The elliptical paraboloid is canonically equal. Paraboloid wrap. Paraboloids near the world

An elіpsoїdom is called a surface that is equal to the current rectangular Cartesian coordinate system Oxyz, it looks like a ^ b ^ > 0. Taken on the Oxz plane, elips and obertatimemo yogo along the Oz axis (Fig. 46). Fig.46 Otriman surface Elіpsoid. Hyperboloids. Paraboloids. The cylinders and the cone are of a different order. - elіpsoїd wrapping - already given a statement about those, like the ruling elіpsoїd of a slanderous look. To take away yoga equal, it is enough to evenly squeeze the ellipsoid wrap. vzdovzh axis Oy with coefficient J ^ !,t.s. replace in yoga equal Jt/5). 10.2. Hyperboloids Turning the hyperbola fl i! \u003d a2 c2 1 on the axis Oz (Fig. 47), it is taken away from the surface, as it is called a single-empty hyperbole wrap. Yogo equal may look * 2 + y; come out in the same way as in the case of the elіpsoida wrap. 5) Elipsoid Rishennya is able to renounce the rinnemic scenari + yj + *j = l "signs Osі oz ~^1. The gap of the rivine risen -sample of the OS KoEFINTA 2^1 OSEMAMIMA, the uninterrupted species of the sagal form. Параболоїди Циліндри і конус другого порядку виходить тим же способом, що і в розібраному вище випадку еліпсоїда Шляхом обертання навколо осі Ог сполученої гіперболи отримаємо двопорожнинний гіперболоїд обертання (рис. 48) Його рівняння а2 С2 Шляхом рівномірного Оу з коефіцієнтом 2 ^ 1 приходимо до двопорожнинного гіперболоїда Zagalny species. Minh wu -I will be oso yogo rivyannya. The rotation of the Uzdovzh OSI OS KOEFITSIT YJ* ^ 1 OIKMOMOMOMEMENTICAL PARARODID. LOMID, in Fig. 50.4. coordinate system Oxyz can be seen de p > 0, q > 0. ny method of reparsing, which is similar to the offensive one: in parallel to the coordinate planes, planes are drawn, reaching the surface, and changing the changes that lead to the result of flat curves to work on the structure of the surface itself. Let's do it again with the planes z = h = const, parallel to the coordinate plane Oxy. At h > 0, hyperbole is taken away at h - hyperbole is caused, and at - a pair of straight lines. Curves are projected onto the Oxy plane. Take away the picture (Fig. 51). Already, this view allows the growth of visnovok about the sidlo-like budov surface (Fig. 52). Fig.51 Fig.52 We can now look at the cross-sections by planes Replacing the equal surfaces on the L, we will take the equalization of the parabolas (Fig.53). A similar picture emerges with a different set of surface planes. In this direction, there are also parabolas of the needles that are straight down (and not uphill, as for cutting with planes y \u003d h) (Fig. 54). Respect. By the method of reparsing it is possible to rozіbratisya at budovі and all earlier looked at the surface of a different order. However, with a wrapping of curves of a different order and an offensive equal pressure to a clear structure, one can come simpler and significantly smarter. On top of another order, which was left out, in fact, already looked at earlier. Cylinders: eliptine and hyperbolic Fig. 56 and a parabolic cone of a different order of appearance, it is possible to take off the wrapping of the bets with a wrapper, straight on the Oz axis and further embossment, or by the method of re-cutting. Zvichayno, in both views it is taken into account that the surface can be examined, looking at the indications in fig. 59. a) calculate the coordinates of foci; , . b) calculate the eccentricity; . c) write equal asymptotes and directrixes; d) write down the obtained hyperbole and calculate the eccentricity. 2. Store canonically equal parabolas, so as to reach the focus to the top of the dot 3. 3. Write the alignment of the dotic to the ellipse ^ + = 1 veto point M(4, 3). 4. It is important to look at that expansion of the curve, assigned to equals: Vіdpovіdі elіps, the whole parallel is great Elіpsoїd. Hyperboloids. Paraboloids. The cylinders and the cone are of a different order. axis Ox; b) hyperbola center O (-1.2), top coefficient of hanging axis X is 3; c) parabola U2 = , vertex (3, 2), axis vector, straightening y of the curve of the parabola, dorsal (-2, -1); d) hyperbola with center, asymptotes parallel to the coordinate axes; e) a pair of lines that overlap f) a pair of parallel lines

There are two kinds of paraboloids: elliptic and hyperbolic.

