Infinite paraboloid. Elipsoid. Hyperboloids. Paraboloids. Raztashuvannya free surface in a bowl
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.
The power of dotism has been brought to a parabola, which is even more important, to that it is screaming from it, that it is changed, that it emerges from the focus of a curved parabolic mirror, that such a mirror, on top of which the wrapping of parabola emerges on about its axis, in order to mirror itself parallel .
The power of parabolic mirrors zastosovuetsya with the power of searchlights, at the headlights of any car, as well as at mirror telescopes. At the same time, in the rest of the fall, back, change, to go into the sky; if they are parallel, they are set in parallel to the focus of the telescope mirror, and so if you change it, to go to the different points of the luminary, richly unparallel, then the stench will be centered to the focus at different points, so that the image of the luminary will appear at the focus, the greater, the greater the parabola of the focal point. This image is already at the microscope (telescope eyepiece). Strictly seeming, only change, strictly parallel to the axis of the mirror, climb to one point (at the focus), parallel to the change, where to go, under the tip to the axis of the mirror, climb more or less to one point, moreover, farther away from the focus, the image more open. Yogo furnishing surrounds the "field of the telescope's dawn".
Let the inner surface of yoga - the mirror surface of the parabolic mirror hang in a beam of light changes parallel to the OS axis. Must change, parallel to the axis of the OU, after the change to turn over in one point of the axis of the OU (focus F). The power of parabolic telescopes was founded on this power. Changes from distant stars come to us at the sight of a parallel beam. Having prepared a parabolic telescope and placing a photographic plate at the focus of Yogo, we may be able to enhance the light signal that is coming from the sky.
This principle lies at the basis of the parabolic antenna, which allows the strength of radio signals. If you place a light at the focus of the parabolic mirror, then if you see the surface of the mirror, the exchange, which goes in the direction of the mirror, does not rise, but climbs from the narrow beam parallel to the axis of the mirror. This fact should be known when preparing projectors and lights, various projectors, mirrors of which are made in the form of paraboloids.
Assigned to the greater optical power of a parabolic mirror, the mirror telescopes, various sleepy heating installations, and floodlights are built up. Having placed at the focus of the parabolic mirror a sharper point of light, we take away the sharp flow of the changes parallel to the axis of the mirror.
When wrapping a parabola, a figure emerges about the axis, which is called a paraboloid. If the inner surface of the paraboloid is mirror-like and direct a beam of changes parallel to the axis of symmetry of the parabola, then the change will be taken at one point, as it is called a focus. At the same hour, as if the light were to be placed at the focus, then the changes in the mirror surface of the paraboloid appear parallel and do not rise.
The first power allows you to take a high temperature at the focus of the paraboloid. Zgidno z legend, tsyu vlastivist vikoristovuvav ancient Greek teachings of Archimedes (287-212 pp. BC). During the defense of Syracuse at the war against the Romans, having incited a system of parabolic mirrors, she allowed the focus of the sony exchanges on the ships of the Romans. As a result, the temperature at the foci of the parabolic mirrors showed a high flooring, which on the ships caught fire and the stench burned.
Another power wins, for example, with the preparation of searchlights and car headlights.
Hyperbole
4. Appointment of hyperbole gives us a simple way to get її without interruption: take two threads, the difference between the lengths of which are more expensive 2a, and attach one end of each thread to point F "and F. Like trimming with your hand, two other threads are moving olіvtsya, dbayuchi about those that the threads were pressed to paper, pulled tight and stuck, pochinayuchi in vіd vіstrі, scho to chair, until the month of z'єdnannya kіntsіv, then in_strya weave a part of one of the heels of hyperbole (there are more, lower threads) (small thread) .).
Changing the roles of the points F "and F, we take away part of the other neck.
For example, on the topic "Curves of the 2nd order" you can look at the following task:
Manager. Two railway stations A and B are located on the same station, one in one. You can deliver the point M vantage from the station A either by direct road transport (the first route), or by zaliznitsi to the station, but by car (another way). The railroad tariff (the price of transportation of 1 ton per 1 km) becomes m rubles, the tariff for motor transport - n rubles, n> m, the tariff for the vanity-distribution - k rubles. Designate the region to the vlivu of the railway station, tobto, that region, it is cheaper to deliver vantage from the station to the yak. designate a geometrically place point, for which another way is the most obvious for the first.
Solution. Significantly AM = r, BM = r, the same delivery variance (transportation and delivery-distribution) by cost AM is nr + k, and the delivery variance by cost ABM is ms + 2k + ng. Then the points M, which determine the difference between the varities of the equal, satisfy the equal nr + k = ms+2k+ng , chi
ms + k = nr - ng
r - g \u003d \u003d const\u003e O,
otzhe, the line that borders the region is one of the main areas of hyperbole | r - r | = Const. For all points of the plane, which lie one side with point A, in the direction of hyperbole, the most visible first path, and for points that lie along the next side, - the other, to that the hyperbole line is called the region of the station.
Option tsієї tasks.
Two railway stations A and B are located on the same railway station, one in one. At point M, the vantage can be delivered from the station A either by direct transport, or by air to the station, or by car (Fig. 49). At the same time, the railroad tariff (the price of transporting 1 ton per 1 km) becomes m rubles, the cost of renting costs k rubles (per 1 ton), and the tariff for motor transport is n rubles (n > m). Significantly, this is the name of the zone of the injection of the railroad station B, because that zone, in the yak, is cheaper to deliver vantage from the A zmіshanim route: by the railroad and then by road.
