For two planes, the following variants of mutual arrangement are possible: they are parallel or intersect in a straight line.
It is known from stereometry that two planes are parallel if two intersecting lines of one plane are respectively parallel to two intersecting lines of another plane. This condition is called a sign of parallel planes.
If two planes are parallel, then they intersect some third plane along parallel lines. Based on this, parallel planes R And Q their traces are parallel straight lines (Fig. 50).
When two planes R And Q parallel to the axis X, their horizontal and frontal traces with an arbitrary mutual arrangement of the planes will be parallel to the x axis, i.e., mutually parallel. Consequently, under such conditions, the parallelism of traces is a sufficient sign characterizing the parallelism of the planes themselves. For the parallelism of such planes, you need to make sure that their profile traces are also parallel. P w and Q w. planes R And Q in figure 51 are parallel, and in figure 52 they are not parallel, despite the fact that P v || Q v , and P h y || Q h .
In the case when the planes are parallel, the horizontals of one plane are parallel to the horizontals of the other. The fronts of one plane must be parallel to the fronts of the other, since these planes have parallel traces of the same name.
In order to construct two planes intersecting with each other, it is necessary to find the line along which the two planes intersect. To construct this line, it is enough to find two points belonging to it.
Sometimes, when the plane is given by traces, it is easy to find these points using a diagram and without additional constructions. Here, the direction of the defined straight line is known, and its construction is based on the use of one point on the plot.
End of work -
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Descriptive geometry. Lecture notes lecture. About projections
Lecture information about projections the concept of projections reading a drawing .. central projection .. an idea of \u200b\u200bthe central projection can be obtained by studying the image that the human eye gives ..
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All topics in this section:
The concept of projections
Descriptive geometry is a science that is the theoretical foundation of drawing. In this science, methods of depicting various bodies and their elements on a plane are studied.
Parallel projection
Parallel projection is a type of projection that uses parallel projecting rays. When constructing parallel projections, you need to set on
Projections of a point onto two projection planes
Consider the projections of points onto two planes, for which we take two perpendicular planes (Fig. 4), which we will call the horizontal frontal and planes. Flat data intersection line
Missing projection axis
To explain how to obtain on the model projections of a point onto perpendicular projection planes (Fig. 4), it is necessary to take a piece of thick paper in the form of an elongated rectangle. It needs to be bent between
Projections of a point onto three projection planes
Consider the profile plane of projections. Projections on two perpendicular planes usually determine the position of the figure and make it possible to find out its real dimensions and shape. But there are times when
Point coordinates
The position of a point in space can be determined using three numbers, called its coordinates. Each coordinate corresponds to the distance of a point from some plane pr
Projection of a straight line
Two points are needed to define a line. A point is defined by two projections on the horizontal and frontal planes, i.e. a straight line is defined using the projections of its two points on the horizontal
Straight traces
The trace of a straight line is the point of its intersection with some plane or surface (Fig. 20). The horizontal trace of a line is a point H
Various positions of the line
A straight line is called a straight line general position, if it is neither parallel nor perpendicular to any of the projection planes. The projections of a line in general position are also neither parallel nor perpendicular.
Mutual arrangement of two straight lines
Three cases of arrangement of lines in space are possible: 1) the lines intersect, that is, they have a common point; 2) the lines are parallel, that is, they do not have a common point, but lie in the same plane
Perpendicular lines
Consider the theorem: if one side right angle parallel to the plane of projections (or lies in it), then the right angle is projected onto this plane without distortion. We present a proof for
Determination of the position of the plane
For an arbitrarily located projection plane, its points fill all three projection planes. Therefore, it makes no sense to talk about the projection of the entire plane, you need to consider only projections
Plane traces
The trace of the plane P is the line of its intersection with a given plane or surface (Fig. 36). The line of intersection of the plane P with the horizontal plane is called
Plane contours and fronts
Among the lines that lie in a certain plane, two classes of lines can be distinguished, which play an important role in solving various problems. These are straight lines, which are called horizontals.
Construction of plane traces
Consider the construction of traces of the plane P, which is given by a pair of intersecting lines I and II (Fig. 45). If a line is in the plane P, then its traces lie on the traces of the same name
Various positions of the plane
A plane in general position is a plane that is neither parallel nor perpendicular to any of the projection planes. The traces of such a plane are also neither parallel nor perpendicular.
Straight line parallel to the plane
There can be several positions of a straight line relative to a certain plane. 1. The line lies in some plane. 2. A line is parallel to some plane. 3. Direct transfer
A straight line that intersects a plane
To find the point of intersection of a line and a plane, it is necessary to construct lines of intersection of two planes. Consider the line I and the plane P (Fig. 54).
