Elliptical paraboloid
Elliptical paraboloid with a \u003d b \u003d 1
Elliptical paraboloid - surface described by a function of the form
,where a and b one sign. The surface is described by a family of parallel parabolas with upward-pointing branches, whose vertices describe a parabola, with upward-pointing branches.
If a a = b then an elliptical paraboloid is a surface of revolution formed by the rotation of a parabola around a vertical axis passing through the vertex of this parabola.
Hyperbolic paraboloid
Hyperbolic paraboloid with a \u003d b \u003d 1
Hyperbolic paraboloid (called "hypar" in construction) is a saddle-shaped surface described in a rectangular coordinate system by an equation of the form
.From the second representation it is seen that the hyperbolic paraboloid is a ruled surface.
The surface can be formed by the movement of a parabola, the branches of which are directed downward, along a parabola, the branches of which are directed upward, provided that the first parabola is in contact with its second vertex.
Paraboloids in the world
In technology
In art
In literature
The device described in Engineer Garin's Hyperboloid was supposed to be paraboloid.
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See what "Elliptical paraboloid" is in other dictionaries:
ELLIPTIC PARABOLOID Big Encyclopedic Dictionary
elliptical paraboloid - one of two types of paraboloids. * * * ELLIPTICAL PARABOLOID ELLIPTIC PARABOLOID, one of two types of paraboloids (see PARABOLOIDS) ... encyclopedic Dictionary
Elliptical paraboloid - one of two types of paraboloids (See Paraboloids) ... Great Soviet Encyclopedia
ELLIPTIC PARABOLOID - non-closed surface of the second order. Canonical. the equation of the E. p. has the form of the E. p. is located on one side of the plane Oxy (see Fig.). Sections of the E. p. Planes parallel to the plane Oxy are ellipses with equal eccentricity (if p ... Encyclopedia of Mathematics
ELLIPTIC PARABOLOID - one of two types of paraboloids ... Natural science. encyclopedic Dictionary
PARABOLOID - (Greek, from parabole parabola, and eidos similarity). A body formed by a rotating parabola. Dictionary of foreign words included in the Russian language. Chudinov AN, 1910. PARABOLOID geometric body formed from the rotation of a parabola, so ... ... Dictionary of foreign words of the Russian language
PARABOLOID - PARABOLOID, paraboloid, husband. (see parabola) (mat.). A second-order surface with no center. Paraboloid of revolution (formed by rotating a parabola around its axis). Elliptical paraboloid. Hyperbolic paraboloid. Dictionary Ushakov ... Ushakov's Explanatory Dictionary
PARABOLOID - PARABOLOID, the surface obtained by the movement of a parabola, the vertex of which slides along another, stationary parabola (with an axis of symmetry parallel to the axis of the moving parabola), while its plane, displaced parallel to itself, remains ... ... Modern encyclopedia
Paraboloid - - type of surface of the second order. A paraboloid can be characterized as an open non-central (that is, having no center of symmetry) second-order surface. Canonical equations of a paraboloid in Cartesian coordinates: if and one ... ... Wikipedia
PARABOLOID - non-closed off-center surface of the second order. Canonical. equations of P .: an elliptic paraboloid (for p \u003d q is called a P. of rotation) and a hyperbolic paraboloid. A. B. Ivanov ... Encyclopedia of Mathematics
There are two types of paraboloids: elliptical and hyperbolic.
Elliptical paraboloidis called a surface that in a certain system of Cartesian rectangular coordinates is determined by the equation
An elliptical paraboloid looks like an infinite convex bowl. It has two mutually perpendicular planes of symmetry. The point at which the origin is aligned is called the vertex of the elliptical paraboloid; the numbers p and q are called its parameters.
A hyperbolic paraboloid is a surface defined by the equation
Hyperbolic paraboloidhas a saddle shape. It has two mutually perpendicular planes of symmetry. The point with which the origin is aligned is called the vertex of the hyperbolic paraboloid; numbers rand qare called its parameters.
