Definition 1. Cylindrical surface is a surface formed by straight lines parallel to each other, called its forming .
If any plane intersecting all generators cylindrical surfaces, intersects it along the line R, then this line is called guide this cylindrical surface.
Theorem . If a Cartesian coordinate system and an equation in the plane are introduced in space xOy is the equation of some line R, then this equation in space is the equation of a cylindrical surface L with guide line R, and the generators are parallel to the axis Oz(Fig. 3.19, a).
Proof. Dot 
lies on a cylindrical surface L if and only if the projection 
points M to the plane xOy parallel to the axis Oz lies on the line R, i.e. if and only if the equation holds 
.
Similar conclusions hold for equations of the form 
(Fig. 3.19, b) and 
(Fig. 3.19, c).
Definition 2 . Cylindrical surfaces whose guides are lines of the second order are called cylindrical surfaces of the second order .
There are three types of second order cylinders: elliptical (Fig. 3.20)
, (5.42)
hyperbolic (Fig. 3.21)
, (5.43)
parabolic (Fig. 3.22)
. (5.44)
Rice. 3.20 Fig. 3.21 Fig. 3.22
For the cylinders given by equations (5.42), (5.43) and (5.44), the guide lines are, respectively, an ellipse
,
hyperbola
,
parabola
,
and the generators are parallel to the axis Oz.
Comment. As we have seen, conical and cylindrical surfaces of the second order have rectilinear generators, and each of these surfaces can be formed by the movement of a straight line in space.
It turns out that among all second-order surfaces, in addition to the cylinder and cone, a single-sheet hyperboloid and a hyperbolic paraboloid also have rectilinear generators, and, just as in the case of a cylinder and a cone, both of these surfaces can be formed by the movement of a straight line in space (see. special literature).
§4. Reducing the general second-order surface equation to canonical form
In the general second order surface equation
a) quadratic form
Where 
;
b) linear form
Where 
;
V) free member
.
To bring equation (5.45) to canonical form, it is necessary, first of all, to carry out such a coordinate transformation 
, and, consequently, the associated orthonormal basis 
, which transforms the quadratic form (5.46) to canonical form(see book 2, chapter 8, §3, clause 3.1).
The matrix of this quadratic form is
,
where, i.e. matrix A– symmetrical. Let us denote by 
eigenvalues, and through 
orthonormal basis composed of eigenvectors of the matrix A. Let
–
transition matrix from basis 
to the base 
, A 
– a new coordinate system associated with this basis.
Then, when transforming coordinates
(5.48)
the quadratic form (5.46) takes the canonical form
Where 
.
Now, applying the coordinate transformation (5.48) to the linear form (5.47), we obtain
Where 
,
– new form coefficients (5.47).
Thus, equation (5.45) takes the form
+.
This equation can be reduced to canonical form using parallel transfer of the coordinate system according to the formulas
or 
(5.49)
After performing the coordinate system transformation by parallel transfer(5.49), general second-order surface equation (5.45) with respect to the Cartesian coordinate system 
will express one of the following seventeen surfaces:
1) ellipsoid 
2) imaginary ellipsoid 
3) single-sheet hyperboloid 
4) two-sheet hyperboloid 
5) cone 
6) imaginary cone 
7) elliptical paraboloid
8) hyperbolic paraboloid
9) elliptical cylinder
10) imaginary elliptic cylinder
11) two imaginary intersecting planes
12) hyperbolic cylinder
13) two intersecting planes
14) parabolic cylinder
15) two parallel planes
16) two imaginary parallel planes
17) two coinciding planes
Example. Determine the type and location of a surface defined relative to a Cartesian rectangular coordinate system 
and the associated orthonormal basis 
equation
Let us give the quadratic form
(5.51)
to the canonical form. The matrix of this form has the form
.
Let us determine the eigenvalues of this matrix from the characteristic equation
From here 1 = 2, 2 = 0, 3 = 3.
