The student most often encounters surfaces of the 2nd order in the first year. At first, tasks on this topic may seem simple, but as you study higher mathematics and deepen into the scientific side, you can finally stop orienting yourself in what is happening. In order to prevent this from happening, it is necessary not only to memorize, but to understand how this or that surface is obtained, how changing the coefficients affects it and its location relative to the original coordinate system, and how to find a new system (one in which its center coincides with the origin coordinates, and the symmetry axis is parallel to one of the coordinate axes). Let's start from the beginning.
Definition
GMT is called a 2nd order surface, the coordinates of which satisfy the general equation of the following form:
F(x, y, z)=0.
It is clear that each point belonging to the surface must have three coordinates in some designated basis. Although in some cases the locus of points may degenerate, for example, into a plane. It only means that one of the coordinates is constant and equals zero in the entire range of acceptable values.
The full painted form of the equality mentioned above looks like this:
A11x2+A22y2 +A33z2+2A12xy+2A23 yz+2A13xz+2A14x+2A24y+2A 34z+A44=0.
Anm – some constants, x, y, z – variables corresponding to affine coordinates of some 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 greatly 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 is observed). The key point in such work is to bring the latter to a canonical form.
The equation written above defines any (all listed below) surfaces of the 2nd order. We will consider examples below.
Types of surfaces of the 2nd order
Equations of surfaces of the 2nd order differ only in the values of the coefficients Anm. From the general view, for 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 simpler figure, or is absent altogether.
Cylinders
This is the simplest type, since a relatively complex curve lies only at the base, acting as a guide. The generators are straight lines perpendicular to the plane in which the base lies.
The graph shows a circular cylinder, a special case of an 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 get it from the general equation, you need to give the coefficients the following values:
Instead of the usual symbols x, y, z, x with a serial number is used - it does not matter.
In fact, 1/a2and the other constants indicated here are the same coefficients indicated in the general equation, but it is customary to write them in this form - this is the canonical representation. Further, only such a notation will be used.
This is how a hyperbolic cylinder is defined. The scheme is the same - the hyperbole will be the guide.
y2=2px
A parabolic cylinder is defined somewhat differently: its canonical form includes a coefficient p, called a parameter. In fact, the coefficient is equal to 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 equationelliptical cylinder, but instead of unit is -1.
Elliptical type
An ellipsoid can be stretched along one of the axes (along which it depends on the values of the constants a, b, c, indicated above; it is obvious that a larger coefficient will correspond to the larger axis).
There is also an imaginary ellipsoid - provided that the sum of the coordinates multiplied by the coefficients is -1:
Hyperboloids
When a minus appears in one of the constants, the ellipsoid equation turns into the equation of a single-sheeted hyperboloid. It must be understood that this minus does not have to be located before 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.
Besides, you need to understand that the equations are presented in canonical form and they can be changed by varying the constants (with the sign preserved!); while their form (hyperboloid, cone, and so on) will remain the same.
This equation is already given by a two-sheeted hyperboloid.
Conical surface
There is no unit in the cone equation - equality to zero.
Only a bounded conical surface is called a cone. The picture below shows that, in fact, there will be two so-called cones on the chart.
Important note: in all considered canonical equations, the constants are taken positive by default. Otherwise, the sign may affect the final chart.
The coordinate planes become the planes of symmetry of the cone, the center of symmetry is located at the origin.
There are only pluses in the imaginary cone equation; it owns one single real point.
Paraboloids
Surfaces of 2nd order in space can take different shapes even with similar equations. For example, there are two types of paraboloids.
x2/a2+y2/b2=2z
An elliptical paraboloid, when the Z axis is perpendicular to the drawing, will be projected into an ellipse.
x2/a2-y2/b2=2z
Hyperbolic paraboloid: sections with planes parallel to ZY will produce parabolas, and sections with planes parallel to XY will produce hyperbolas.
Intersecting planes
There are cases when surfaces of the 2nd order degenerate into a plane. These planes can be arranged in various ways.
First consider the intersecting planes:
x2/a2-y2/b2=0
This modification of the canonical equation results in just two intersecting planes (imaginary!); all real points are on the axis of the coordinate that is missing in the equation (in the canonical - the Z axis).
Parallel planes
y2=a2
When there is only one coordinate, the surfaces of the 2nd order degenerate into a pair of parallel planes. Remember, any other variable can take the place of Y; then planes parallel to other axes will be obtained.
y2=−a2
In this case, they become imaginary.
Coinciding planes
y2=0
With such a simple equation, a pair of planes degenerate into one - they coincide.
Don't forget that in the case of a three-dimensional basis, the above equation does not define the straight line y=0! It lacks the other two variables, but that just means that their value is constant and equal to zero.
Building
One of the most difficult tasks for a student is the construction of surfaces of the 2nd order. It is even more difficult to move from one coordinate system to another, given the angles of inclination of the curve relative to the axes and the offset of the center. Let's repeat how to consistently determine the future view of the drawing with an analyticalway.
To build a 2nd order surface, you need:
- bring the equation to canonical form;
- determine the type of surface under study;
- construct based on coefficient values.
Below are all the types considered:
To consolidate, let's describe in detail one example of this type of task.
Examples
Suppose there is an equation:
3(x2-2x+1)+6y2+2z2+ 60y+144=0
Let's bring it to the canonical form. Let us single out the full squares, that is, we arrange the available terms in such a way that they are the expansion of the square of the sum or difference. For example: if (a+1)2=a2+2a+1 then a2+2a +1=(a+1)2. We will carry out the second operation. In this case, it is not necessary to open the brackets, since this will only complicate the calculations, but it is necessary to take out the common factor 6 (in brackets with the full square of the Y):
3(x-1)2+6(y+5)2+2z2=6
The variable z occurs in this case only once - you can leave it alone for now.
We analyze the equation at this stage: all unknowns are preceded by a plus sign; when divided by six, one remains. Therefore, we have an equation that defines an ellipsoid.
Note that 144 was factored into 150-6, after which the -6 was moved to the right. Why did it have to be done this way? Obviously, the largest divisor in this example is -6, so that after dividing by itone is left on the right, it is necessary to “postpone” exactly 6 from 144 (the fact that one should be on the right is indicated by the presence of a free term - a constant not multiplied by an unknown).
Divide everything by six and get the canonical equation of the ellipsoid:
(x-1)2/2+(y+5)2/1+z2 /3=1
In the classification of surfaces of the 2nd order used earlier, a particular 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 parenthesis 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).
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 two. The major axis parallel to the x-axis is two roots of two. The minor axis of the second ellipse, parallel to the Y axis, remains the same - it is equal to two. And the major axis, parallel to the Z axis, is equal to two roots of three.
With the help of the data obtained from the original equation, by converting to the canonical form of the data, we can draw an ellipsoid.
Summing up
Covered in this articlethe topic is quite extensive, but, in fact, as you can now see, not very complicated. Its development, in fact, ends at the moment when you memorize the names and equations of surfaces (and, of course, how they look). In the example above, we have discussed each step in detail, but bringing the equation to the canonical form requires minimal knowledge of higher mathematics and should not cause any difficulties for the student.
Analysis of the future schedule on the existing equality is already a more difficult task. But for its successful solution, it is enough to understand how the corresponding second-order curves are built - ellipses, parabolas, and others.
Degeneracy cases - 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 the constants, turning the graph into one or another figure, the topic will be mastered.
Success in your studies!
