94-592: The AIM-9 Sidewinder ("AIM" for "Air Interception Missile") is a short-range air-to-air missile . Entering service with the United States Navy in 1956 and the Air Force in 1964, the AIM-9 is one of the oldest, cheapest, and most successful air-to-air missiles. Its latest variants remain standard equipment in most Western-aligned air forces. The Soviet K-13 (AA-2 "Atoll"), a reverse-engineered copy of
188-585: A Riemannian manifold , as well as the structure of an abelian Lie group. Perhaps the simplest example of this is when L = Z 2 {\displaystyle \mathbb {Z} ^{2}} : R 2 / Z 2 {\displaystyle \mathbb {R} ^{2}/\mathbb {Z} ^{2}} , which can also be described as the Cartesian plane under the identifications ( x , y ) ~ ( x + 1, y ) ~ ( x , y + 1) . This particular flat torus (and any uniformly scaled version of it)
282-572: A SARH (semi-active radar homing) variant (AIM-9C) and an IR (AIM-9D) in 1963. The AIM-9C's semi-active radar was exclusively tied to the F-8 Crusader 's radar and fire control system (FCS). A total of around 1,000 AIM-9C missiles were launched from 1965 to 1967, but their usage in Vietnam war proved unsuccessful, downing no enemies. A filter modification program for reworked units (to allow high altitude capability up to 18,288m (60,000 feet) This
376-449: A closed path that circles the torus' "hole" (say, a circle that traces out a particular latitude) and then circles the torus' "body" (say, a circle that traces out a particular longitude) can be deformed to a path that circles the body and then the hole. So, strictly 'latitudinal' and strictly 'longitudinal' paths commute. An equivalent statement may be imagined as two shoelaces passing through each other, then unwinding, then rewinding. If
470-514: A fiber bundle over S (the Hopf bundle ). The surface described above, given the relative topology from R 3 {\displaystyle \mathbb {R} ^{3}} , is homeomorphic to a topological torus as long as it does not intersect its own axis. A particular homeomorphism is given by stereographically projecting the topological torus into R 3 {\displaystyle \mathbb {R} ^{3}} from
564-593: A maximal torus ; that is, a closed subgroup which is a torus of the largest possible dimension. Such maximal tori T have a controlling role to play in theory of connected G . Toroidal groups are examples of protori , which (like tori) are compact connected abelian groups, which are not required to be manifolds . Automorphisms of T are easily constructed from automorphisms of the lattice Z n {\displaystyle \mathbb {Z} ^{n}} , which are classified by invertible integral matrices of size n with an integral inverse; these are just
658-752: A product of a Euclidean open disk and a circle. The volume of this solid torus and the surface area of its torus are easily computed using Pappus's centroid theorem , giving: A = ( 2 π r ) ( 2 π R ) = 4 π 2 R r , V = ( π r 2 ) ( 2 π R ) = 2 π 2 R r 2 . {\displaystyle {\begin{aligned}A&=\left(2\pi r\right)\left(2\pi R\right)=4\pi ^{2}Rr,\\[5mu]V&=\left(\pi r^{2}\right)\left(2\pi R\right)=2\pi ^{2}Rr^{2}.\end{aligned}}} These formulas are
752-534: A solid rocket motor for propulsion, similar to most conventional missiles, a continuous-rod fragmentation warhead , and an infrared seeker . The seeker tracks a difference in temperatures detected and uses proportional guidance to achieve impact. Older variants such as the AIM-9B with uncooled seeker heads could only track the high temperatures of engine exhaust , making them strictly rear aspect. Later variants, however, featured liquid nitrogen coolant bottles in
846-521: A 1/3 twist (120°): the 3-dimensional interior corresponds to the points on the 3-torus where all 3 coordinates are distinct, the 2-dimensional face corresponds to points with 2 coordinates equal and the 3rd different, while the 1-dimensional edge corresponds to points with all 3 coordinates identical. These orbifolds have found significant applications to music theory in the work of Dmitri Tymoczko and collaborators (Felipe Posada, Michael Kolinas, et al.), being used to model musical triads . A flat torus
940-663: A contract to support Sidewinder operations through to 2055. Air Force spokeswoman Stephanie Powell said that its relatively low cost, versatility, and reliability mean it is "very possible that the Sidewinder will remain in Air Force inventories through the late 21st century". The AIM-9 was a product of the US Naval Weapons Center at China Lake in the Mojave Desert . It features a lightweight, compact design with cruciform canards and tail fins. It uses
1034-492: A flat torus into 3-dimensional Euclidean space R 3 {\displaystyle \mathbb {R} ^{3}} was found. It is a flat torus in the sense that, as a metric space, it is isometric to a flat square torus. It is similar in structure to a fractal as it is constructed by repeatedly corrugating an ordinary torus at smaller scales. Like fractals, it has no defined Gaussian curvature. However, unlike fractals, it does have defined surface normals , yielding