Prism and pyramid
Consider a straight prism that stands on a horizontal plane (Fig. 56). Her side grains
Cylinder and cone
A cylinder is a figure whose surface is obtained by rotating the straight line m around the i-axis, located in the same plane with this straight line. In the case when the line m
Ball, torus and ring
When some axis of rotation I is the diameter of a circle, then a spherical surface is obtained (Fig. 66).
Lines used in drawing
Three main types of lines (solid, dashed and dash-dotted) of various thicknesses are used in drawing (Fig. 76).
Location of views (projections)
In drawing, six types are used, which are shown in Figure 85. The figure shows the projections of the letter "L".
Deviation from the above rules for the arrangement of views
In some cases, deviations from the rules for constructing projections are allowed. Among these cases, the following can be distinguished: partial views and views located without a projection connection with other views.
Number of projections that define this body
The position of bodies in space, shape and size are usually determined by a small number of appropriately selected points. If, when depicting a projection of a body, pay attention
Rotation of a point about an axis perpendicular to the plane of projections
Figure 91 shows the axis of rotation I, which is perpendicular to the horizontal plane, and a point A arbitrarily located in space. When rotating about the axis I, this point describes
Determination of the natural length of a segment by rotation
A segment parallel to any projection plane is projected onto it without distortion. If you rotate the segment so that it becomes parallel to one of the projection planes, then you can define
The construction of projections of the section figure can be done in two ways
1. You can find the meeting points of the edges of the polyhedron with the cutting plane, and then connect the projections of the found points. As a result of this, projections of the desired polygon will be obtained. In this case,
Pyramid
Figure 98 shows the intersection of the pyramid surface with the frontal projection plane P. Figure 98b shows the frontal projection a of the meeting point of the rib KS with the plane
oblique sections
Oblique sections are understood as a range of tasks for constructing natural types of sections of the body under consideration by the projected plane. To perform an oblique section, it is necessary to dismember
Hyperbola as a section of the surface of a cone by the frontal plane
Let it be required to construct a section of the surface of a cone standing on a horizontal plane by the plane P, which is parallel to the plane V. Figure 103 shows the frontal
Section of the surface of the cylinder
There are the following cases of a section of the surface of a right circular cylinder by a plane: 1) a circle, if the secant plane P is perpendicular to the axis of the cylinder, and it is parallel to the bases
Section of the surface of the cone
In the general case, a circular conical surface includes two completely identical cavities that have a common vertex (Fig. 107c). The generators of one cavity are a continuation of the
Section of the surface of the ball
Any section of the surface of the ball by a plane is a circle, which is projected without distortion only if the cutting plane is parallel to the plane of projections. In the general case, we
oblique sections
Let it be required to construct a natural view of the section by the frontally projecting plane of the body. Figure 110a considers a body bounded by three cylindrical surfaces (1, 3 and 6), the surface
Pyramid
To find traces of a line on the surface of some geometric body, you need to draw through a straight auxiliary plane, then find the section of the surface of the body by this plane. The desired will be
Cylindrical helix
Formation of a helix. Consider Figure 113a where the point M moves uniformly along a certain circle, which is a section of a circular cylinder by the plane P. Here this plane
Two bodies of revolution
The method of drawing auxiliary planes is used when constructing a line of intersection of the surfaces of two bodies of revolution. The essence of this method is as follows. Carry out an auxiliary plane
Sections
There are some definitions and rules that apply to sections. The section is flat figure, which was obtained as a result of the intersection of this body with some
cuts
Definitions and rules that apply to cuts. A cut is such a conditional image of an object when its part, located between the eye of the observer and the cutting plane
Partial cut or tear
The cut is called complete if the depicted object is cut in its entirety, the remaining cuts are called partial, or cuts. In Figure 120, in the left view and on the plan, full sections are made. hairstyle
Plane, line, point - the basic concepts of geometry. It is difficult for us to give them clear definitions, but intuitively we understand what they are. The plane has only two dimensions. She has no depth. A straight line has only one dimension, and a point has no dimensions at all - no length, no width, no height.
The plane is infinite. Therefore, in tasks we draw only part of the plane. You have to portray it somehow.
And what does all this look like in space? Very simple. A sheet of thick paper will serve as a "model" of the plane. You can take another flat object, for example, a CD, a plastic card. Pencils may well depict straight lines. All axioms and theorems of stereometry can be shown "on the fingers", that is, with the help of improvised materials. Read - and immediately build such a "model".
Two planes in space are either parallel or intersect. Examples in the surrounding area are easy to find.
If two planes have a common point, then they intersect in a straight line.
We do not consider separately the case of "planes coincide". Since they coincide, it means that this is one plane, not two.
Angle between planes
Let the planes and be given by the equations and , respectively. It is required to find the angle between these planes.