Now we find eigenvectors matrices A: 1) let 
, then from the equation 
or in coordinate form
find where 
– any number, and therefore 
, A 
. From the entire set of collinear vectors 
choose a vector 
, whose modulus 
, i.e. normalize the vector 
.
2) for 
we have
.
From here 
, Where 
– any number. Then 
, A 
. Normalizing the vector 
, find the unit vector 
:
,
Where 
.
3)
, then for the components 
vector 
we have a system
From where, where 
– any number, and therefore 
, A 
. Normalizing the vector 
, find the unit vector 
for the direction given by the vector 
:
Where 
.
Let us now move from the orthonormal basis 
to an orthonormal basis 
, composed of eigenvectors of the matrix A and connect with the last basis a new Cartesian rectangular coordinate system 
. The transition matrix for such a transformation has the form
,
and the coordinates are converted according to the formulas
(5.52)
Applying this coordinate transformation to the quadratic form (5.51), we reduce it to the canonical form
, Where 
.
Let us now determine what form the linear formula has
, Where 
,
if the coordinates are transformed according to formulas (5.52). We have
Thus, if the coordinate system 
transform using formulas (5.52), then relative to the new coordinate system 
the second-order surface under consideration is given by the equation
Equation (5.53) is reduced to canonical form using parallel transfer of the coordinate system according to the formulas
after which, the equation of the surface relative to the coordinate system 
takes the form
or 
This equation expresses an elliptic cylinder whose directing ellipse is located in the coordinate plane 
, and the generating lines are parallel to the axis 
Comment. Reduction scheme general equation of a second-order surface to canonical form, presented in this section, can also be applied to reducing the general equation of a second-order curve to canonical form.
Students most often encounter surfaces of the 2nd order in the first year. At first, problems on this topic may seem simple, but as you study higher mathematics and deepening into the scientific side, you can finally lose your bearings on what is happening. In order to prevent this from happening, you need to not just memorize, but understand how this or that surface is obtained, how changing coefficients affects it and its location relative to the original coordinate system, and how to find new system(one in which its center coincides with the origin of coordinates, and is parallel to one of coordinate axes). Let's start from the very beginning.
Definition
A 2nd order surface is called a GMT, the coordinates of which satisfy the general equation of the following form:
It is clear that each point belonging to the surface must have three coordinates in some designated basis. Although in some cases locus points can degenerate, for example, into a plane. This only means that one of the coordinates is constant and equal to zero throughout the entire range of permissible values.
The full written form of the above equality looks like this:
A 11 x 2 +A 22 y 2 +A 33 z 2 +2A 12 xy+2A 23 yz+2A 13 xz+2A 14 x+2A 24 y+2A 34 z+A 44 =0.
A nm - some constants, x, y, z - variables corresponding affine coordinates any point. In this case, at least one of the constant factors must not be equal to zero, that is, not any point will correspond to the equation.
In the vast majority of examples, many numerical factors are still identically equal to zero, and the equation is significantly simplified. In practice, determining whether a point belongs to a surface is not difficult (it is enough to substitute its coordinates into the equation and check whether the identity holds). The key point in such work is to bring the latter to canonical form.
The equation written above defines any (all listed below) 2nd order surfaces. Let's look at examples below.
Types of surfaces of 2nd order
The equations of 2nd order surfaces differ only in the values of the coefficients A nm. From general view at certain values of the constants, various surfaces can be obtained, classified as follows:
- Cylinders.
- Elliptical type.
- Hyperbolic type.
- Conical type.
- Parabolic type.
- Planes.
Each of the listed types has a natural and imaginary form: in the imaginary form, the locus of real points either degenerates into a more a simple figure, or is absent altogether.
Cylinders
This is the simplest type, as the relatively complex curve lies only at the base, acting as a guide. The generators are straight lines, perpendicular planes, in which the base lies.
The graph shows circular cylinder - special case elliptical cylinder. In the XY plane, its projection will be an ellipse (in our case, a circle) - a guide, and in XZ - a rectangle - since the generators are parallel to the Z axis. To obtain it from the general equation, it is necessary to give the following values to the coefficients:
Instead of the usual symbols x, y, z, x's with serial number- it doesn't matter.