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#17327803531671128-527: A heat-homing rocket. The name Sidewinder was selected in 1950 and is the common name of Crotalus cerastes , a rattlesnake , which uses infrared sensory organs to hunt warm-blooded prey. It did not receive official funding until 1951 when the effort was mature enough to show to Admiral William "Deak" Parsons , the Deputy Chief of the Bureau of Ordnance (BuOrd). It subsequently received designation as
1222-705: A new nose dome and superior optical filtering. Conversions were done to European AIM-9B to upgrade them to the FGW.2 standard. The official designation is the AIM-9B FGW.2 but it is known as the AIM-9F in US nomenclature. The AIM-9G was very similar to the AIM-9D in most aspects, and did not differ externally. The AIM-9G was an AIM-9D that used an improved AIM-9D seeker head with SEAM (Sidewinder Extended Acquisition Mode), this allowed
1316-641: A program in 1952. Originally called the Sidewinder 1 , the first live firing was on 3 September 1952. The missile intercepted a drone for the first time on 11 September 1953. The missile carried out 51 guided flights in 1954, and in 1955 production was authorized. In 1954, the US Air Force carried out trials with the original AIM-9A and the improved AIM-9B at the Holloman Air Development Center. The first operational use of
1410-481: A radio proximity fuze could be used. These improvements were all added into AIM-9D and went into service with the USN. Around 1,000 AIM-9D units were produced from 1965 to 1969. The primary problem of the AIM-9D was breakup during launch. The AIM-9D was eventually developed into AIM-9G. ATM-9D (USN) : AIM-9D used for captive flight target acquisition training. GDU-1/B : AIM-9D used for firing practice. The AIM-9E "Echo"
1504-429: A rectangle together, choosing the other two sides instead will cause the same reversal of orientation. The first homology group of the torus is isomorphic to the fundamental group (this follows from Hurewicz theorem since the fundamental group is abelian ). The 2-torus is a twofold branched cover of the 2-sphere, with four ramification points . Every conformal structure on the 2-torus can be represented as such
1598-1206: A rectangular strip of flexible material such as rubber, and joining the top edge to the bottom edge, and the left edge to the right edge, without any half-twists (compare Klein bottle ). Torus is a Latin word for "a round, swelling, elevation, protuberance". A torus of revolution in 3-space can be parametrized as: x ( θ , φ ) = ( R + r cos θ ) cos φ y ( θ , φ ) = ( R + r cos θ ) sin φ z ( θ , φ ) = r sin θ {\displaystyle {\begin{aligned}x(\theta ,\varphi )&=(R+r\cos \theta )\cos {\varphi }\\y(\theta ,\varphi )&=(R+r\cos \theta )\sin {\varphi }\\z(\theta ,\varphi )&=r\sin \theta \\\end{aligned}}} using angular coordinates θ , φ ∈ [ 0 , 2 π ) , {\displaystyle \theta ,\varphi \in [0,2\pi ),} representing rotation around
1692-406: A regular torus. For example, in the following map: If R and P in the above flat torus parametrization form a unit vector ( R , P ) = (cos( η ), sin( η )) then u , v , and 0 < η < π /2 parameterize the unit 3-sphere as Hopf coordinates . In particular, for certain very specific choices of a square flat torus in the 3-sphere S , where η = π /4 above, the torus will partition
1786-466: A ring torus is homeomorphic to the Cartesian product of two circles : S 1 × S 1 {\displaystyle S^{1}\times S^{1}} , and the latter is taken to be the definition in that context. It is a compact 2-manifold of genus 1. The ring torus is one way to embed this space into Euclidean space , but another way to do this is the Cartesian product of
1880-454: A so-called "smooth fractal". The key to obtaining the smoothness of this corrugated torus is to have the amplitudes of successive corrugations decreasing faster than their "wavelengths". (These infinitely recursive corrugations are used only for embedding into three dimensions; they are not an intrinsic feature of the flat torus.) This is the first time that any such embedding was defined by explicit equations or depicted by computer graphics. In
1974-544: A sphere — by adding one additional point that represents the limiting case as a rectangular torus approaches an aspect ratio of 0 in the limit. The result is that this compactified moduli space is a sphere with three points each having less than 2π total angle around them. (Such a point is termed a "cusp", and may be thought of as the vertex of a cone, also called a "conepoint".) This third conepoint will have zero total angle around it. Due to symmetry, M* may be constructed by glueing together two congruent geodesic triangles in
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#17327803531672068-420: A spinning disk with lines painted on it, alternately known as a "reticle" or "chopper". The reticle spun at a fixed speed, causing the output of the photocell to be interrupted in a pattern, and the precise timing of the resulting signal indicated the bearing of the target. Although Hamburg and similar devices like Madrid were essentially complete, the work of mating them to a missile had not been carried out by