Elliptic paraboloid the surface is called, as in the current system of Cartesian rectangular coordinates it is assigned equal

An elliptical paraboloid may look like an inexhaustible swollen bowl. Vіn maє dvі mutually perpendicular to the plane of symmetry. Krapka, with some cob of coordinates, is called the top of an elliptic paraboloid; the numbers p and q are called i-parameters.

A hyperbolic paraboloid is called a surface, as it stands for equal

Hyperbolic paraboloid make a saddle shape. Vіn maє dvі mutually perpendicular to the plane of symmetry. Krapka, with some cob of coordinates, is called the vertex of a hyperbolic paraboloid; numbers Rі q are called yoga parameters.

Right 8.4. Let's take a look at the hyperbolic paraboloid mind

Let it be necessary to induce a part of the paraboloid that lies in the ranges: xО[–3; 3], atО[–2; 2] with a crop D=0.5 for both changes.

vikonannya. On the back of your head z. At the butt

Enter the value of the change X at the stovpets BUT. For whom in the middle A1 input character X. At the middle A2 be entered before the value of the argument - left between the range (–3). At the middle A3- another meaning of the argument - the left between the range plus the prompt (–2,5). Potim, having seen the block of the middle A2:AZ, autocomplete takes all the values ​​of the argument (for the right lower cut, the block can be extended to the middle A14).

Significance of the change at put in a row 1 . For whom in the middle IN 1 enter before the value of the change - left between the range (-2). At the middle Z 1- other value of the change - the left between the range plus the wake-up call (- 1,5). Potim, having seen the block of the middle B1:C1, autocomplete takes all the values ​​of the argument (for the right lower cut, the block can be extended to J1).

Then enter the value of the change z. For which tabular cursor should be placed in the table IN 2 and enter the formula - = $A2^2/18 -B$1^2/8, why press the key Enter. At the middle IN 2 is 0. Now it is necessary to copy the function from the room IN 2. For this autocompletion (stretching to the right) copy the formula back to the range B2:J2, after what (stretched down) - y range Q2:J14.

As a result, in the range Q2:J14 the table of points of the hyperbolic paraboloid appears.

To encourage diagrams on the toolbar standard need to press the button Meister Diagram. At the dialogue vіknі, what happened. Meister diagram (croc 1 of 4): diagram type indicate the type of diagrams - on top, and looking - Drotov (clearance) surface(Right upper diagram near right window). After what we press the button Dali at the dialogue window.

At the dialogue vіknі, what happened. Meister diagram (croc 2 of 4): dzherelo danih Diagrams need to select the tab Range give it to the field Range give the mouse an interval of data Q2:J14.

Dali is necessary to indicate in the rows of cleanliness, rows of data are stashed. Choose the orientation of the axes Xі y. At the butt of the jumper Rows in for the assistance of the misha's indicator, we will put it in the position of the stumps.

We select the Row i tab in the field X axis signatures indicate the range of signatures. For the next field, activate the field by clicking in the new mouse, and enter the axis signature range X -A2: A14.

Enter the value of the signature of the axis y. For whom at the working field Row we take the first record Row 1 the one that activated the working field Im'ya misha's guide, we introduce the first value of the change y: -2. Let's sweat by the field Row picking up another record Row 2 i in the working field Im'ya enter another value of the change y: -1.5. Repeat in this order until the rest of the record - Row 9.

When the required records appear, press the button Dali.

In the third window, it is necessary to enter the title of the diagrams and the names of the axes. For which you need to select the tab Titles, clicking on it with a mouse. After what the working field Called diagrams enter the name from the keyboard: Hyperbolic paraboloid. Then, similarly, enter in the work fields All X (categories),All Y (rows of data)і Weight Z (value) appropriate names: x, yі z.

To the surface of the 2nd order, there is also a hyperbolic paraboloid. Tsya surface can be taken away by the zastosuvannym algorithm vikoristovu wrapping such a line as a non-destructive axis.

To inspire a hyperbolic paraboloid, there is a special model. This model includes two parabolas, which are arranged in two mutually perpendicular planes.