In fact, 1/a 2 and the other constants indicated here are the same coefficients indicated in the general equation, but it is customary to write them in exactly this form - this is canonical representation. In what follows, this type of entry will be used exclusively.
This defines a hyperbolic cylinder. The scheme is the same - the hyperbole will be the guide.
A parabolic cylinder is defined slightly differently: its canonical form includes a coefficient p, called a parameter. In fact, the coefficient is q=2p, but it is customary to divide it into the two factors presented.
There is another type of cylinder: imaginary. No real point belongs to such a cylinder. It is described by the equation of an elliptic cylinder, but instead of one there is -1.
Elliptical type
The ellipsoid can be stretched along one of the axes (along which it depends on the values of the constants a, b, c indicated above; obviously, the larger axis will correspond to a larger coefficient).
There is also an imaginary ellipsoid - provided that the sum of the coordinates multiplied by the coefficients is equal to -1:
Hyperboloids
When a minus appears in one of the constants, the equation of the ellipsoid turns into the equation of a single-sheet hyperboloid. You must understand that this minus does not have to be located in front of the x3 coordinate! It only determines which of the axes will be the axis of rotation of the hyperboloid (or parallel to it, since when additional terms appear in the square (for example, (x-2) 2), the center of the figure shifts, as a result, the surface moves parallel to the coordinate axes). This applies to all 2nd order surfaces.
In addition, you need to understand that the equations are presented in canonical form and they can be changed by varying the constants (while maintaining the sign!); at the same time, their appearance (hyperboloid, cone, and so on) will remain the same.
Such an equation is given by a two-sheet hyperboloid.
Conical surface
In the cone equation, there is no unity - it is equal to zero.
A cone is only a limited conical surface. The picture below shows that, in fact, there will be two so-called cones on the chart.
Important note: in all considered canonical equations, constants are assumed to be positive by default. Otherwise, the sign may affect the final graph.
The coordinate planes become planes of symmetry of the cone, the center of symmetry is located at the origin.
In the equation of an imaginary cone there are only pluses; it owns one single real point.
Paraboloids
Surfaces of 2nd order in space can take various shapes even with similar equations. For example, paraboloids come in two types.
x 2 /a 2 +y 2 /b 2 =2z
An elliptical paraboloid, when the Z axis is perpendicular to the drawing, will be projected into an ellipse.
x 2 /a 2 -y 2 /b 2 =2z
Hyperbolic paraboloid: in sections with planes parallel to ZY, parabolas will be obtained, and in sections with planes parallel to XY, hyperbolas will be obtained.
Intersecting planes
There are cases when 2nd order surfaces degenerate in the plane. These planes can be arranged in various ways.
First let's look at intersecting planes:
x 2 /a 2 -y 2 /b 2 =0
With this modification of the canonical equation, we simply get two intersecting planes (imaginary!); all real points are located on the axis of the coordinate that is absent in the equation (in the canonical one - the Z axis).
Parallel planes
If there is only one coordinate, 2nd order surfaces degenerate into a pair parallel planes. Don't forget, any other variable can take the place of the player; then planes parallel to other axes will be obtained.
In this case they become imaginary.
Coincident planes
With this simple equation a pair of planes degenerates into one - they coincide.
Don't forget that in the case of a three-dimensional basis, the above equation does not specify the straight line y=0! It's missing the other two variables, but that just means their value is constant and equal to zero.
Construction
One of the most difficult tasks for a student is precisely the construction of 2nd order surfaces. It is even more difficult to move from one coordinate system to another, taking into account the angles of inclination of the curve relative to the axes and the offset of the center. Let's review how to consistently determine future view drawing in an analytical way.
To construct a 2nd order surface, you need to:
- bring the equation to canonical form;
- determine the type of surface under study;
- build based on the values of the coefficients.
Below are all the types considered:
To reinforce this, we will describe in detail one example of this type of task.
