With the difference that instead of "flat" graphs, we will consider the most common spatial surfaces, and also learn how to correctly build them by hand. I have been looking for software tools for building 3D drawings for quite some time and found a couple of good applications, but despite all the ease of use, these programs do not solve an important practical issue well. The fact is that in the foreseeable historical future, students will continue to be armed with a ruler with 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 paid attention to the technique of manual construction, and a significant part of the illustrations on the page is a handmade product.
How is this reference material different from analogues?
With decent practical experience, I know very well which surfaces are most often dealt with in real problems of higher mathematics, and I hope that this article will help you quickly replenish your luggage with relevant knowledge and applied skills, which are 90-95% cases should suffice.
What do you need to know right now?
The most elementary:
First, you need to be able build right 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 materials of the lesson, you will learn how 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 not everything fits in your head from the 1st reading - you can always return to any paragraph as needed later.
Information is within the power of everyone - to master it, you do not need any super-knowledge, special artistic talent and spatial vision.
Begin!
In practice, the spatial surface is usually given function of two variables or an equation of the form (the constant of 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, in essence, is implicitly given function of 2 variables, which in typical cases can be easily reduced to the form . I remind you of the simplest example c :
–plane equation kind.
is the plane function in explicitly .
Let's start with it:
Common Plane Equations
Typical options for the arrangement of planes in a rectangular coordinate system are discussed in detail at the very beginning of the article. Plane equation. Nevertheless, once again we will dwell on equations that are of great importance for practice.
First of all, you have to fully recognize the equations of planes that are parallel to the 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), while it is desirable that the point at which the coordinate axis “pierces” the plane is the center of symmetry: 
Strictly speaking, the coordinate axes in some places should have been depicted with a dotted line, but in order to avoid confusion, we will neglect this nuance.
– (left drawing) the inequality defines the half-space farthest from us, excluding the plane itself;
– (medium drawing) the inequality defines the right half-space, including the plane ;
– (right drawing) a double inequality specifies a "layer" located between the planes , including both planes.
For self workout:
Example 1
Draw a body bounded by planes
Compose a system of inequalities that define the given body.
An old acquaintance should come out 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, DO NOT NEGLECT learning tasks, even if they seem too simple. Otherwise, it may turn out that they missed it once, missed it twice, and then spent an hour grinding out a three-dimensional drawing in some real example. In addition, mechanical work will help to learn the material much more efficiently and develop intelligence! It is no coincidence that in kindergarten and elementary school, children are loaded with drawing, modeling, designers and other tasks for fine motor skills of fingers. Forgive the digression, but my two notebooks on developmental psychology should not disappear =)
We will conditionally call the following group of planes “direct proportions” - these are planes passing through the coordinate axes:
2) the equation of the form defines a plane passing through the axis;
3) the equation of the form defines a plane passing through the axis.
Although the formal sign is obvious (which variable is missing in the equation - the plane passes through that axis), it is always useful to understand the essence of the events taking place:
Example 2
Build Plane
What is the best way to build? I propose the following algorithm:
First, we rewrite the equation in the form , from which it is clearly seen that the “y” can take any values. We fix the value , that is, we will consider the coordinate plane . The equations set spatial line lying in the given coordinate plane. Let's draw this line on the drawing. The line passes through the origin, so to construct it, it is enough to find one point. Let . Set aside a point and draw a line.
Now back to the equation of the plane. Since the "y" takes any values, then the straight line constructed in the plane is continuously “replicated” to the left and to the right. This is how our plane is formed, passing through the axis. To complete the drawing, to the left and to the right of the straight line we set aside two parallel lines and “close” the symbolic parallelogram with transverse horizontal segments: 
Since the condition did not impose additional restrictions, the fragment of the plane could be depicted slightly smaller or slightly larger.
Once again, we repeat the meaning of the spatial linear inequality using the example. How to determine the half-space that it defines? Let's take a point not owned plane, for example, a point from the half-space closest to us and substitute its coordinates into the inequality:
Received correct inequality, which means that the inequality defines the lower (with respect to the plane ) half-space, while the plane itself is not included in the solution.
