or billions of years with a loosely bound chaotic orbit until it was perturbed inward.
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In these methods, it is considered that the body is always moving in a conic section, however the conic section is constantly changing due to the perturbations. If all perturbations were to cease at any particular instant, the body would continue in this (now unchanging) conic section indefinitely; this conic is known as the osculating orbit and its orbital elements at any particular time are what are sought by the methods of general perturbations. General perturbations takes advantage of the fact that in many problems of celestial mechanics, the two-body orbit changes rather slowly due to the perturbations; the two-body orbit is a good first approximation. General perturbations is applicable only if the perturbing forces are about one order of magnitude smaller, or less, than the gravitational force of the primary body.
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Animation depicting evolution of a Cornu spiral with the tangential circle with the same radius of curvature as at its tip, also known as an osculating circle. To travel along a circular path, an object needs to be subject to a centripetal acceleration (e.g.: the moon circles around the earth because of gravity; a car turns its front wheels inward to generate a centripetal force). If a vehicle traveling on a straight path were to suddenly transition to a tangential circular path, it would require centripetal acceleration suddenly switching at the tangent point from zero to the required value; this would be difficult to achieve (think of a driver instantly moving the steering wheel from straight line to turning position, and the car actually doing it), putting mechanical stress on the vehicle's parts, and causing much discomfort (due to lateral jerk).
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But in that case (apart from the "equinox of date" case described above), two dates will be associated with the data: one date is the epoch for the time-dependent expressions giving the values, and the other date is that of the coordinate system in which the values are expressed. For example, orbital elements, especially osculating elements for minor planets, are routinely given with reference to two dates: first, relative to a recent epoch for all of the elements: but some of the data are dependent on a chosen coordinate system, and then it is usual to specify the coordinate system of a standard epoch which often is not the same as the epoch of the data. An example is as follows: For minor planet (5145) Pholus, orbital elements have been given including the following data:Harvard Minor Planet Center, data for Pholus Epoch 2010 Jan. 4.0 TT . . .
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Razin developed a closed-form solution for a spherical earth. Williams and Last extended Razin's solution to an osculating sphere earth model. When necessitated by the combination of vehicle-station distance (e.g., hundreds of miles or more) and required solution accuracy, the ellipsoidal shape of the earth must be considered. This has been accomplished using the Gauss–Newton NLLSMinimum Performance Standards (MPS) Automatic Co- ordinate Conversion Systems, Report of RTCM Special Committee No. 75, Radio Technical Commission for Marine Services, Washington, D.C, 1984 method in conjunction with ellipsoid algorithms by Andoyer,"Formule donnant la longueur de la géodésique, joignant 2 points de l'ellipsoide donnes par leurs coordonnées geographiques", Marie Henri Andoyer, Bulletin Geodsique, No. 34 (1932), pages 77–81 Vincenty"Direct and Inverse Solutions of Geodesics on the Ellipsoids with Applications of Nested Equations", Thaddeus Vincenty, Survey Review, XXIII, Number 176 (April 1975) and Sodano.
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Quetzálcoatl is an Amor asteroid – a subgroup of near-Earth asteroids that approach the orbit of Earth from beyond, but do not cross it – and a member of the Alinda family of highly eccentric asteroids. It orbits the Sun at a distance of 1.1–4.0 AU with a period of around 4 years. The osculating orbit as of 2017 has a period just over 4 years, but the period varies because Quetzálcoatl is near the 3:1 orbital resonance with Jupiter (and possibly because it is near the 1:4 resonance with Earth). Its orbit has an eccentricity of 0.57 and an inclination of 20° with respect to the ecliptic. When it was discovered in March 1953 it had a magnitude around 15, but in recent times its magnitude rarely dips below 20 because even when it is near perihelion it is far from Earth. After the 1953 close approach there were others every four years until March 1981, but the next one will not be until 77 years (19 orbits averaging 4,05 years) later, in February 2062, when its magnitude will be about 17.
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