2162-461: A torus is a closed surface defined as the product of two circles : S × S . This can be viewed as lying in C and is a subset of the 3-sphere S of radius √2. This topological torus is also often called the Clifford torus . In fact, S is filled out by a family of nested tori in this manner (with two degenerate circles), a fact which is important in the study of S as
2256-425: A torus is punctured and turned inside out then another torus results, with lines of latitude and longitude interchanged. This is equivalent to building a torus from a cylinder, by joining the circular ends together, in two ways: around the outside like joining two ends of a garden hose, or through the inside like rolling a sock (with the toe cut off). Additionally, if the cylinder was made by gluing two opposite sides of
2350-426: A torus is the product of two circles, a modified version of the spherical coordinate system is sometimes used. In traditional spherical coordinates there are three measures, R , the distance from the center of the coordinate system, and θ and φ , angles measured from the center point. As a torus has, effectively, two center points, the centerpoints of the angles are moved; φ measures the same angle as it does in
2444-422: A torus of revolution include swim rings , inner tubes and ringette rings . A torus should not be confused with a solid torus , which is formed by rotating a disk , rather than a circle, around an axis. A solid torus is a torus plus the volume inside the torus. Real-world objects that approximate a solid torus include O-rings , non-inflatable lifebuoys , ring doughnuts , and bagels . In topology ,
2538-563: A torus without stretching the paper (unless some regularity and differentiability conditions are given up, see below). A simple 4-dimensional Euclidean embedding of a rectangular flat torus (more general than the square one) is as follows: where R and P are positive constants determining the aspect ratio. It is diffeomorphic to a regular torus but not isometric . It can not be analytically embedded ( smooth of class C , 2 ≤ k ≤ ∞ ) into Euclidean 3-space. Mapping it into 3 -space requires one to stretch it, in which case it looks like
2632-469: A two-sheeted cover of the 2-sphere. The points on the torus corresponding to the ramification points are the Weierstrass points . In fact, the conformal type of the torus is determined by the cross-ratio of the four points. The torus has a generalization to higher dimensions, the n-dimensional torus , often called the n -torus or hypertorus for short. (This is the more typical meaning of
2726-875: A wide variety of missile projects were underway, from huge systems like the Bell Bomi rocket-powered bomber to small systems like air-to-air missiles. By the early 1950s, both the US Air Force and Royal Air Force had started major IR seeker missile projects. The development of the Sidewinder missile began in 1946 at the Naval Ordnance Test Station (NOTS), Inyokern, California, now the Naval Air Weapons Station China Lake , as an in-house research project conceived by William B. McLean . McLean initially called his effort "Local Fuze Project 602" using laboratory funding, volunteer help and fuze funding to develop what they called
2820-459: Is R n {\displaystyle \mathbb {R} ^{n}} modulo the action of the integer lattice Z n {\displaystyle \mathbb {Z} ^{n}} (with the action being taken as vector addition). Equivalently, the n -torus is obtained from the n -dimensional hypercube by gluing the opposite faces together. An n -torus in this sense is an example of an n- dimensional compact manifold . It
2914-461: Is a member of the Lie group SO(4). It is known that there exists no C (twice continuously differentiable) embedding of a flat torus into 3-space. (The idea of the proof is to take a large sphere containing such a flat torus in its interior, and shrink the radius of the sphere until it just touches the torus for the first time. Such a point of contact must be a tangency. But that would imply that part of
AIM-9 Sidewinder - Misplaced Pages Continue
3008-411: Is a torus with the metric inherited from its representation as the quotient , R 2 {\displaystyle \mathbb {R} ^{2}} / L , where L is a discrete subgroup of R 2 {\displaystyle \mathbb {R} ^{2}} isomorphic to Z 2 {\displaystyle \mathbb {Z} ^{2}} . This gives the quotient the structure of
3102-473: Is a very limited weapon, but it had no serious competitors and counters when it was introduced, causing it to be adopted by the USAF and NATO as a standard weapon, with around 80,000 units being produced from 1958 to 1962. The viewing angle of the AIM-9B's sensor was a minuscule 4 degrees, So at launch, the pilot had to accurately aim the aircraft's sight over or above the target (to account for drag). The speed of
3196-457: Is also an example of a compact abelian Lie group . This follows from the fact that the unit circle is a compact abelian Lie group (when identified with the unit complex numbers with multiplication). Group multiplication on the torus is then defined by coordinate-wise multiplication. Toroidal groups play an important part in the theory of compact Lie groups . This is due in part to the fact that in any compact Lie group G one can always find