Example 3
Build Planes
a) ;
b) .
These are tasks for self-construction, in case of difficulty, use similar reasoning. Brief instructions and drawings at the end of the lesson.
In practice, planes parallel to the axis are especially common. A special case, when the plane passes through the axis, was just in paragraph "b", and now we will analyze a more general problem:
Example 4
Build Plane
Solution: the variable "z" does not explicitly participate 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 plane equation in the form 
from which it is clear that "Z" can take any values. Let's fix it and in the "native" plane draw the usual "flat" straight line. To build it, it is convenient to take reference points.
Since "Z" takes all values, then the constructed straight line continuously "multiplies" up and down, thereby forming the desired plane 
. Carefully draw up a parallelogram of reasonable size: 
Ready.
Equation of a plane in segments
The most important applied variety. If a all odds general equation of the plane different from zero, then it can be represented as 
, which is called plane equation in segments. Obviously, the plane intersects the coordinate axes at points , and the great advantage of such an equation is the ease of drawing:
Example 5
Build Plane
Solution: first, we compose the equation of the plane in segments. Throw the free term to the right and divide both parts by 12: 
No, this is not a typo and all things happen in space! We examine the proposed surface by the same method that was recently used for planes. We rewrite the equation in the form 
, from which it follows that "Z" takes any values. We fix and construct an ellipse in the plane. Since "Z" takes all values, then the constructed ellipse is continuously "replicated" up and down. It is easy to understand that the surface endless:
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 generating cylinder (which literally form it). 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 defines the "inside" of the infinite "pipe", including the cylindrical surface itself, and, accordingly, the opposite inequality defines the set of points outside the cylinder.
In practical problems, the most popular case is when guide cylinder is circle:
Example 8
Construct the surface given by the equation
It is impossible to depict an endless “pipe”, therefore art is limited, as a rule, to “cutting”.
First, it is convenient to build a circle of radius in the plane, and then a couple more circles above and below. The resulting circles ( guides cylinder) neatly connected by four parallel straight lines ( generating cylinder): 
Don't forget to use dotted lines for invisible lines.
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 outer part. For a better understanding, I recommend to consider several specific points in space and see for yourself.
Example 9
Construct a surface and find its projection onto a plane
We rewrite the equation in the form 
from which it follows that "x" takes any values. Let us fix and draw in the plane circle– centered at the origin, unit radius. Since "x" continuously takes 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 ( generating cylinder). In some places, overlays turned out, but what to do, such a slope: 
This time I limited myself to a piece of the 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, it turned out 6 generatrices - two additional straight lines "close" the surface from the upper left and lower right corners.
Now let's deal with the projection of the cylinder onto the plane. Many readers understand what a projection is, but, nevertheless, let's spend another five-minute physical education. Please stand up and tilt your head over the drawing so that the tip of the axis looks perpendicular to your forehead. What the cylinder looks like from this angle is its projection onto the plane. But it seems to be an endless strip, enclosed between straight lines, including the straight lines themselves. This projection is exactly domain functions (upper "gutter" of the cylinder), (lower "gutter").
By the way, let's clarify the situation with projections onto other coordinate planes. Let the rays of the sun shine on the cylinder from the side of 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 on the plane is somewhat different. If you look at the cylinder from the tip of the axis, then it is projected into a circle of unit radius 
with which we started the construction.
Example 10
Construct a surface and find its projections on coordinate planes
This is a task to solve on your own. If the condition is not very clear, square both sides and analyze the result; find out exactly what part of the cylinder the function specifies. Use the construction technique that has been repeatedly used above. Brief 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:
(on the familiar grounds of an 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 come across quite rarely, and it is quite unbelievable to meet a cylindrical surface “oblique” with respect 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 on the coordinate planes.
Couldn't resist this example =)
Solution: We follow the beaten path. Let's rewrite the equation in the form , from which it follows that "Z" can take any value. Let us fix and construct an ordinary parabola on the plane , having previously marked the trivial reference points . Since "Z" takes all values, then the constructed parabola is continuously "replicated" up and down to infinity. We set aside the same parabola, say, at a height (in the plane) and carefully connect them with parallel lines ( generators of the cylinder): 
I remind useful technique: if initially there is no confidence in the quality of the drawing, then it is better to first draw the lines thinly and 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 we 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 can't talk about domains of a function of two variables- for the reason that the equation of the cylinder is not reducible to the functional form .