Exercise 8.4.Consider the construction of a hyperbolic paraboloid of the form
Let it be necessary to construct a part of the paraboloid lying in the ranges: xÎ [–3; 3], atÎ [–2; 2] with a step D \u003d 0.5 for both variables.
Performance... First, it is necessary to solve the equation with respect to the variable z.In the example
Let's introduce the values \u200b\u200bof the variable xinto column AND... To do this, in the cell A1 enter the symbol x.Into the cell A2 the first value of the argument is entered - the left border of the range (–3). Into the cell A3 - the second value of the argument is the left border of the range plus a construction step (–2,5). Then, by selecting a block of cells A2: AZ, by autocomplete we get all the values \u200b\u200bof the argument (we extend from the lower right corner of the block to the cell A14).
Variable values atenter in the line 1 ... To do this, in the cell IN 1 the first value of the variable is entered - the left boundary of the range (–2). Into the cell C1 - the second value of the variable is the left border of the range plus a construction step (- 1,5). Then, by selecting a block of cells B1: C1, by autocomplete we get all the values \u200b\u200bof the argument (for the lower right corner of the block we extend to the cell J1).
Next, enter the values \u200b\u200bof the variable z.To do this, the table cursor must be placed in a cell AT 2 and enter the formula - \u003d $ A2 ^ 2/18 -B $ 1 ^ 2/8,then press the key Enter... In a cell AT 2 appears 0. Now you need to copy the function from the cell AT 2... To do this, by auto-filling (dragging to the right), copy this formula first into the range B2: J2, after which (by pulling down) - into the range B2: J14.
As a result, in the range B2: J14 a table of points of the hyperbolic paraboloid appears.
To build a chart on the toolbar Standard you must press the button Chart Wizard... In the dialog box that appears Chart Wizard (Step 1 of 4): Chart Type specify the type of chart - Surface, and the view - Wire (transparent) surface (top right diagram in the right window). Then press the button Further in the dialog box.
In the dialog box that appears Chart Wizard (Step 2 of 4): Data Source charts need to select tab Range data and field Range specify the data interval with the mouse B2: J14.
Further, it is necessary to indicate the rows of data in the rows or columns. This will determine the orientation of the axes xand at.In the example, the switch Ranks in use the mouse pointer to position the columns.
Select the Row tab and in the field X-axis labels specify the range of signatures. To do this, activate this field by clicking on it with the mouse pointer and enter the range of axis labels x -A2: A14.
Enter the values \u200b\u200bof the axis labels at.To do this, in the working area Row select the first record Row 1and by activating the working field Name with the mouse pointer, enter the first value of the variable y: –2.Then in the field Row select the second entry Row 2 and into the working field Name enter the second value of the variable y: -1.5.We repeat this way until the last entry - Row 9.
After the required entries appear, press the button Further.
In the third window, you need to enter the title of the chart and the names of the axes. To do this, select the tab Headingsby clicking on it with the mouse pointer. Then in the working field Chart title enter the name from the keyboard: Hyperbolic paraboloid.Then, in the same way, enter into the working fields X-axis (categories),Y-axis (data series)and Z-axis (values) relevant titles: x, yand z.
Ellipsoid - surface in three-dimensional spaceobtained by deformation of a sphere along three mutually perpendicular axes. The canonical equation of an ellipsoid in Cartesian coordinates coinciding with the deformation axes of the ellipsoid:.
The quantities a, b, c are called the semiaxes of the ellipsoid. Also called an ellipsoid is a body bounded by the surface of an ellipsoid. An ellipsoid is one of the possible second-order surface shapes.
In the case when a pair of semiaxes has the same length, an ellipsoid can be obtained by rotating the ellipse around one of its axes. Such an ellipsoid is called an ellipsoid of revolution or a spheroid.
An ellipsoid more accurately than a sphere reflects the idealized surface of the Earth.
Ellipsoid volume :.