3290-447: Is known as the "square" flat torus. This metric of the square flat torus can also be realised by specific embeddings of the familiar 2-torus into Euclidean 4-space or higher dimensions. Its surface has zero Gaussian curvature everywhere. It is flat in the same sense that the surface of a cylinder is flat. In 3 dimensions, one can bend a flat sheet of paper into a cylinder without stretching the paper, but this cylinder cannot be bent into
3384-581: Is more accurate and somewhat more resistant to countermeasures. The new rocket motor burns longer and the redesigned body makes the R-13M more maneuverable. K-13M1/R-13M1 : Improved R-13M with new forward fins introduced in 1976. The lackluster performance of the AIM-9B caused the Navy to look for successor. And in 1963 the AAM-N-7 Sidewinder IC was designed, It was developed in two variations:
3478-545: Is the n -fold product of the circle, the n -torus is the configuration space of n ordered, not necessarily distinct points on the circle. Symbolically, T n = ( S 1 ) n {\displaystyle \mathbb {T} ^{n}=(\mathbb {S} ^{1})^{n}} . The configuration space of unordered , not necessarily distinct points is accordingly the orbifold T n / S n {\displaystyle \mathbb {T} ^{n}/\mathbb {S} _{n}} , which
3572-491: Is the quotient of the torus by the symmetric group on n letters (by permuting the coordinates). For n = 2, the quotient is the Möbius strip , the edge corresponding to the orbifold points where the two coordinates coincide. For n = 3 this quotient may be described as a solid torus with cross-section an equilateral triangle , with a twist ; equivalently, as a triangular prism whose top and bottom faces are connected with
3666-400: Is the standard 2-torus, T 2 {\displaystyle \mathbb {T} ^{2}} . And similar to the 2-torus, the n -torus, T n {\displaystyle \mathbb {T} ^{n}} can be described as a quotient of R n {\displaystyle \mathbb {R} ^{n}} under integral shifts in any coordinate. That is, the n -torus
3760-513: The AIM-95 Agile and SRAAM that were intended to replace it. The Sidewinder is the most widely used air-to-air missile in the West, with more than 110,000 missiles produced for the U.S. and 27 other nations, of which perhaps one percent have been used in combat. It has been built under license by Sweden and other nations. The AIM-9 has an estimated 270 aircraft kills. In 2010, Boeing won
3854-454: The Euler characteristic of the n -torus is 0 for all n . The cohomology ring H ( T n {\displaystyle \mathbb {T} ^{n}} , Z ) can be identified with the exterior algebra over the Z - module Z n {\displaystyle \mathbb {Z} ^{n}} whose generators are the duals of the n nontrivial cycles. As the n -torus
AIM-9 Sidewinder - Misplaced Pages Continue
3948-418: The conical scan was very slow, additionally, the uncooled missile had a low sensitivity and was liable to extraneous heat. The AIM-9B was recommended for use on non-threatening targets (like bombers), only from behind (so it can lock on the thermal radiation from the target engines) and only with the sun behind or to the side of your aircraft (as the missile would lock onto it due to its thermal radiation). It
4042-430: The embedding of S 1 {\displaystyle S^{1}} in the plane with itself. This produces a geometric object called the Clifford torus , a surface in 4-space . In the field of topology , a torus is any topological space that is homeomorphic to a torus. The surface of a coffee cup and a doughnut are both topological tori with genus one. An example of a torus can be constructed by taking
4136-478: The hyperbolic plane along their (identical) boundaries, where each triangle has angles of π/2, π/3, and 0. (The three angles of a hyperbolic triangle T determine T up to congruence.) As a result, the Gauss-Bonnet theorem shows that the area of each triangle can be calculated as π - (π/2 + π/3 + 0) = π/6, so it follows that the compactified moduli space M* has area equal to π/3. The other two cusps occur at
4230-454: The square root gives a quartic equation , ( x 2 + y 2 + z 2 + R 2 − r 2 ) 2 = 4 R 2 ( x 2 + y 2 ) . {\displaystyle \left(x^{2}+y^{2}+z^{2}+R^{2}-r^{2}\right)^{2}=4R^{2}\left(x^{2}+y^{2}\right).} The three classes of standard tori correspond to
4324-433: The " moduli space " of the torus to contain one point for each conformal equivalence class, with the appropriate topology. It turns out that this moduli space M may be identified with a punctured sphere that is smooth except for two points that have less angle than 2π (radians) around them: One has total angle = π and the other has total angle = 2π/3. M may be turned into a compact space M* — topologically equivalent to
4418-406: The 3-sphere into two congruent solid tori subsets with the aforesaid flat torus surface as their common boundary . One example is the torus T defined by Other tori in S having this partitioning property include the square tori of the form Q ⋅ T , where Q is a rotation of 4-dimensional space R 4 {\displaystyle \mathbb {R} ^{4}} , or in other words Q