Let the parabola I roztashovuєtsya at the flat that is unruly. Parabola II zdіysnuє folding movement:

▫ її spat position zbіgaєtsya from the flat
, moreover, the vertex of the parabola zbіgaєtsya with the cob of coordinates: =(0,0,0);

▫ distance parabola parallel transfer, moreover, її vertex
zdіysnyuє trajectory, scho zbіgaєtsya with parabola I;

▫ Two different positions of the parabola II are seen: one is the pins of the parabola uphill, the other is the pins down.

Let's write down the alignment: for the first parabola I:
- Postiyno; for another parabola II:
- Pochatkove position, rіvnyannya Rukh:
It doesn't matter bachichi, what's the point
may coordinates:
. Oscilki need to represent the law of the point
: if the point is to lay parabola I, then it is necessary to constantly win the line: =
і
.

From the geometric features of the model, it is easy to bachiti, that the ruhoma is a parabola note deaku surface. In such a time, the surface, which is described by parabola II, can be seen:

either→
. (1)

form
. There are two possibilities:

one). Signs of quantities pі q avoid: parabolas I and II are folded on one side of the plane OXY. Acceptable: p = a 2 і q = b 2 . Todі otrimuєmo vіvnyannja vіdomoї surfіnі:

→ elliptic paraboloid . (2)

2). Signs of quantities pі q different: parabolas I and II are arranged along the different sides of the plane OXY. Come on p = a 2 і q = - b 2 . Now it is necessary to equalize the surface:

→hyperbolic paraboloid . (3)

Reveal the geometric shape of the surface, as if it were equal to (3) it doesn’t matter, so as to guess the kinematic model of the interplay of two parabolas, which would take the fate of Russia.

Parabola I is mentally shown on the little one with a red color. Through those that the shape of the surface is strikingly stretching on the cavalry saddle, the outskirts of qiu are often called. saddle .

In physicist, with the increase in the stability of the processes, introduce types of equalities: stіyke - hole, swell down, stubble - swelled up the surface uphill and in the middle - seat. Jealousy of the third type is also referred to the type of non-stable jealousy, moreover, only on the red line (parabola I) can be jealousy.

§ 4. Cylindrical surfaces.

When looking at the surface of the wrapping, they identified the simplest cylindrical surface - the wrapping cylinder, which is a circular cylinder.

In elementary geometry, the cylinder of appointments is analogous to the main appointments of a prism. It’s more folding to milk it:

▫ let me have a flat bagatokutnik near the space
- Significantly yak , and with it a bagatokutnik
- Significantly yak
;

▫ zastosovuєmo to the bagatokutnik
move parallel: dots
move along trajectories parallel to the given straight line ;

▫ yakscho
, then yoga flat
parallel to the plane ;

▫ the surface of the prism is called: ,
– imagine prisms, as well as parallelograms
,
,... – bichna surface prism.

At speeding up to the elementary designation of the prism for the purpose of inspiring a larger zagalny designation of the prism and її superficial, and itself, it is different:

▫ not surrounded by a prism - all rich-faceted body, surrounded by ribs ,,... that flats between the ribs;

▫ the prism is surrounded by a rich-faceted body, surrounded by ribs ,,... and parallelograms
,
,...; bіchna surface of the zієї prism - the collection of parallelograms
,
,...; the foundations of the prism - the sukupnіst bahatokutnikov ,
.

Let me have an unrestricted prism: ,,... Let's move the prism with a large area . Let's move the prism with another area
. At the peretina we take off the bagatokutnik
. At the scorched slope, it’s important that the flat
not parallel to the plane . Tse means that the prism is not inspired by parallel transfers of the bagatokutnik .

The proponated prisms include not only those straight prisms, but be albeit truncated.

In the analytical geometry, the cylindrical surface of the rozumitely lining is marked, that the uncircumscribed cylinder includes an uncircumscribed prism like an ozone drop: you don’t have to let it go, that the bagatokutnik can be replaced with a long line, not ob'yazkovo closed - directly cylinder. straight name satisfy cylinder.

From what has been said, it is clear: for the designation of a cylindrical surface, it is necessary to set a straight line and a straight line.

Cylindrical surfaces are built on the basis of plane curves of the 2nd order, services direct for appease .