Surface area of \u200b\u200ban ellipsoid of revolution:
Hyperboloid- this is the type of surface of the second order in three-dimensional space, specified in Cartesian coordinates by the equation - (one-sheet hyperboloid), where a and b are real semiaxes, and c is an imaginary semiaxis; or - (two-sheet hyperboloid), where a and b are imaginary semiaxes, and c is a real semiaxis.
If a \u003d b, then such a surface is called a hyperboloid of revolution. A one-sheet hyperboloid of revolution can be obtained by rotating a hyperbola around its imaginary axis, a two-sheet hyperboloid - around a real one. A two-sheeted hyperboloid of revolution is also the locus of points P, the modulus of the difference between the distances from which to two given points A and B is constant: | AP - BP | \u003d const. In this case, A and B are called the foci of the hyperboloid.
A one-sheet hyperboloid is a double-ruled surface; if it is a hyperboloid of revolution, then it can be obtained by rotating a straight line around another straight line intersecting with it.
Paraboloid - type of surface of the second order. A paraboloid can be characterized as an open non-central (that is, having no center of symmetry) second-order surface.
Canonical equations of a paraboloid in Cartesian coordinates:
· If a and b are of the same sign, then the paraboloid is called elliptic.
· If a and b are of opposite signs, then the paraboloid is called hyperbolic.
· If one of the coefficients is equal to zero, then the paraboloid is called a parabolic cylinder.
ü - elliptical paraboloid, where a and b are of the same sign. The surface is described by a family of parallel parabolas with upward-pointing branches, whose vertices describe a parabola, with upward-pointing branches. If a \u003d b then the elliptical paraboloid is a surface of revolution formed by the rotation of the parabola around the vertical axis passing through the vertex of this parabola.
ü - hyperbolic paraboloid.
On around its axis, you can get an ordinary elliptical. It is a hollow isometric body whose sections are ellipses and parabolas. An elliptical paraboloid is defined as:
x ^ 2 / a ^ 2 + y ^ 2 / b ^ 2 \u003d 2z
All main sections of a paraboloid are parabolas. When cutting the XOZ and YOZ planes, only parabolas are obtained. If you cut a perpendicular section relative to the Xoy plane, you can get an ellipse. Moreover, the sections, which are parabolas, are set by equations of the form:
x ^ 2 / a ^ 2 \u003d 2z; y ^ 2 / a ^ 2 \u003d 2z
The sections of the ellipse are given by other equations:
x ^ 2 / a ^ 2 + y ^ 2 / b ^ 2 \u003d 2h
An elliptical paraboloid with a \u003d b turns into a paraboloid of revolution. The construction of a paraboloid has a number of certain features that must be taken into account. Start the operation by preparing the base - drawing the function graph.
In order to start building a paraboloid, you first need to build a parabola. Draw a parabola in the Oxz plane as shown. Give the future paraboloid a specific height. To do this, draw a straight line so that it touches the upper points of the parabola and is parallel to the Ox axis. Then draw a parabola in the Yoz plane and draw a straight line. You will get two paraboloid planes perpendicular to each other. Then, in the Xoy plane, draw a parallelogram to help you draw the ellipse. In this parallelogram, write an ellipse so that it touches all its sides. After these transformations, erase the parallelogram, and the volumetric image of the paraboloid remains.
There is also a hyperbolic paraboloid that is more concave than elliptical. Its sections also have parabolas, and in some cases, hyperbolas. The main sections along Oxz and Oyz, as in the case of an elliptic paraboloid, are parabolas. They are given by equations of the form:
x ^ 2 / a ^ 2 \u003d 2z; y ^ 2 / a ^ 2 \u003d -2z
If you draw a section about the Oxy axis, you can get a hyperbola. When constructing a hyperbolic paraboloid, be guided by the following equation:
x ^ 2 / a ^ 2-y ^ 2 / b ^ 2 \u003d 2z - the equation of a hyperbolic paraboloid
Initially, construct a fixed parabola in the Oxz plane. Draw a movable parabola in the Oyz plane. Then set the height of the paraboloid h. To do this, mark two points on the fixed parabola, which will be the vertices of two more moving parabolas. Then draw another coordinate system O "x" y "to draw hyperbolas. The center of this coordinate system should coincide with the height of the paraboloid. After all construction, draw those two moving parabolas mentioned above so that they touch the extreme points of the hyperbolas. the result is a hyperbolic paraboloid.