4512-560: The AIM-9B, but is worse than the "D". The canard design was changed to a squared tip double delta planform, this helped improve canard behaviour at higher angles of attack (AOA). Over 5,000 AIM-9B's were rebuilt into AIM-9E's. The AIM-9E appeared in Vietnam after the conclusion of the Operation Rolling Thunder in 1968, with the U.S. Air Force (USAF), becoming one of their main missile armaments. Up until Operation Linebacker in 1972 intense air-to-air activity in Vietnam
4606-554: The AIM-9B, was also widely adopted. Low-level development started in the late 1940s, emerging in the early 1950s as a guidance system for the modular Zuni rocket . This modularity allowed for the introduction of newer seekers and rocket motors, including the AIM-9C variant, which used semi-active radar homing and served as the basis of the AGM-122 Sidearm anti-radar missile . Due to the Sidewinder's infrared guidance system,
4700-487: The Navy opted for a different approach after Walt Freitag, a USN engineer proposed a full change to solid-state in one missile. The "H" variant had major changes over the AIM-9D/G, which had multiple issues with reliability. One of the issues was the intolerance of the vacuum tubes to repeated 20ft/sec sink rate landings by US Navy aircraft on carrier decks. The "H" was the first Sidewinder to be fully solid state, replacing
4794-652: The Taiwan strait resulted in a AIM-9B becoming lodged in a MiG-17 without exploding, allowing it to be removed after landing. The Soviets later became aware that the Chinese had at least one Sidewinder, and after some wrangling, were able to persuade the Chinese to send them one of the captured missiles. K-13/R-3 (AA-2) Variants : K-13/R-3 (Object 300) (AA-2 Atoll): It was the standard variant and entered limited service only two years later in 1960. K-13A/R-3S (Object 310) (AA-2A Atoll) : This entered service in 1962. The R-3S
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#17327803531674888-575: The USAF did not use) ATM-9G (USN) : AIM-9G used for captive flight target acquisition training. Within December 1965, two designers McLean and LaBerge (who were employed by Philco-Ford) came together to create ways to improve the AIM-9G's reliability. One submission was to advance all the remaining missile electronic components from vacuum to solid-state gradually.The US Air Force adhered to this steady replacement of their AIM-9's to solid-state, however
4982-404: The aircraft, rendering it inoperable. The continuous rod warhead features rods welded together to form a cylindrical outer shell, with explosive filler inside. Upon detonation, the rods are scattered in a toroidal shape, ensuring that at least some portion of the shrapnel hits enemy aircraft. Newer models of the AIM-9 sought to increase the range that the seeker head's gimbal can turn, allowing
5076-446: The amount of energy devoted to actuating control surfaces, the AIM-9 does not use active roll stabilization. Instead, it uses rollerons , small metal discs protruding out of the aft end of the tips of the tail fins which spin as the missile flies through the air, providing gyroscopic stabilization. The AIM-9 uses a passive infrared proximity fuze to detonate its warhead near an enemy aircraft, scattering shrapnel that aims to damage
5170-412: The axis of revolution passes twice through the circle, the surface is a spindle torus (or self-crossing torus or self-intersecting torus ). If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a double-covered sphere . If the revolved curve is not a circle, the surface is called a toroid , as in a square toroid. Real-world objects that approximate
5264-615: The brevity code " Fox two " is used when firing the AIM-9. Originally a tail-chasing system, early models saw extensive use during the Vietnam War , but had a low success rate (8% hit rate with the AIM-9E variant). This led to all-aspect capability in the L (Lima) version, which proved an effective weapon during the 1982 Falklands War and Operation Mole Cricket 19 in Lebanon. Its adaptability has kept it in service over newer designs like
5358-416: The circle. The main types of toruses include ring toruses, horn toruses, and spindle toruses. A ring torus is sometimes colloquially referred to as a donut or doughnut . If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution , also known as a ring torus . If the axis of revolution is tangent to the circle, the surface is a horn torus . If
5452-620: The designation AIM-9E-2 As the Sidewinder was being acquired by NATO forces, licensed production was given to West Germany and they would produce around 15,000 units. Like the Americans, the West Germans sought to improve the AIM-9B design due to its limitations. The only visible exterior difference is a greenish sensor window, but many tech improvements were added beneath the shell. Unnoticed improvements include solid state electronics (instead of vacuum tubes), carbon dioxide seeker cooling,
