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Meaning of the word hyperboloid
hyperboloid in the crossword dictionary
Explanatory dictionary of the Russian language. D.N. Ushakov
hyperboloid
hyperboloid, m. (mat.). The surface formed by rotating a hyperbola (in 1 value).
New explanatory dictionary of the Russian language, T. F. Efremova.
hyperboloid
m. An open surface formed by rotating a hyperbola (2*) around one of its axes (in geometry).
Encyclopedic Dictionary, 1998
Hyperboloid
Hyperboloid(from - hyperbole, and - appearance, appearance). In mathematics, a hyperboloid is a type of second-order surface in three-dimensional space, given in Cartesian coordinates by the equation
$(x^2 \over a^2) + (y^2 \over b^2) - (z^2 \over c^2)=1$ ( single-sheet hyperboloid),
Where a And b- real semi-axes, and c- imaginary semi-axis;
$- (x^2 \over a^2) - (y^2 \over b^2) + (z^2 \over c^2)=1$ ( two-sheet hyperboloid),
Where a And b- imaginary semi-axes, and c- real semi-axis.
If a = b, then such a surface is called hyperboloid of revolution. A one-sheet hyperboloid of revolution can be obtained by rotating the hyperbola around its imaginary axis, and a two-sheet hyperboloid around its real axis. A two-sheet hyperboloid of revolution is also the locus of points P, the modulus of the difference in distances from which to two given points A and B is constant: ∣ AP − BP∣ = const. In this case, A and B are called the foci of the hyperboloid.
A one-sheet hyperboloid is a doubly ruled surface; if it is a hyperboloid of revolution, then it can be obtained by rotating a line around another line intersecting it.
Hyperboloid (disambiguation)
Hyperboloid:
- A hyperboloid is a type of second-order surface in three-dimensional space.
- “Engineer Garin's Hyperboloid” (1925-1927) - a science fiction novel by A. N. Tolstoy, which describes a fictional device called a hyperboloid, based on its main operating element - a concave mirror in the shape of a hyperboloid, which made it possible to concentrate a heat beam and transmit it over a distance, which turned the hyperboloid into a weapon of fantastic destructive power.
- “Engineer Garin's Hyperboloid” is a film (1965) based on the novel of the same name by A. N. Tolstoy. The role of engineer Garin was played by Evgeny Evstigneev.
Examples of the use of the word hyperboloid in literature.
Big milling machine was busy processing the last large beryllium part - a half-meter metal hyperboloid twenty centimeters at the widest part.
The waves crashed against the steps and plinths on the sides of the stairs, where, instead of the usual statues or vases, there were four bronze lattice towers supporting gilded balls - they contained charged hyperboloids threatening the approaches from the ocean.
Three hundred meters deeper than that, at the bottom of the mine, they were buzzing. hyperboloids and there were short, continuous explosions of hot soil cooled by compressed air.
Its interior consisted of a complex system of blower and outlet pipes, fasteners, a network of wires, duralumin wells, inside which moved elevator scoops, pulleys, platforms for elevator transmission and platforms where liquid air and hyperboloids.
Garin has billions, newspapers all over the world are shouting about Garin,” said Levi, “he managed to build hyperboloid, he has taken possession of an island in the Pacific Ocean and is preparing for big things.
Zoya didn't know if she managed to knock down the big one hyperboloid, or only the stars were no longer visible behind the smoke.
Garin was going to use hyperboloid for transmitting electricity wirelessly.
Driving along the causeway, Hackworth opened it, wanting to see if he could fit a bowler hat inside without folding, crushing, twisting, or otherwise damaging the elegant hyperboloid fields.
From a small height it resembled a sculptural structure, twisted hyperboloid, tapering downwards.
Beam hyperboloid danced madly among this destruction, groping for the most important thing - warehouses of explosive semi-finished products.
Anyone who even touches the top casing hyperboloid, subject to death penalty.
In the left wing of the castle, adjacent to the foot of the tower of the large hyperboloid, they set up furnaces and smeared earthenware crucibles where the gold was supposed to evaporate.
Jansen grabbed him - he smelled of Madame Lamole's perfume. Then he remembered that from the office there was an underground passage to the large elevator hyperboloid and there must be a secret door somewhere here.
Madame Lamole said: - The elevator of the big hyperboloid does not work, the elevator is raised to the very top.
They climbed the rock, where the lattice tower of a large hyperboloid, and a long bundle was lowered at its foot.
Single-sheet hyperboloid. Surface defined by equation
called a one-sheet hyperboloid. This surface has three planes of symmetry - coordinate planes, since the current coordinates y and z are included in equation (55) in even powers.
Intersecting a one-sheet hyperboloid with a plane we obtain a hyperbola ABCD lying in the plane (Fig. 97)
Similarly, in a section of a single-sheet hyperboloid by a plane, we obtain the hyperbola EFGH
in-plane
When a single-sheet hyperboloid is intersected by a plane, the result is an ellipse BFCG, the equations of which have the form:
The semi-axes of this ellipse increase with increasing absolute value h.