2) The projection of the cylinder onto the 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
Construct parabolic cylinders:
a) , restrict ourselves to a fragment of the surface in the near half-space;
b) in between
In case of difficulties, we are not in a hurry and argue by analogy with the previous examples, fortunately, the technology has been thoroughly worked out. It is not critical if the surfaces turn out to be a little clumsy - it is important to correctly display the fundamental picture. I myself don’t particularly bother with the beauty of the lines, if I get a tolerable “C grade” drawing, I usually don’t redo it. In the sample solution, by the way, one more technique was used to improve the quality of the drawing ;-)
Hyperbolic cylinders
guides such cylinders are hyperbolas. This type of surface, according to my observations, is much rarer than the 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 surfaces of the 2nd order, and now we will continue to get acquainted with other representatives of this group:
Ellipsoid. Sphere and ball
The canonical equation of an ellipsoid in a rectangular coordinate system has the form 
, where are positive numbers ( axle shafts ellipsoid), which in the general case different. An ellipsoid is called surface, and body bounded by this surface. The body, as many have guessed, is given by the inequality 
and the coordinates of any interior point (as well as any surface point) necessarily satisfy this inequality. The design is symmetrical with respect to the coordinate axes and coordinate planes: 
The origin of the term "ellipsoid" is also obvious: if the surface is "cut" by coordinate planes, then three different (in the general case)
A cylindrical surface is a surface composed of all lines intersecting a given line L and parallel to a given line I. In this case, the line L is called the guide of the cylindrical surface, and each of the lines that make up this surface and parallel to the line is called a generatrix (Fig. 89). In the future, we will consider only such cylindrical surfaces, the guides of which lie in one of the coordinate planes, and the generators are parallel to the coordinate axis perpendicular to this plane.
Let us consider some line L in the Oxy plane, which has the equation in the Oxy coordinate system
Let's build a cylindrical surface with generators parallel to the Oz axis and the guide L (Fig. 90). Let us show that equation (39) will be the equation of this surface if it is related to the coordinate system in space . Let be any fixed point of the constructed cylindrical surface.
Denote by N the point of intersection of the guide L and the generatrix passing through the point M. The point will obviously be the projection of the point M onto the plane. Therefore, the points M and N have the same abscissa and the same ordinate y. But the point N lies on the curve L, and its x and y coordinates satisfy the equation (39) of this curve. Therefore, the coordinates of the point also satisfy this equation, since it does not contain . Thus, the coordinates of any point of this cylindrical surface satisfy equation (39). The coordinates of points that do not lie on this surface do not satisfy equation (39), since these points are projected onto a plane outside the curve
Thus, not containing the equation, if it is referred to the coordinate system in space, is the equation of a cylindrical surface with generators parallel to the axis and the guide L, which in the plane is given by the same equation
In space, the guide L is determined by a system of two equations:
Similarly, it can be shown that the equation not containing y and the equation not containing define cylindrical surfaces in the Oxy space with generatrixes parallel to the axes, respectively.
Consider examples of cylindrical surfaces.
1. Surface defined by the equation
is cylindrical and is called an elliptical cylinder (Fig. 91).
Its generators are parallel to the axis and the guide is an ellipse with semi-axes a and b, lying in the plane. In particular, if then the guide is a circle and the surface is a right circular cylinder. His equation
2. Cylindrical surface defined by the equation
is called a hyperbolic cylinder (Fig. 92). The generators of this surface are parallel to the axis and the hyperbola located in the plane with the real semi-axis a and the imaginary semi-axis b serves as a guide.
3. Cylindrical surface defined by the equation
is called a parabolic cylinder (Fig. 93). Its guide is a parabola lying in the plane , and the generators are parallel to the Ox axis.