5546-410: The details below. Request from 172.68.168.133 via cp1102 cp1102, Varnish XID 572466768 Upstream caches: cp1102 int Error: 429, Too Many Requests at Thu, 28 Nov 2024 07:52:33 GMT Torus In geometry , a torus ( pl. : tori or toruses ) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with
5640-528: The early AIM-9A & B was that a non-propulsive attachment (NPA) for their MK 15 motor was provided, assuming an assembled missile would be less dangerous to ground crew and material if the rocket motor was ignited. This same NPA was used in the AIM-9B Sidewinder as well. The AIM-9B is very similar to the AIM-9A, but the "B" has a more sophisticated rear and more aerodynamical front fins. The AIM-9B
5734-406: The integral matrices with determinant ±1. Making them act on R n {\displaystyle \mathbb {R} ^{n}} in the usual way, one has the typical toral automorphism on the quotient. The fundamental group of an n -torus is a free abelian group of rank n . The k -th homology group of an n -torus is a free abelian group of rank n choose k . It follows that
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#17327803531675828-472: The internal wiring harnesses. These improvements facilitated a better 100 Hz reticle rate, and a 16.5 deg/sec tracking rate. The most significant design change was the addition of cooling for the PbS detector, adding Peltier (thermoelectric) cooling, giving the advantage of unlimited cooling when positioned on the launch rail, but is only active when electrical power is present. The AIM-9E gives greater range over
5922-400: The last sighting. So if the target remained at 5 degrees left between two rotations of the mirror, the electronics would not output any signal to the control system. Consider a missile fired at right angles to its target; if the missile is flying at the same speed as the target, it should "lead" it by 45 degrees, flying to an impact point far in front of where the target was when it was fired. If
6016-479: The launchers, allowing the missile to track any part of the aircraft heated by air resistance due to high speed flight, giving modern Sidewinders all-aspect capabilities. The nose canards provide maneuverability for the AIM-9, with the AIM-9X using thrust vectoring to augment this. The hot gases generated were used to actuate the nose canards in older models, while newer variants use thermal batteries . To minimize
6110-528: The little-used US Navy AIM-9C Sidewinder. This took longer to develop, and did not enter service until 1966. K-13M/R-13M (Object 380) (AA-2D Atoll) : The R-13M is a much improved version of the R-3S and has capabilities similar to the AIM-9G Sidewinder. The R-13M is still a tail engagement missile only but is far more capable than the R-3S due to its new seeker and rocket motor. The new cooled seeker
6204-422: The missile automatically got pre-launch instructions. The conical scanning speed was also increased greatly. The seeker head was now able to seek in a 25˚ circular scan. This allowed the AIM-9G to have an improved chance of acquiring the target than earlier models. This, along with other upgraded solid-state modules, culminated in the AIM-9G. The improvement was substantial enough that an order of 5,000 AIM-9D seekers
6298-506: The missile is traveling four times the speed of the target, it should follow an angle about 11 degrees in front. In either case, the missile should keep that angle all the way to interception, which means that the angle that the target makes against the detector is constant. It was this constant angle that the Sidewinder attempted to maintain. This " proportional pursuit " system is straightforward to implement and offers high-performance lead calculation almost for free and can respond to changes in
6392-457: The missile to track aircraft at greater angles from its direct line of sight, or boresight. Models such as the AIM-9L, AIM-9M, and AIM-9X feature high off-boresight capabilities, meaning they are able to track targets at high seeker gimbal angles, or highly distant from its boresight. The Sidewinder is not guided by the actual position recorded by the detector, but by the change in position since
6486-546: The missile was by Grumman F9F-8 Cougars and FJ-3 Furies of the United States Navy in the middle of 1956. Nearly 100,000 of the first generation (AIM-9B/C/D/E) of the Sidewinder were produced with Raytheon and General Electric as major subcontractors. Philco-Ford produced the guidance and control sections of the early missiles. The NATO version of the first-generation missile was built under license in Germany by Bodenseewerk Gerätetechnik ; 9,200 examples were built. AIM-9A
6580-430: The north pole of S . The torus can also be described as a quotient of the Cartesian plane under the identifications or, equivalently, as the quotient of the unit square by pasting the opposite edges together, described as a fundamental polygon ABA B . The fundamental group of the torus is just the direct product of the fundamental group of the circle with itself: Intuitively speaking, this means that