When you get an ellipse lying in the plane and having the smallest semi-axes a and b. When we obtain a one-sheet hyperboloid of revolution
When planes intersect it, circles will be obtained
In paragraphs 2 and 3, cylindrical and conical surfaces were considered, each of which is composed of straight lines. It turns out that a one-sheet hyperboloid can also be considered as a surface composed of straight lines. Consider the straight line defined by the equations
in which a, b and c are the semi-axes of a single-sheet hyperboloid, and k is an arbitrarily chosen number
Multiplying these equations term by term, we obtain the equation
i.e., the equation of a one-sheet hyperboloid.
Thus, the equation of a one-sheet hyperboloid is a consequence of the system of equations (59). Therefore, the coordinates of any point satisfying the system of equations (59) also satisfy equation (55) of a one-sheet hyperboloid. In other words, all points of the line (59) belong to the hyperboloid (55). By changing the values of k, we get a whole family of lines lying on the surface (55). Similarly, it can be shown that a one-sheet hyperboloid contains all direct families
where is an arbitrary parameter.
It can also be shown that through each point of a one-sheet hyperboloid there passes one straight line from each of the indicated families. Thus, a single-sheet hyperboloid can be considered as a surface composed of straight lines (Fig. 98). These lines are called rectilinear generators of a one-sheet hyperboloid.
The ability to compose the surface of a single-sheet hyperboloid from straight lines is used in construction technology.
So, for example, according to the design proposed by engineer V. G. Shukhov, a radio mast was built in Moscow using beams located along the rectilinear generatrices of a single-cavity hyperboloid.
Two-sheet hyperboloid. Surface defined by equation
called a two-sheet hyperboloid.
Coordinate planes are planes of symmetry for a two-sheet hyperboloid.
Intersecting this surface with coordinate planes we obtain, respectively, hyperbolas
- (Greek, from hyperbole hyperbole, and eidos similarity). An open curved surface of the 2nd order, resulting from the rotation of a hyperbola. Dictionary of foreign words included in the Russian language. Chudinov A.N., 1910. HYPERBOLOID Greek, from hyperbole, ... ... Dictionary of foreign words of the Russian language
hyperboloid- a, m. hyperboloïde m. mat. An open surface formed by rotating a hyperbola around one of its axes. BAS 2. Hyperboloid of engineer Garin. Lex. Jan. 1803: hyperboloid; SAN 1847: hyperbolic/d: BAS 1954: hyperbolic/id... Historical Dictionary Gallicisms of the Russian language
HYPERBOLOID- HYPERBOLOID, hyperboloid, male. (mat.). The surface formed by rotating a hyperbola (in 1 value). Ushakov's explanatory dictionary. D.N. Ushakov. 1935 1940 … Ushakov's Explanatory Dictionary
hyperboloid- noun, number of synonyms: 2 conoid (4) surface (32) ASIS Dictionary of Synonyms. V.N. Trishin. 2013… Dictionary of synonyms
Hyperboloid- Single-sheet hyperboloid. HYPERBOLOID (from hyperbola and Greek eidos view), a surface that is obtained by rotating a hyperbola around one of the axes of symmetry. In one case a two-sheet hyperboloid is formed, in the other a single-sheet... ... Illustrated Encyclopedic Dictionary
hyperboloid- hiperboloidas statusas T sritis fizika atitikmenys: engl. hyperboloid vok. Hyperboloid, m rus. hyperboloid, m pranc. hyperboloïde, m … Fizikos terminų žodynas
Hyperboloid- (mat.) Two types of second-order surfaces are known under this name. 1) Unisexual geometrical structure. This surface, related to the axes of symmetry, has the equation x2/a2 + y2/b2 z2/c2 = 1. Unisexual geometrical structure is a ruled surface and there are two systems on it... ... Encyclopedic Dictionary F. Brockhaus and I.A. Efron
Hyperboloid- m. An open surface formed by the rotation of a hyperbola [hyperbola II] around one of its axes (in geometry). Ephraim's explanatory dictionary. T. F. Efremova. 2000... Modern explanatory dictionary Russian language Efremova
hyperboloid- hyperboloid, hyperboloids, hyperboloid, hyperboloids, hyperboloid, hyperboloids, hyperboloid, hyperboloids, hyperboloid, hyperboloids, hyperboloid, hyperboloids (Source: “Complete accentuated paradigm according to A. A. Zaliznyak”) ... Forms of words
HYPERBOLOID- open central surface of the second order. There are two types of gas: single-sheet gas and double-sheet gas. In the proper coordinate system (see figure), the equation of a single-sheet gas has the form: and a two-sheet gas has the form: Numbers a, b and c (and segments such ... ... Mathematical Encyclopedia
Books
- , Alexey Tolstoy. The book includes science fiction novels by A. N. Tolstoy, created in the 20s of the last century... Buy for 1000 rubles
- Hyperboloid of engineer Garin. Aelita, Alexey Tolstoy. The novel "Engineer Garin's Hyperboloid" and the story "Aelita" marked the beginning of Soviet science fiction literature. They differ in that fantastic themes are given in combination with...
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