The hyperbolic paraboloid also belongs to the surfaces of the second order. This surface cannot be obtained by applying an algorithm using the rotation of some line about a fixed axis.
A special model is used to construct a hyperbolic paraboloid. This model includes two parabolas located in two mutually perpendicular planes.
Let parabola I be in the plane and motionless. Parabola II makes a complex movement:
▫ its initial position coincides with the plane 
, and the vertex of the parabola coincides with the origin: 
=(0,0,0);
▫ further, this parabola makes a parallel transfer movement, and its top 
makes a trajectory coinciding with parabola I;
▫ two different initial positions of parabola II are considered: one - branches of the parabola upward, the second - branches downward.
Let us write the equations: for the first parabola I: 
- invariable; for the second parabola II: 
- initial position, equation of motion: 
It is not hard to see that the point 
has coordinates: 
... Since it is necessary to display the law of motion of a point 
: this point belongs to parabola I, then the following relations must be constantly satisfied: 
=
and 
.
It is easy to see from the geometric features of the model that the movable parabola sweeps up some surface. In this case, the equation of the surface described by parabola II has the form:
or → 
. (1)
The shape of the resulting surface depends on the distribution of the parameter signs 
... Two cases are possible:
1). Signs of quantities p and q coincide: parabolas I and II are located on the same side of the plane OXY... Let's take: p = a 2 and q = b 2 ... Then we get the equation of the known surface:
→
elliptical paraboloid
. (2)
2). Signs of quantities p and q are different: parabolas I and II are located on opposite sides of the plane OXY... Let be p = a 2 and q = - b 2 ... Now we get the equation of the surface:
→hyperbolic paraboloid
. (3)
It is not difficult to represent the geometric shape of the surface determined by equation (3) if we recall the kinematic model of the interaction of two parabolas participating in the motion.
In the figure, parabola I is conventionally shown in red. Only the vicinity of the surface at the origin is shown. Due to the fact that the shape of the surface expressively hints at a cavalry saddle, this neighborhood is often called - saddle .
In physics, when studying the stability of processes, the following types of equilibrium are introduced: stable - a hole, convex downward, unstable - a surface convex upward, and intermediate - a saddle. Equilibrium of the third type is also referred to the type of unstable equilibrium, and only on the red line (parabola I) equilibrium is possible.
§ 4. Cylindrical surfaces.
When considering surfaces of revolution, we have defined the simplest cylindrical surface - a cylinder of revolution, that is, a circular cylinder.
In elementary geometry, the cylinder is defined by analogy with general definition prisms. It's quite complicated:
▫ let us have a flat polygon in space 
- denote as 
, and the polygon coincides with it 
- denote as 
;
▫ applicable to polygon 
parallel translation movement: points 
move along trajectories parallel to a given direction 
;
▫ if you stop moving the polygon 
, then its plane 
parallel to plane 
;
▫ the surface of a prism is called: a collection of polygons 
,
– grounds
prisms and parallelograms 
,
,...
– side surface
prisms.
IN 
we will use the elementary definition of a prism to construct a more general definition of a prism and its surface, namely, we will distinguish:
▫ unlimited prism is a multifaceted body bounded by edges 
,
, ... and the planes between these edges;
▫ bounded prism is a polyhedral body bounded by edges 
,
, ... and parallelograms 
,
, ...; the lateral surface of this prism is a set of parallelograms 
,
, ...; prism bases - a collection of polygons 
,
.