Examples
Let's say we have the equation:
3(x 2 -2x+1)+6y 2 +2z 2 +60y+144=0
Let's bring it to canonical form. Let's select complete squares, that is, we will arrange the available terms in such a way that they are a decomposition of the square of the sum or difference. For example: if (a+1) 2 =a 2 +2a+1, then a 2 +2a+1=(a+1) 2. We will perform a second operation. Parentheses in in this case it is not necessary to disclose, since this will only complicate the calculations, but to bring out common multiplier 6 (in brackets with perfect square game) you need:
3(x-1) 2 +6(y+5) 2 +2z 2 =6
The variable zet appears in this case only once - you can leave it alone for now.
Let's analyze the equation at this stage: all unknowns have a plus sign in front of them; Dividing by six leaves one. Consequently, we have before us an equation defining an ellipsoid.
Notice that 144 was factored into 150-6, and then -6 was moved to the right. Why did it have to be done this way? Obviously the most big divisor V in this example-6, therefore, in order for a unit to remain on the right after dividing by it, it is necessary to “set aside” exactly 6 from 144 (the fact that the unit should be on the right is indicated by the presence of a free term - a constant not multiplied by an unknown).
Let's divide everything by six and get the canonical equation of the ellipsoid:
(x-1) 2 /2+(y+5) 2 /1+z 2 /3=1
In the previously used classification of 2nd order surfaces, a special case is considered when the center of the figure is at the origin of coordinates. In this example it is offset.
We assume that each bracket with unknowns is a new variable. That is: a=x-1, b=y+5, c=z. In the new coordinates, the center of the ellipsoid coincides with the point (0,0,0), therefore, a=b=c=0, whence: x=1, y=-5, z=0. In the initial coordinates, the center of the figure lies at the point (1,-5,0).
The ellipsoid will be obtained from two ellipses: the first in the XY plane and the second in the XZ plane (or YZ - it doesn’t matter). The coefficients by which the variables are divided are squared in the canonical equation. Therefore, in the above example, it would be more correct to divide by the root of two, one and the root of three.
The minor axis of the first ellipse, parallel to the Y axis, is equal to two. The major axis is parallel to the X axis - two roots of two. The minor axis of the second ellipse, parallel to the Y axis, remains the same - it is equal to two. A major axis, parallel to the Z axis, is equal to two roots of three.
Using the data obtained from the original equation by converting it to canonical form, we can draw an ellipsoid.
Summing up
The topic covered in this article is quite extensive, but in fact, as you can now see, it is not very complicated. Its development, in fact, ends at the moment when you memorize the names and equations of surfaces (and, of course, what they look like). In the example above, we examined each step in detail, but bringing the equation to canonical form requires minimal knowledge of higher mathematics and should not cause any difficulties for the student.
Analysis of the future schedule based on existing equality is already more than difficult task. But to solve it successfully, it is enough to understand how the corresponding second-order curves are constructed - ellipses, parabolas and others.
Cases of degeneration are an even simpler section. Due to the absence of some variables, not only the calculations are simplified, as mentioned earlier, but also the construction itself.
As soon as you can confidently name all types of surfaces, vary constants, turning a graph into one shape or another, the topic will be mastered.
Good luck in your studies!
With the difference that instead of “flat” graphs, we will consider the most common spatial surfaces, and also learn how to competently build them by hand. I spent quite a long time selecting software tools for creating three-dimensional drawings and found a couple of good applications, but despite all the ease of use, these programs do not solve the important practical question. The fact is that in the foreseeable historical future, students will still be armed with a ruler and a pencil, and even having a high-quality “machine” drawing, many will not be able to correctly transfer it to checkered paper. Therefore, in the manual special attention is devoted to the technique of manual construction, and a significant part of the page’s illustrations is a handmade product.
What's different about this reference material from analogues?
Having a decent practical experience, I know very well which surfaces I most often have to deal with in real problems higher mathematics, and I hope that this article will help you in as soon as possible replenish your luggage with relevant knowledge and applied skills, which should be enough in 90-95% of cases.