At the cob stage, the crowning of cylindrical surfaces is acceptable to reduce the allowance:

▫ Do not let the cylindrical surface straight ahead and roztashovuetsya in one of the coordinate planes;

▫ directly satisfying zbіgaєtsya z one z axes of coordinates, that is perpendicular to the plane, in which it is assigned directly.

Accepting the exchange does not lead to the loss of sleepiness, the shards are deprived of the possibility of rahunok to choose overcut by flats і
be more geometric shapes: straight, slender, shortened cylinders.

Eliptic cylinder .

Let the cylinder straight ahead, they took the elips :
, spreading at the coordinate plane

: eliptic cylinder.

Hyperbolic cylinder .

:

, but directly affirming everything
. In this direction, the alignment of the cylinder is the same line : hyperbolic cylinder.

Parabolic cylinder .

Let the straight cylinder take the hyperbole :
, expanded in the coordinate plane
, but directly affirming everything
. In this direction, the alignment of the cylinder is the same line : parabolic cylinder.

Respect: vrakhovuyuchi global rules encourage the alignment of cylindrical surfaces, as well as the presentation of private butts of an elliptical, hyperbolic and parabolic cylinders, it is significant: the need for a cylinder for whether it is somehow satisfying, for those who accept forgiveness of minds, is not guilty of everyday hardships!

Let's look now at the deep mind, inspire the alignment of cylindrical surfaces:

▫ straight cylindrical surface roztashovuetsya at a sufficient area of ​​space
;

▫ directly satisfying the adopted coordinate system is sufficient.

Accept the imaginative little one.

▫ straight cylindrical surface roztashovuetsya near a large area space
;

▫ coordinate system
taken from the coordinate system
parallel transfers;

▫ directly at the flat best: for a curve of the 2nd order, it is important that the cob of coordinates spіvpadє z center symmetry of the curve, what is seen;

▫ directly satisfying dovilne (may be given by any of the methods: vector, direct and in).

Please note that the coordinate systems
і
run away. Tse means that the 1st krok of the cover algorithm induces cylindrical surfaces, which reflects the parallel transfer:
→
, In front of the vicons.

Guessing, as if you are afraid of being parallel to the transference at the infamous swing, looking at a simple butt.

butt 6–13 : Coordinate system
at the sight:
=0. Write down the direct line to the system
.

Solution:

one). Significantly good point
: in the system
yak
, i in the system
yak
.

2). Let's write the vector equality:
=
+
. In the coordinate form, you can write in the view:
=
+
. But at the sight:
=
–
, or:
=.

3). Let's write down the alignment of the straight cylinder at the coordinate system
:

Verify: straight line conversion: =0.

Also, remember that the center of the curve, which directly represents the cylinder, must always be placed on the cob of coordinates of the system
at the flat .

Rice. At . Basic drawing when the cylinder is stimulated.

More one more allowance, which will let you know the remaining crumbs of the cylindrical surface. Scattered around the coordinate system, it doesn’t matter to go straight to the axis
coordinate systems
from the normal of the area , and straight axes
і
with axes of symmetry straight , then we will take into account that the situation is direct may be crooked, ripped at the flat
, moreover, one її all symmetry zbіgaєtsya z vіssyu
, and a friend of mine
.

Respect: so, as the operation is parallel to the transfer and the wrapping of the rather unbreakable axis of the operation, it is easy to do, then the acceptance of the allowance does not sound like a zastosuvannya to the algorithm of stimulating the cylindrical surface in the most infamous fall!

Mi Bachili spread out near the flat
, and the twirl is parallel to the axis
, enough to signify only directly .

Since a cylindrical surface can be unambiguously assigned to a given line, which is taken into account in the cut of the surface by a fairly flat area, then it is acceptable to use such a wild algorithm for solving problems:

1 ▫ . Let me straighten up cylindrical surface is given by vector . Projected directly , given equals:
\u003d 0, on a plane, perpendicular to a straight line, which makes , then on the plane
. As a result, the cylindrical surface will be given in the coordinate system
equals:
=0.

2 ▫
on the axis
on kut
: smist kuta
get in touch with the system
, and the alignment of the final surface turns into alignment:
=0.