6674-458: The original vacuum tubes. The AIM-9H also included a new lead sulphide detector, using nitrogen cooling. The new guidance package was built using semiconductors. When the engineers redesigned these electronics, they essentially kept the AIM-9G's optical system, but the tracking rate increased further, from the original 12˚ to 20˚ degrees per second, this complementing the more powerful 120 lb.ft actuators that had been installed. They also replaced
6768-473: The other 7 were MiG-21s. This was due to the missile design and USN fighter pilot training at TOPGUN . The United States Air Force attempted to attain AIM-9Gs from the USN, due to bad experience with their AIM-9 Sidewinders models (B, E, and J), but they were incompatible with US Air Force's Sidewinder launchers due to the different cooling mechanisms. (the USN used a nitrogen gas container on the launcher, which
6862-431: The same as for a cylinder of length 2π R and radius r , obtained from cutting the tube along the plane of a small circle, and unrolling it by straightening out (rectifying) the line running around the center of the tube. The losses in surface area and volume on the inner side of the tube exactly cancel out the gains on the outer side. Expressing the surface area and the volume by the distance p of an outermost point on
6956-412: The slewing of the optics through a search pattern to acquire the enemy (most likely using a rosette scan ), it also allowed the slaving of the optics to a radar or helmet sight. This was connected to the onboard computer of the aircraft, which gave the capability of capturing the target using the data coming from the airborne radar. This meant that the target could be locked without being in the sights, and
7050-491: The spherical system, but is known as the "toroidal" direction. The center point of θ is moved to the center of r , and is known as the "poloidal" direction. These terms were first used in a discussion of the Earth's magnetic field, where "poloidal" was used to denote "the direction toward the poles". In modern use, toroidal and poloidal are more commonly used to discuss magnetic confinement fusion devices. Topologically ,
7144-449: The study of Riemann surfaces , one says that any two smooth compact geometric surfaces are "conformally equivalent" when there exists a smooth homeomorphism between them that is both angle-preserving and orientation-preserving. The Uniformization theorem guarantees that every Riemann surface is conformally equivalent to one that has constant Gaussian curvature . In the case of a torus, the constant curvature must be zero. Then one defines
7238-1057: The surface of the torus to the center, and the distance q of an innermost point to the center (so that R = p + q / 2 and r = p − q / 2 ), yields A = 4 π 2 ( p + q 2 ) ( p − q 2 ) = π 2 ( p + q ) ( p − q ) , V = 2 π 2 ( p + q 2 ) ( p − q 2 ) 2 = 1 4 π 2 ( p + q ) ( p − q ) 2 . {\displaystyle {\begin{aligned}A&=4\pi ^{2}\left({\frac {p+q}{2}}\right)\left({\frac {p-q}{2}}\right)=\pi ^{2}(p+q)(p-q),\\[5mu]V&=2\pi ^{2}\left({\frac {p+q}{2}}\right)\left({\frac {p-q}{2}}\right)^{2}={\tfrac {1}{4}}\pi ^{2}(p+q)(p-q)^{2}.\end{aligned}}} As
7332-474: The target's flight path, which is much more efficient and makes the missile "lead" the target. During World War II , various researchers in Germany designed infrared guidance systems of various complexity. The most mature development of these, codenamed Hamburg , was intended for use by the Blohm & Voss BV 143 glide bomb in an anti-ship role. Hamburg used a single IR photocell as its detector along with
7426-439: The term " n -torus", the other referring to n holes or of genus n . ) Just as the ordinary torus is topologically the product space of two circles, the n -dimensional torus is topologically equivalent to the product of n circles. That is: The standard 1-torus is just the circle: T 1 = S 1 {\displaystyle \mathbb {T} ^{1}=\mathbb {S} ^{1}} . The torus discussed above
7520-617: The thermal battery with a turbo-alternator. The AIM-9H also included a continuous-rod bundle warhead, improving its destructive capability. The AIM-9H was the last and most manoeuvrable of the rear-aspect USN Sidewinders, with USN moving to the all-aspect AIM-9L. The AIM-9H was actually used at the very end of the Vietnam war, with it being introduced into the US navy service in 1972 and being used in Operation Linebacker . A total of around 7,700 AIM-9H units would be manufactured from 1972-1974 by Philco-Ford and Raytheon. The AIM-9H
7614-426: The three possible aspect ratios between R and r : When R ≥ r , the interior ( x 2 + y 2 − R ) 2 + z 2 < r 2 {\displaystyle {\textstyle {\bigl (}{\sqrt {x^{2}+y^{2}}}-R{\bigr )}^{2}}+z^{2}<r^{2}} of this torus is diffeomorphic (and, hence, homeomorphic) to
7708-458: The time the war ended. In the immediate post-war era, Allied military intelligence teams collected this information, along with many of the engineers working on these projects. Several lengthy reports on the various systems were produced and disseminated among the Western aircraft firms, while a number of the engineers joined these companies to work on various missile projects. By the late 1940s