What you need to know at the moment?
The most basic:
Firstly, you need to be able to build correctly spatial Cartesian coordinate system (see the beginning of the article Graphs and properties of functions ) .
What will you gain after reading this article?
Bottle After mastering the lesson materials, you will learn to quickly determine the type of surface by its function and/or equation, imagine how it is located in space, and, of course, make drawings. It’s okay if you don’t get everything in your head after the first reading - you can always return to any paragraph later as needed.
Information is within the power of everyone - to master it you do not need any super knowledge, special artistic talent or spatial vision.
Let's begin!
In practice, the spatial surface is usually given function of two variables or an equation of the form (the constant on the right side is most often equal to zero or one). The first designation is more typical for mathematical analysis, the second – for analytical geometry . The equation is essentially implicitly given a function of 2 variables, which in typical cases can easily be reduced to the form . I remind you simplest example c:
– plane equation kind .
– plane function in explicitly .
Let's start with it:
Common equations of planes
Typical options arrangement of planes in rectangular system coordinates are discussed in detail at the very beginning of the article Plane equation . However, let us once again dwell on the equations that have great importance for practice.
First of all, you must fully automatically recognize the equations of planes that are parallel to coordinate planes. Fragments of planes are standardly depicted as rectangles, which in the last two cases look like parallelograms. By default, you can choose any dimensions (within reasonable limits, of course), but it is desirable that the point at which the coordinate axis “pierces” the plane is the center of symmetry: 
Strictly speaking, the coordinate axes should be depicted with dotted lines in some places, but in order to avoid confusion we will neglect this nuance.
– (left drawing) the inequality specifies the half-space farthest from us, excluding the plane itself;
– (middle drawing) the inequality specifies the right half-space, including the plane;
– (right drawing) the double inequality defines a “layer” located between the planes, including both planes.
For self-warm-up:
Example 1
Draw a body bounded by planes
Create a system of inequalities that define a given body.
An old acquaintance should emerge from under the lead of your pencil. cuboid . Do not forget that invisible edges and faces must be drawn with a dotted line. Finished drawing at the end of the lesson.
Please, DON'T NEGLECT learning objectives, even if they seem too simple. Otherwise, it may happen that you missed one, missed two, and then spent a solid hour trying out a three-dimensional drawing in some real example. Besides, mechanical work will help you learn the material much more effectively and develop your intelligence! It is no coincidence that kindergarten And elementary school children are loaded with drawing, modeling, construction kits and other tasks for fine motor skills fingers. Sorry for the digression, but don’t let my two notebooks go to waste developmental psychology =)
We will conditionally call the next group of planes “direct proportionality” - these are planes passing through the coordinate axes:
2) an equation of the form specifies a plane passing through the axis ;
3) an equation of the form specifies a plane passing through the axis.
Although the formal sign is obvious (which variable is missing from the equation – the plane passes through that axis), it is always useful to understand the essence of the events taking place:
Example 2
Construct plane
What is the best way to build? I suggest next algorithm:
First, let’s rewrite the equation in the form , from which it is clearly seen that the “y” can take any meanings. Let us fix the value, that is, we will consider the coordinate plane. Equations set space line , lying in a given coordinate plane. Let's depict this line in the drawing. The straight line passes through the origin of coordinates, so to construct it it is enough to find one point. Let . Set aside a point and draw a straight line.
Now we return to the equation of the plane. Since the "Y" accepts any values, then the straight line constructed in the plane is continuously “replicated” to the left and to the right. This is exactly how our plane is formed, passing through the axis. To complete the drawing, to the left and right of the straight line we put two parallel lines and “close” the symbolic parallelogram with transverse horizontal segments: 
Since the condition did not impose additional restrictions, a fragment of the plane could be depicted in slightly smaller or slightly larger sizes.