3 ▫ . Coordinate system wrapping is customizable
on the axis
on kut
: smist kuta a lot of intelligence from a little one. Last wrapping coordinate system
get in touch with the system
, And the equalization of the final surface turns into
=0. Tse i є vnyannya cylindrical surface, which had direct tasks. and tvirna at the coordinate system
.

The application below is an illustration of the implementation of the recorded algorithm and the calculation of the difficulties of similar tasks.

butt 6–14 : Coordinate system
the alignment of the straight cylinder is specified at the sight:
=9. Fold the cylinder so that it is parallel to the vector =(2,–3,4).

R
Yeshenya:

one). Projected directly on the cylinder on a perpendicular plane . It seems that such a transformation of a given task, I transform it into an elіps, the axes of which will be: great =9, but small =
.

Tsey little ones illustrating the design of a stake given in a plane
to the coordinate plane
.

2). The result of the design of the stake is elips:
=1, otherwise
. Our viewpoint is:
, de
==.

3
). Again, alignment of the cylindrical surface in the coordinate system
taken away. Shards for the mental responsibility of the mother of the alignment of the cylinder in the coordinate system
, then it is no longer possible to stop the conversion of coordinates, which translates the coordinate system
y coordinate system
, contagion and equalization of the cylinder:
equal, expressed through changes
.

four). hurry up basic small and write down all the necessary trigonometric values ​​for the solution of the problem:

==,
==,
==.

5). We write down the formula for the transformation of coordinates for the transition to the system
to the system
:
(AT)

6). We write down the formula for the transformation of coordinates for the transition to the system
to the system
:
(FROM)

7). Submitting changes
from system (B) to system (C), as well as the reverse values ​​of trigonometric functions that are victorious, we write:

=
=
.

=
=
.

eight). Lack of knowledge і at straight line cylinder :
at the coordinate system
. Vikonavshi carefully all reworkings of algebra, necessarily equal to the finite surface in the coordinate system
: =0.

Vidpovid: cone alignment: =0.

butt 6–15 : Coordinate system
the alignment of the straight cylinder is specified at the sight:
=9, =1. Fold the cylinder so that it is parallel to the vector =(2,–3,4).

Solution:

one). It doesn’t matter if you remember that this butt is blowing in front of the front only, which was directly moved in parallel to 1 uphill.

2). Tse means that in spіvvіdnannyah (B) should be accepted: =-one. Vrahovyuchi virazi system (C), soon record for change :

=
.

3). The change is easily repaired by the correction of the last record of alignment for the cylinder from the front butt:

Vidpovid: cone alignment: =0.

Respect: it is not important to remember that the main difficulties in the case of different transformations of coordinate systems in problems with cylindrical surfaces are neatness і vitrivality in the margaphones of algebra: let the system of enlightenment live, adopted in our richly-suffering country!

Elliptic paraboloid

Elliptic paraboloid for a=b=1

Elliptic paraboloid- Surface, which is described by the function of the mind

de aі b one sign. The surface is described by a family of parallel parabolas with needles, straight up the hill, the tops of which describe a parabola, with needles, also straight up the hill.

Yakscho a = b then an elliptical paraboloid is the surface wrapping, the parabolic wrapper is placed on the vertical axis, which passes through the top of this parabola.

Hyperbolic paraboloid

Hyperbolic paraboloid for a=b=1

Hyperbolic paraboloid(called in everyday life "gipar") - a simplistic surface, which is described in a rectangular coordinate system equal to the mind

From another manifestation, it is clear that the hyperbolic paraboloid is a linear surface.

The surface can be covered with the movement of a parabola, the needles of which are straightened down, with a parabola, the needles of which are straightened uphill, for the mind that the first parabola sticks to its other peak.

Paraboloids near the world

At the technical

At the mystic

Literature

Attached, descriptions by Hyperboloid engineer Garin maw buti paraboloid.

Wikimedia Foundation. 2010 .