Comment. As is known, a straight line in space can be given by the equations of various pairs of planes intersecting along this straight line. Similarly, a curve in space can be defined using the equations of various surfaces intersecting along this curve.
CYLINDRICAL SURFACES
| Parameter name | Meaning |
| Article subject: | CYLINDRICAL SURFACES |
| Rubric (thematic category) | Maths |
SURFACES
Let G be a line and - a non-zero vector not parallel to the plane of the line Г (if Г is a flat line.
Definition 10. Cylindrical surface with guide Г and generators parallel to the vector , it is customary to call the set of points of all possible lines parallel to the vector and crossing the line G.
The main problem to be solved: how to find the equation of a cylindrical surface, if the equations of the line Г and the coordinates of the vector are given .
(28)
It remains to exclude the parameter t from these equations.
We have obtained the following rules for compiling the equation of a cylindrical surface:
If the direction of the cylindrical surface is given by equations (27) and the generators are parallel to the vector , then to compose the equation of the surface, it is enough in equations (27) to replace x with x - mt, y with y - nt, z with z - pt and exclude the parameter from the resulting equations.
Example 1 Write an equation for a cylindrical surface, if the generators are parallel to the vector = (3, 2, -1) and the guide G has the equations 
Example 2. Write an equation for a cylindrical surface if the guide is a line lying in the plane (HOY), and the generators are parallel to the axis (ОZ).
Solution. A vector parallel to the generators is a vector. We replace x in the equations of the guide with x - 0‣‣‣t, ᴛ.ᴇ. x is replaced by x. Similarly, y is replaced by y. But z is replaced by z - t. We obtain From the second equation z = t. This means that z can take on all possible real values regardless of x and y, and x and y are related by the same equation f(x, y) = 0 as in the equation of the guide. The equation of a cylindrical surface in this case will be f(x, y) = 0.
Consequence. Equations , , y 2 = 2px define cylindrical surfaces with guides ellipse, hyperbola and parabola, respectively. Their generators are parallel to the axis (ОZ).
If the guide of the cylindrical surface is a line of the second order, then the surface is usually called second order cylinder.
Comment. Pay attention to the fact that the equations f(x, y) = 0, f(x, z) = 0, f(y, z) = 0, define on the planes (XOY), (XOZ) and (YOZ), respectively, some lines. But in an affine coordinate system in space, they define cylinders with generators parallel to the axis (ОZ), (ОУ) and (ОХ), respectively.
CYLINDRICAL SURFACES - concept and types. Classification and features of the category "CYLINDRICAL SURFACES" 2017, 2018.
Lesson number 10.
Topic:Surfaces of revolution.
Cylindrical surfaces
Theoretical information.
1. Surfaces of revolution.
Limit. A surface of revolution is a surface formed by the rotation of a plane line around an axis lying in the plane of this line.
Let 
, then it can be given by the equations
The equation of the surface formed by the rotation of the line around the axis Oz will look like:
(1)
2. Cylindrical surfaces.
Let some flat line be given in space and a vector 
not parallel to the plane of this line.
Definition. A cylindrical surface is a set of points in space that lie on straight lines parallel to a given vector and intersect a given line .
The line is called the guide of the cylindrical surface, the straight lines are called generators.
Consider a special case: the guide line lies in the plane xOy: and is given by the equations: 
and the direction vector of generators has coordinates 
,
.
In this case, the equation of a cylindrical surface has the form
. (2)
Get the equation of the surface of revolution (1).
Get the equation of a cylindrical surface (2).
Compilation of the equation of the surface of revolution according to the equations of the guide and the axis of revolution.
Compilation of the equation of a cylindrical surface according to the equations of the guide and the guide vector of generators.
Exercises.
Basic typical tasks.
Examples of problem solving.
Task 1. In plane yOz given a circle centered at the point (0; 4; 0) of radius 1. Write the equation for the surface formed by the rotation of this circle around the axis Oz.
Yeshenie.
Equations of a circle lying in a plane yOz centered at the point (0; 4; 0) of radius 1, have the form
(3)
When this circle rotates around the Oz axis, a surface is obtained, called a torus. Let M is an arbitrary point on the torus. Let's pass through the point M plane , perpendicular to the axis of rotation, i.e. axes Oz, in the section we get a circle. Denote the center of this circle P, and the point of intersection of the plane with the circle forming the surface of revolution is N.