Let us have an unlimited prism: 
,
, ... Let's cross this prism with an arbitrary plane 
... Let's cross the same prism with another plane 
... In the section we get a polygon 
... In the general case, we assume that the plane 
not parallel to plane 
... This means that the prism is not built by parallel translation of the polygon. 
.
The proposed construction of a prism includes not only straight and inclined prisms, but also any truncated ones.
In analytic geometry, we will understand cylindrical surfaces in such a generalized way that an unbounded cylinder includes an unbounded prism as a special case: one has only to assume that a polygon can be replaced by an arbitrary line, not necessarily closed - guide
cylinder. Direction 
called generatrix
cylinder.
From all that has been said, it follows that to define a cylindrical surface, it is necessary to specify a guide line and a direction of the generator.
Cylindrical surfaces are obtained on the basis of 2 nd order plane curves serving guides for generators .
At the initial stage of the study of cylindrical surfaces, we make simplifying assumptions:
▫ let the guide of a cylindrical surface always be located in one of the coordinate planes;
▫ direction of the generatrix 
coincides with one of the coordinate axes, that is, perpendicular to the plane in which the guideline is defined.
The adopted restrictions do not lead to a loss of generality, since there remains a possibility due to the choice of sections by planes 
and 
build arbitrary geometric shapes: straight, inclined, truncated cylinders.
Elliptical cylinder .
Let an ellipse be taken as a cylinder guide 
:
located in the coordinate plane 
: elliptical cylinder.
Hyperbolic cylinder .
:
, and the direction of the generator determines the axis 
... In this case, the equation of the cylinder is the line itself 
: hyperbolic cylinder.
Parabolic cylinder .
Let the hyperbola be taken as a cylinder guide 
:
located in the coordinate plane 
, and the direction of the generator determines the axis 
... In this case, the equation of the cylinder is the line itself 
: parabolic cylinder.
Comment: Considering general rules the construction of equations of cylindrical surfaces, as well as the presented particular examples of elliptic, hyperbolic and parabolic cylinders, we note: the construction of a cylinder for any other generator, for the adopted simplifying conditions, should not cause any difficulties!
Let us now consider more general conditions for constructing equations of cylindrical surfaces:
▫ the guide of a cylindrical surface is located in an arbitrary plane of space 
;
▫ direction of the generatrix 
in the adopted coordinate system arbitrarily.
The accepted conditions are shown in the figure.
▫ cylindrical surface guide 
located in an arbitrary plane 
space 
;
▫ coordinate system 
obtained from the coordinate system 
parallel transfer;
▫ guide location 
in plane 
the most preferable: for a second-order curve, we will assume that the origin 
coincides with center
symmetry of the curve under consideration;
▫ direction of the generatrix 
arbitrary (can be specified in any of the ways: vector, straight line, etc.).
In what follows, we will assume that the coordinate systems 
and 
match. This means that the 1st step of the general algorithm for constructing cylindrical surfaces, reflecting parallel translation: 
→
is pre-executed.
Let us recall how parallel transfer is taken into account in the general case, considering a simple example.
Example 6–13
: In coordinate system 
as: 
\u003d 0. Write down the equation of this rail in the system 
.
Decision:
1). We denote an arbitrary point 
: in system 
as 
, and in the system 
as 
.
2). We write down the vector equality: 
=
+
... In coordinate form, this can be written as: 
=
+
... Or in the form: 
=
–
, or: 
=.
3). We write the equation of the cylinder guide 
in coordinate system 
:
Answer: Transformed guide equation: \u003d 0.
So, we will assume that the center of the curve representing the cylinder guide is always located at the origin of the coordinates of the system 
in plane 
.
Figure: IN ... Basic drawing when building a cylinder.
Let's make one more assumption to simplify the final steps of constructing the cylindrical surface. Since using the rotation of the coordinate system, it is easy to align the direction of the axis 
coordinate systems 
with plane normal 
, and the directions of the axes 
and 
with axes of symmetry of the guide 
, then we will assume that as the initial position of the guide 
we have a curve located in the plane 
, and one of its axis of symmetry coincides with the axis 
, and the second with the axis 
.