• Elon Menachem
• Eltang

Marvel at such an "Eliptic paraboloid" in other dictionaries:

ELLIPTIC PARABOLOYD Great Encyclopedic Dictionary

elliptic paraboloid- one of two types of paraboloids. * * * ELLIPTIC PARABOLOID ELLIPTIC PARABOLOY, one of two types of paraboloids (div. PARABOLOID) ... Encyclopedic dictionary

Elliptic paraboloid- one of two types of paraboloids. Great Radianska Encyclopedia

ELLIPTIC PARABOLOYD- Unclosed surface of a different order. Canonical rivnyannya E. p. maє looked at E. p. roztashovaniya on one side of the Ohu area (div. fig.). Pererizi E. p. Mathematical Encyclopedia

ELLIPTIC PARABOLOYD- one of two types of paraboloids. Natural science. Encyclopedic dictionary

PARABOLIC- (Greek, vіd parabole parabola, i eidos podіbnіst). The body, which becomes a parabola, which wraps around. Glossary of inshomonic words that have gone up to the stock of Russian language. Chudinov A.N., 1910. A PARABOLID is a geometrical body, which has hid itself in the form of a wrapping of a parabola, so ... Dictionary of foreign words of Russian language

PARABOLIC- PARABOLOYD, paraboloid, human. (div. parabola) (mat.). On top of another order does not mean the center. Parabolic wrapping (the wrappings of the parabola are settled on the її axis). Elliptic paraboloid. Hyperbolic paraboloid. Tlumachny dictionary of Ushakov. Tlumachny dictionary of Ushakov

PARABOLIC- PARABOLOD, surface, which is taken from the Russian parabola, the apex of which is forged along the other, non-robust parabola (from the whole symmetry, parallel axis parabolas that are collapsing), then the same plane, moving parallel to itself, is being abandoned ... Modern Encyclopedia

Paraboloid- is a surface type of a different order. A paraboloid can be characterized as a non-closed non-central surface of a different order (so it does not have a center of symmetry). The canonical alignment of the paraboloid in Cartesian coordinates: even one ... ... Wikipedia

PARABOLIC- non-closed non-central surface of a different order. Canonical Rivnyannia P.: elliptic paraboloid (when p = q is called P. wrapper) and hyperbolic paraboloid. A. B. Ivanov ... Mathematical Encyclopedia

Elipsoid- On the surface in a trivial space, deformed by the deformation of the sphere, there are three mutually perpendicular axes. The canonical alignment of the ellipsoid in Cartesian coordinates, which avoids the axes of the ellipsoid deformation: .

Values ​​a, b, c are called ellipsoid pivos. The body is also called an elіpsoid, surrounded by the surface of an elіpsoid. Elіpsoїd є one of the possible forms on top of another order.

As a pair of pivos may have the same length, the ellipse can be taken away from the ellipse wrappers for about one of the yogo axes. Such an ellipsoid is called an ellipsoid wrapper or a spheroid.

Elipsoid is more precise, lower sphere, reflecting the idealized surface of the Earth.

Volume of ellipsoid:.

Surface area of ​​elіpsoida wrap:

Hyperboloid- the view of the surface in a different order in a trivi-worldly space, which is specified in Cartesian coordinates equal - (single-spaced hyperboloid), where a and b are real lines, and c - clear lines; abo - (double-spread hyperboloid), de a and b - vyavn_ pіvosі, and c - diysna pіvvіs.

If a = b, then such a surface is called a hyperbole wrap. A single-empty hyperboloid wrapper can be taken away from hyperbolic wrappers on the її obvious axis, a double-empty wrapper - on the її obvious axis. A two-dimensional hyperbolic wrapper is also a geometrical point P, the module of the difference is up to two set point A and B are constant: | AP−BP | = Const. In this case, A and B are called foci of the hyperboloid.

Single-ported hyperboloid є double linear surface; as if it is a hyperboloid wrapper, then wine can be taken away from the wrappings directly on the other side of the line that crosses with it.

Paraboloid is a surface type of a different order. A paraboloid can be characterized as a non-closed, non-central surface of a different order (which does not have a center of symmetry).

The canonical alignment of the paraboloid in Cartesian coordinates:

· if a and b have the same sign, then the paraboloid is called elliptic.

aka a and b different sign, Parabolic is called hyperbolic.

· If one of the coefficients is equal to zero, then the paraboloid is called a parabolic cylinder.

ü - elіptichny paraboloid, de a and b of the same sign. The surface is described by a family of parallel parabolas with needles, straight up the hill, the tops of which describe a parabola, with needles, also straight up the hill. Like a = b, then an elliptical paraboloid is a superficial wrapping around the vertical axis of the parabola, which passes through the top of this parabola.

ü is a hyperbolic paraboloid.