Let us repeat once again the meaning of spatial linear inequality by example. How to determine the half-space it defines? Let's take some point not belonging to plane, for example, a point from the half-space closest to us and substitute its coordinates into the inequality:
Received true inequality, which means that the inequality specifies the lower (relative to the plane) half-space, while the plane itself is not included in the solution.
Example 3
Construct planes
A) ;
b) .
These are tasks for self-building, in case of difficulties, use similar reasoning. Brief instructions and drawings at the end of the lesson.
In practice, planes parallel to the axis are especially common. The special case when the plane passes through the axis was just discussed in point “be”, and now we will analyze more common task:
Example 4
Construct plane
Solution: the variable “z” is not explicitly included in the equation, which means that the plane is parallel to the applicate axis. Let's use the same technique as in the previous examples.
Let us rewrite the equation of the plane in the form 
from which it is clear that “zet” can take any meanings. Let’s fix it and draw a regular “flat” straight line in the “native” plane. To construct it, it is convenient to take reference points.
Since "Z" accepts All values, then the constructed straight line continuously “multiplies” up and down, thereby forming the desired plane 
. We carefully draw up a parallelogram of a reasonable size: 
Ready.
Equation of a plane in segments
The most important applied variety. If All odds general equation of the plane
non-zero, then it can be represented in the form 
which is called equation of the plane in segments. It is obvious that the plane intersects the coordinate axes at points , and the great advantage of such an equation is the ease of constructing a drawing:
Example 5
Construct plane
Solution: First, let's create an equation of the plane in segments. Let's throw the free term to the right and divide both sides by 12: 
No, there is no typo here and all things happen in space! We examine the proposed surface using the same method that was recently used for planes. Let's rewrite the equation in the form 
, from which it follows that “zet” takes any meanings. Let us fix and construct an ellipse in the plane. Since "zet" accepts All values, then the constructed ellipse is continuously “replicated” up and down. It is easy to understand that the surface infinite:
This surface is called elliptical cylinder
. An ellipse (at any height) is called guide cylinder, and parallel lines passing through each point of the ellipse are called forming cylinder (which are literally words form it). The axis is axis of symmetry surface (but not part of it!).
The coordinates of any point belonging to a given surface necessarily satisfy the equation 
.
Spatial the inequality specifies the “inside” of the infinite “pipe”, including the cylindrical surface itself, and, accordingly, opposite inequality defines the set of points outside the cylinder.
IN practical problems the most popular special case is when guide cylinder is circle :
Example 8
Build a surface given by the equation
It is impossible to depict an endless “pipe”, so art is usually limited to “trimming”.
First, it is convenient to construct a circle of radius in the plane, and then a couple more circles above and below. The resulting circles ( guides cylinder) carefully connect with four parallel straight lines ( forming cylinder): 
Don't forget to use dotted lines for lines that are invisible to us.
The coordinates of any point belonging to a given cylinder satisfy the equation 
. The coordinates of any point lying strictly inside the “pipe” satisfy the inequality 
, and the inequality 
defines a set of points of the external part. For a better understanding, I recommend considering several specific points space and see for yourself.
Example 9
Construct a surface and find its projection onto the plane
Let's rewrite the equation in the form 
from which it follows that "x" takes any meanings. Let us fix and depict in the plane circle
– with center at the origin, unit radius. Since "x" continuously accepts All values, then the constructed circle generates a circular cylinder with an axis of symmetry. Draw another circle ( guide cylinder) and carefully connect them with straight lines ( forming cylinder). In some places there were overlaps, but what to do, such a slope: 
This time I limited myself to a piece of a cylinder in the gap, and this is not accidental. In practice, it is often necessary to depict only a small fragment of the surface.
Here, by the way, there are 6 generatrices - two additional straight lines “cover” the surface from the upper left and lower right corners.
Now let's look at the projection of a cylinder onto a plane. Many readers understand what projection is, but, nevertheless, let’s conduct another five-minute physical exercise. Please stand and bow your head over the drawing so that the point of the axis points perpendicular to your forehead. What a cylinder appears to be from this angle is its projection onto a plane. But it seems to be an endless strip, enclosed between straight lines, including the straight lines themselves. This projection- that's exactly domain of definition functions (upper “gutter” of the cylinder), (lower “gutter”).