Denote the coordinates of the point M(x,
y,
z), then P(0,
0, z), while N(0, 
,
z). Since the points M and N lie on the circle centered at the point P, then
,
.
We write the last equality in coordinates
. (4)
The point N lies on a circle, during the rotation of which a torus is formed, which means that its coordinates must satisfy equations (3), we write the first equation of system (3)
,
,
.
Let's square the last equation.
and substitute the expression for 
from equality (4), we obtain
Equation (5) is the required one.
Task 2. Write an equation for a cylindrical surface if the guide lies in a plane xOy and has the equation 
, and the generators are parallel to the vector (1; 2; –1).
Let the point M(x,
y,
z) is an arbitrary point of the cylindrical surface. Let's pass through the point M generating l, it intersects the guide at the point 
. Since the guide lies in the plane xOy, then 
. Compose the canonical equations of the straight line l
.
Equate the first and second fractions to the last
(6)
Point N lies on the guide, so its coordinates satisfy its equation:
.
Substituting expressions for 
and from system (6), we obtain
. (7)
(7) is the required equation.
a) an ellipse 
;
b) hyperbolas 
;
c) parabolas 
.
a) The guide lies in the plane 
and has the equation , and the generators are parallel to the vector (1; 0; 1);
b) the guide lies in the plane yOz and has the equation 
, and the generators are parallel to the axis Ox;
c) the guide lies in the xOz plane and is a circle 
, and the generators are parallel to the Oy axis.
Write the equation for a cylindrical surface if:
a) the guide is given by the equations 
and the generatrix is parallel to the vector 
;
b) the guide is given by the equations 
and the generatrix is parallel to the line x=
y=
z.
a) 
,
,
,
M(2; 0; 1);
b) l:
,
M(2; –1; 1).
Lesson number 11.
Topic:conical surfaces.
Theoretical information.
Let some flat line and a point be given in space S not lying in the plane of this line.
Definition. A conical surface is a set of points in space that lie on lines passing through a given point. S and intersecting this line .
The line is called the guide of the conical surface, the point S- a vertex, the lines are called generators.
Consider a special case: the vertex S coincides with the origin, the guide line lies in a plane parallel to the plane xOy:
z=
c, and is given by the equation: 
.
In this case, the equation of the conical surface has the form
. (1)
If the guide is an ellipse centered on the axis Oz,
then we get a surface called a cone of the second order, the equation of this surface has the form:
. (2)
Axis Oz in this case is the axis of the cone of the second order.
Sections of a cone of the second order:
Let the plane not pass through the vertex of the cone of the second order, then the plane intersects the cone:
a) along an ellipse, if intersects all generators of the cone;
b) by hyperbola, if is parallel to two generators of the cone;
c) along a parabola, if is parallel to one generatrix of the cone.
Exercises.
Get the equation of the conical surface (1).
Obtain the second order conic surface equation (2).
Basic typical tasks.
Compilation of the equation of a conical surface by the coordinates of the vertex and the equation of the guide.
Examples of problem solving.
Task 1. Write an equation for a conical surface whose vertex is at the origin and whose guideline is given by the equations 
Let the point M(x,
y,
z) is an arbitrary point of the conical surface. Let us draw a generatrix through this point l, it will intersect the guide at the point 
. We write the canonical equations of the straight line l, as the equation of a straight line passing through a point N and the vertex of the cone O(0, 0, 0)
,
.
Let us express from the last system and : 
,
. Because dot N lies on the guide conical surface, then its coordinates must satisfy the equations of the guide:
(3)
Let us substitute the found expressions into the second equation of system (3)
,
,
,
. (4)
,
. (5)
We substitute (4) and (5) into the first equation of system (3)
,
.
The resulting equation is the desired equation of the conical surface.; Linear dependence vectors. Coordinate system. Orthonormal basis. Linear operations above vectors in coordinates. scalar work vectors. vector work vectors ...
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