Solution. Variant of delivery 1 t vantage for the cost AM to become r n, de r = AM, and for the cost AOM it is more expensive 1m + k + r n. We need to overcome the underlying unevenness r n 1m+ k+ r n і to determine how to subdivide points on the plane (x, y), to which it is cheaper to deliver vantage either by the first or the other way.
We know the equal lines that establish a cordon between these two zones, that is a geometrically space point, for such offenses the paths are “equally visible”:
r n = 1m+ k+ r n
З цієї mind otrimuєmo r - r = = const.
The same line divided the hyperbole. For all external points of hyperbole, the most prominent path is the first one, and for internal ones it is another. Therefore, hyperbole label the inflow zone of station B. Another hyperbole label the inflow zone of station A (the vantage is delivered from station B). We know the parameters of our hyperbole. Її great vsіs 2a = , and between foci (such as stations A and B) at times 2с = l.
In such a rank, the mind's ability to think of the task, which is considered to be a< с, будет
Tse zavdannya pov'yazuє abstract geometric understanding of hyperbole z transport and economic zavdannyam.
Shukane geometrical place is a point, an impersonal point, which lies in the middle of the right hyperbole, to avenge the point.
6. In the know " Silgospmashin important operational characteristics of a tractor that works on the scale, which show its stability, є kut late sickness and lateral roll.
Let's take a look at the wheel tractor for simplicity. On top, de pratsyuє tractor (prinaimnі, її to mow a small part), vvazhatimutsya flat (flat ruhu). The later height of the tractor is called the projection of the straight line, which is the rear of the middle of the front and rear axles, onto the plane of the floor. A kut of a transverse roll is called a kut, with a horizontal plane of a straight line, perpendicular to the later axis and lying in the plane of the floor.
When studying in the course of mathematics with the topics “Straight and flat in the open space”, the task is considered:
a) Know the cut of the late tractor’s sickness, which is collapsing with a skill, so that it can lead to a new skill and a cut to reinforce the tractor’s trajectory in the direction of the later straight.
b) The limiting cut of the transverse roll of the tractor is the largest allowable cut of the sickle, across which the tractor can stand without throwing. Yaki parameters of the tractor are enough to know for the designation of the boundary kut of the transverse nahilu; how to know
kut?
7. The presence of straight-line victorious victors is found in everyday technology. Volodymyr Grigorovich Shukhov (1853-1939), a Russian engineer, was the founder of practical zastosuvannya tsgogo є vіdomiy rosіyskiy іnzhenier. V. G. Shukhov designed the construction of shackles, hangers and supports, folded from metal beams, which are laid out along straight one-shot hyperbole wrap. The high quality of such constructions has a greater lightness, a low versatility of preparation and thinning ensure a wide extension of them in the daily routine.
8. LEGALIZE THE ROUGH OF THE VILLA SOLID TIL
For a free body, one can see everything, but it still does not mean that the hand of a free body is handless, not subject to any laws; navpak, the forward movement of a solid body is independent of the yogo ovnishnoi form to be confused by the law about the center of the mass and lead to the movement of one point, and the obertal one is the so-called head axes of inertia, but ellipsoid of inertia. So, the club, thrown into the open space, or the grain, which flies out of sorting, etc., collapses progressively, like one point (the center of the mass), and at once it wraps around the center of the mass. Zagalom with forwardal Russian, whether it be firm, the body is independent of its form, or the folding car can be replaced with one point (the center of the mass), and with the overt, with an elіpsoїdom іnertsії. , radii-vectors of such equal - de / - moment of inertia of the body of any axes that pass through the center of the ellipsoid.
As the moment of inertia of the body under the hour of the wrapping may change, the speed of the wrapping will also change. For example, after an hour of a haircut over the head, the acrobats squeeze into the chest, through which the moment of inertia of the body changes, and the speed of the wrap increases, which is necessary for the success of the haircut. Just like that, after licking, people move their hands to the sides, through which the moment of inertia increases, and the swirlness of the wrapper changes. So the very change is the moment of inertia of the rake zhneї of the vertical axis of the hour and the turn of the horizontal axis.
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 of 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 - is clear; 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 hyperbole wrappers on the її obvious axis, a double-empty wrapper - on the її obvious axis. A two-way hyperboloid wrapping a geometrical space point P, a module of difference between any of up to two given points A and B is 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).
Canonical equality 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 the elliptical paraboloid is the surface wrapping, the parabola wrapped around the vertical axis, which passes through the top of this parabola.
ü is a hyperbolic paraboloid.
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 prom_zhne - saddle. Jealousy of the third type can also be brought to the type of unstable 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, but in 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 fluctuations in the choice of overlapping 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 it go like a straight cylinder, they took a 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 first step of the cryptic algorithm induces cylindrical surfaces, which reflects the parallel transfer: 
→
, In front of the vicons.
Guessing, how to be afraid of being parallel to the transference at the infamous swing, having looked 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 somewhat 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 change 
.
four). hurry up basic small and write down all the necessary trigonometric values for the solution of the problem:
=
=
,
=
=
,
=
=
.
5). Let's write down the formula for the transformation of coordinates for the transition to the system 
to the system 
:
(AT)
6). Let's 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.
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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 PARABOLOID, one of two types of paraboloids (div. PARABOLOIDS) ... 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. with flats, parallel planes Wow, with ellipses with equal eccentricity (like r... 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 (which does not have a center of symmetry). The canonical alignment of the paraboloid in Cartesian coordinates: even one ... ... Wikipedia
PARABOLIC- Unclosed 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
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