Comment: since the execution of the operations parallel transfer and rotation around a fixed axis of the operation are quite simple, the assumptions made do not limit the applicability of the developed algorithm for constructing a cylindrical surface in the most general case!
We saw that when constructing a cylindrical surface in the case when the guide 
located in the plane 
, and the generator is parallel to the axis 
, it is enough to define only the guide 
.
Since a cylindrical surface can be uniquely determined by specifying any line obtained in the section of this surface by an arbitrary plane, we will accept the following general algorithm for solving the problem:
1
▫
... Let the direction of the generator 
cylindrical surface is given by the vector 
... We will design a guide 
given by the equation: 
\u003d 0, on a plane perpendicular to the direction of the generatrix 
, that is, on the plane 
... As a result, the cylindrical surface will be specified in the coordinate system 
equation: 
=0.
2
▫
around the axis 
at the corner 
: meaning of angle 
compatible with the system 
, and the equation of the conical surface is transformed into the equation: 
=0.
3
▫
... Apply the rotation of the coordinate system 
around the axis 
at the corner 
: meaning of angle 
is quite clear from the figure. As a result of rotation, the coordinate system 
compatible with the system 
, and the equation of the conical surface is transformed into 
\u003d 0. This is the equation of a cylindrical surface, for which a guide was given 
and generating 
in coordinate system 
.
The example below illustrates the implementation of the written algorithm and the computational difficulties of such tasks.
Example 6–14
: In coordinate system 
the equation of the cylinder guide is given 
as: 
\u003d 9. Make an equation for a cylinder whose generatrices are parallel to the vector 
=(2,–3,4).
R 
solution:
1). Let's design the cylinder guide on a plane perpendicular to 
... It is known that such a transformation turns a given circle into an ellipse, the axes of which will be: 
\u003d 9, and small 
=
.
This figure illustrates the design of a circle defined in a plane 
on the coordinate plane 
.
2). The result of designing a circle is an ellipse: 
\u003d 1, or 
... In our case, these are: 
where 
=
=
.
3
). So, the equation of a cylindrical surface in the coordinate system 
received. Since, according to the condition of the problem, we must have the equation of this cylinder in the coordinate system 
, then it remains to apply a coordinate transformation that transfers the coordinate system 
to coordinate system 
, along with the equation of the cylinder: 
into an equation expressed in terms of variables 
.
4). We will use basic figure, and write down all the trigonometric values \u200b\u200bnecessary for solving the problem:
=
=
,
=
=
,
=
=
.
five). Let us write down the coordinate transformation formulas when passing from the system 
to the system 
:
(IN)
6). Let us write down the coordinate transformation formulas when passing from the system 
to the system 
:
(FROM)
7). Substituting variables 
from system (B) to system (C), and also taking into account the values \u200b\u200bof the used trigonometric functions, we write:
=
=
.
=
=
.
eight). It remains to substitute the found values 
and 
into the cylinder guide equation 
:
in coordinate system 
... After completing carefully
all algebraic transformations, we obtain the equation of the conical surface in the coordinate system 
:
=0.
Answer: cone equation: \u003d 0.
Example 6–15
: In coordinate system 
the equation of the cylinder guide is given 
as: 
=9,
\u003d 1. Make an equation for a cylinder whose generatrices are parallel to the vector 
=(2,–3,4).
Decision:
1). It is easy to see that this example differs from the previous one only in that the guide was parallelly moved 1 up.
2). This means that in relations (B) one should take: 
=
-1. Taking into account the expressions of the system (C), we correct the notation for the variable 
:
=
.
3). The change is easily taken into account by correcting the final notation of the equation for the cylinder from the previous example:
Answer: cone equation: \u003d 0.
Comment: it is easy to see that the main difficulty with multiple transformations of coordinate systems in problems with cylindrical surfaces is accuracy and endurance in algebraic marathons: long live the educational system adopted in our long-suffering country!