By the way, let’s clarify the situation with projections onto other coordinate planes. Let the sun's rays shine on the cylinder from the tip and along the axis. The shadow (projection) of a cylinder onto a plane is a similar infinite strip - a part of the plane bounded by straight lines (- any), including the straight lines themselves.
But the projection onto the plane is somewhat different. If you look at the cylinder from the tip of the axis, then it will be projected into a circle of unit radius 
, with which we began the construction.
Example 10
Construct a surface and find its projections onto coordinate planes
This is a task for independent decision. If the condition is not very clear, square both sides and analyze the result; find out which part of the cylinder is specified by the function. Use the construction technique repeatedly used above. Quick Solution, drawing and comments at the end of the lesson.
Elliptical and other cylindrical surfaces can be offset relative to the coordinate axes, for example:
(based on familiar motives of the article about 2nd order lines
)
– a cylinder of unit radius with a line of symmetry passing through a point parallel to the axis. However, in practice, such cylinders are encountered quite rarely, and it is absolutely incredible to encounter a cylindrical surface that is “oblique” relative to the coordinate axes.
Parabolic cylinders
As the name suggests, guide such a cylinder is parabola .
Example 11
Construct a surface and find its projections onto coordinate planes.
I couldn't resist this example =)
Solution: Let's go along the beaten path. Let's rewrite the equation in the form, from which it follows that “zet” can take any value. Let us fix and construct an ordinary parabola on the plane, having previously marked the trivial support points. Since "Z" accepts All values, then the constructed parabola is continuously “replicated” up and down to infinity. We lay the same parabola, say, at a height (in the plane) and carefully connect them with parallel straight lines ( forming the cylinder): 
I remind you useful technique: if you are initially unsure of the quality of the drawing, then it is better to first draw the lines very thinly with a pencil. Then we evaluate the quality of the sketch, find out the areas where the surface is hidden from our eyes, and only then apply pressure to the stylus.
Projections.
1) The projection of a cylinder onto a plane is a parabola. It should be noted that in this case it is impossible to talk about domain of definition of a function of two variables – for the reason that the cylinder equation is not reducible to functional view.
2) The projection of a cylinder onto a plane is a half-plane, including the axis
3) And finally, the projection of the cylinder onto the plane is the entire plane.
Example 12
Build parabolic cylinders:
a) limit yourself to a fragment of the surface in the near half-space;
b) in the interval
In case of difficulties, we do not rush and reason by analogy with previous examples; fortunately, the technology has been thoroughly developed. It is not critical if the surfaces turn out a little clumsy - it is important to correctly display the fundamental picture. I myself don’t really bother with the beauty of the lines; if I get a passable drawing with a C grade, I usually don’t redo it. By the way, the sample solution uses another technique to improve the quality of the drawing ;-)
Hyperbolic cylinders
Guides such cylinders are hyperboles. This type of surface, according to my observations, is much less common than previous types, so I will limit myself to a single schematic drawing of a hyperbolic cylinder: 
The principle of reasoning here is exactly the same - the usual school hyperbole
from the plane continuously “multiplies” up and down to infinity.
The considered cylinders belong to the so-called 2nd order surfaces, and now we will continue to get acquainted with other representatives of this group:
Ellipsoid. Sphere and ball
Canonical equation an ellipsoid in a rectangular coordinate system has the form 
, Where - positive numbers (axle shafts ellipsoid), which in general case different. An ellipsoid is called surface, so body, limited by a given surface. The body, as many have guessed, is determined by inequality 
and coordinates of any internal point(as well as any point on the surface) necessarily satisfy this inequality. The design is symmetrical with respect to coordinate axes and coordinate planes: 
The origin of the term “ellipsoid” is also obvious: if the surface is “cut” coordinate planes, then the sections will have three different ones (in the general case)