7802-484: The torus, since it has zero curvature everywhere, must lie strictly outside the sphere, which is a contradiction.) On the other hand, according to the Nash-Kuiper theorem , which was proven in the 1950s, an isometric C embedding exists. This is solely an existence proof and does not provide explicit equations for such an embedding. In April 2012, an explicit C (continuously differentiable) isometric embedding of
7896-534: The torus. The typical doughnut confectionery has an aspect ratio of about 3 to 2. An implicit equation in Cartesian coordinates for a torus radially symmetric about the z {\displaystyle z} - axis is ( x 2 + y 2 − R ) 2 + z 2 = r 2 . {\displaystyle {\textstyle {\bigl (}{\sqrt {x^{2}+y^{2}}}-R{\bigr )}^{2}}+z^{2}=r^{2}.} Algebraically eliminating
7990-420: The tube and rotation around the torus' axis of revolution, respectively, where the major radius R {\displaystyle R} is the distance from the center of the tube to the center of the torus and the minor radius r {\displaystyle r} is the radius of the tube. The ratio R / r {\displaystyle R/r} is called the aspect ratio of
8084-559: Was a pre-production of the Sidewinder, first fired successfully in September 1953. Missile production began in 1955, and the first models entered the Navy's fleet service in 1956. Generally, it was a prototype production run, with 240 pieces being produced, and mainly intended for training pilots in air combat techniques. The AIM-9A was initially called the AAM-N-7 before the tri-service designation change in 1962. An interesting fact about
8178-453: Was added for the fuze, being the first in the world. This enhanced the missile's head sensitivity. Maneuverability was also improved with a faster tracking rate, as well as a new actuator system. The Sidewinder's range was improved as well, with the new Hercules MK 36 solid-fuel rocket motor allowing the missile to fly up to 18km. Finally, a new Mk 48 continuous-rod warhead was fitted to the missile for increased damage; this also meant infrared or
8272-432: Was famously the first Sidewinder variant to be fired in anger as on 24 September 1958, it achieved the world's first successful kill with a air-to-air missiles, when Taiwanese F-86Fs shot down Communist Chinese MiG-15s using AIM-9Bs supplied and fitted by the U.S. Navy (USN). RB24 : A Swedish AIM-9B Sidewinder. K-13/R-3 (AA-2) : The K-13/R-3 was a reversed engineered AIM-9B Sidewinder, A engagement on 28 September 1958 in
8366-550: Was not present. There were 71 AIM-9E launch attempts from January to October 1972, however, only 6 missiles managed to down an aircraft, with 1 other hitting an aircraft, but not causing complete destruction. Reasons for the poor success rate was listed as "poor air crew training, launches out of the envelope, the tactical situation, marginal tone, tone discrimination, the missile going ballistic, and other malfunctions". AIM-9E : Standard production model. AIM-9E-2 : Some "E" models are equipped with reduced-smoke rocket motors and have
8460-525: Was stopped at 1,850 units, with the rest being ordered to AIM-9G seeker specifications instead. Around 2120 AIM-9G were built by Raytheon from 1970 to 1972. The AIM-9G would be used with its predecessor, the AIM-9D, during the Vietnam War, as the US Navy's choice of IR missile. A 46% hit rate with the AIM-9G during Operation Linebackers I and II in 1972 was achieved, of which 14 aircraft were MiG-17s and
8554-543: Was the basis for the all-aspect USAF/USN AIM-9L. ATM-9H : Was a training version of the AIM-9H for captive flight target acquisition. The AIM-9K was a planned U.S. Navy (USN) upgrade to the AIM-9H, but the development was abandoned in favour of USAF/USN joint AIM-9L. Air-to-air missile Too Many Requests If you report this error to the Wikimedia System Administrators, please include
8648-491: Was the first version developed solely by the U.S. Air Force (USAF). The AIM-9E allows the expansion of the weapons acquisition envelope, especially at low-altitude, increasing its Probability of Kill (P[k]). It achieved this using a new low-drag conical nose head, being a distinguishing feature of USAF Sidewinders. A magnesium fluoride seeker dome was introduced, along with a more compact optical assembly, an improved guidance control system, new electronics, and significant changes to
8742-441: Was the first version to enter widespread production, in spite of a very long seeker settling time around 22 seconds, as opposed to 11 seconds for the original version. PL-2 : Chinese produced R-3S. A-91 : Romanian produced R-3S. K-13R/R-3R (Object 320) (AA-2B/C Atoll) : While the R-3S was being introduced in 1961, work started on a semi-active radar homing (SARH) version for high-altitude use, with 8 km range, similar to
8836-427: Was the only planned modification. Recognizing the limitations of the initial AIM-9B, the US Navy (USN) worked to improve the missile's performance. They changed the missile nose to an aerodynamical ogival nose. The seeker was improved with a wider field of view beyond 25 degrees and a reduced instantaneous field of view of 2.5 degrees, to reduce foreign thermal interference (from flares). A better nitrogen cooling system
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