The planes, intersecting, form four dihedral angles (Fig. 11.6): two obtuse and two acute or four straight lines, both obtuse angles being equal to each other, and both acute angles are also equal to each other. We will always look for an acute angle. To determine its value, we take a point on the line of intersection of the planes and at this point in each of the planes we draw perpendiculars to the line of intersection. We also draw normal vectors and planes and with origins at a point (Fig. 11.6).
Fig. 11.6. Angle between planes
If a plane is drawn through a point perpendicular to the line of intersection of the planes and , then the lines and and images of the vectors and will lie in this plane. Let's make a drawing in a plane (two options are possible: Fig. 11.7 and 11.8).
Fig. 11.7. The angle between normal vectors is acute
Fig.11.8. The angle between normal vectors is obtuse
In one version (Fig. 11.7) and , therefore, the angle between the normal vectors is equal to the angle , which is the linear angle of the acute dihedral angle between the planes and .
In the second variant (Fig. 11.8) , and the angle between the normal vectors is . Because
then in both cases 
.
By definition of the scalar product 
. Where
and correspondingly
If the planes are parallel, then their normal vectors are collinear. We get the condition of parallel planes
| (11.6) |
where is any number.
23.Different types of equations of a straight line in space Vector-parametric equation of a straight line where  - a fixed point lying on a straight line; - direction vector. In coordinates (parametric equations): Equations of a straight line on two points
 24. Various types of equations of a straight line in space Canonical equations of the straight line
 Parametric equations straight we obtain by equating each of the relations (3.4) with the parameter t: x = x 1 + mt , y = y 1 + nt , z = z 1 + р t . 25. Mutual position of lines Two lines in space can intersect, intersect and can be parallel. 1. Intersecting lines
Intersecting lines are those lines that have one common point. It follows from invariant property 5 that the projection of the intersection point of the projections of the lines a and b is the intersection point of these lines (Fig. 3.4).  . Rice. 3.4. intersecting lines 2. Parallel lines
On fig. 3.5 shows parallel lines - lines intersecting at an improper point (lines lying in the same plane and intersecting at a point at infinity). It follows from invariant property 6 that the projections of parallel lines a and b are parallel. 3. Crossed lines
Crossing lines are lines that do not lie in the same plane; they are lines that do not have a single common point. In the complex drawing (Fig. 3.6), the intersection points of the projections of these lines do not lie on the same perpendicular to the X axis (unlike intersecting lines, see Fig. 3.4).  . Rice. 3.5. Image of parallel lines  . Rice. 3.6. Intersecting lines The distance from a point to a line is equal to the length of the perpendicular dropped from the point to the line. If the line is parallel to the projection plane (h | | P 1), then in order to determine the distance from point A to line h, it is necessary to lower the perpendicular from point A to the horizontal h.
|
Two planes in space can either be mutually parallel, in a particular case coinciding with each other, or intersect. Mutually perpendicular planes are special case intersecting planes.
1. Parallel planes. Planes are parallel if two intersecting lines of one plane are respectively parallel to two intersecting lines of another plane.
This definition is well illustrated by the task, through point B, to draw a plane parallel to the plane given by two intersecting straight lines ab (Fig. 61).
A task. Given: a plane in general position given by two intersecting straight lines ab and point B.
It is required through point B to draw a plane parallel to the plane ab and set it by two intersecting lines c and d.
According to the definition, if two intersecting lines of one plane are respectively parallel to two intersecting lines of another plane, then these planes are parallel to each other.
In order to draw parallel lines on the diagram, it is necessary to use the property of parallel projection - the projections of parallel lines are parallel to each other
d//a, с//b Þ d1//a1,с1//b1; d2//a2 ,с2//b2; d3//a3,с3//b3.
Figure 61. Parallel planes
2. Intersecting planes, a special case - mutually perpendicular planes. The line of intersection of two planes is a straight line, for the construction of which it is enough to determine its two points common to both planes, or one point and the direction of the line of intersection of the planes.
Consider the construction of the line of intersection of two planes, when one of them is projecting (Fig. 62).
A task. Given: the plane in general position is given by the triangle ABC, and the second plane is horizontally projecting a.
It is required to construct a line of intersection of planes.
The solution of the problem is to find two points common to these planes through which a straight line can be drawn. The plane defined by the triangle ABC can be represented as straight lines (AB), (AC), (BC). The point of intersection of the line (AB) with the plane a - point D, line (AC) -F. The segment defines the line of intersection of the planes. Since a is a horizontally projecting plane, the projection D1F1 coincides with the trace of the plane aP1, so it remains only to construct the missing projections on P2 and P3.
Figure 62. Intersection of a plane of general position with a horizontally projecting plane
Let's move on to general case. Let two generic planes a(m,n) and b (ABC) be given in space (Fig.63)
Figure 63. Intersection of planes in general position
Consider the sequence of constructing the line of intersection of the planes a(m//n) and b(ABC). By analogy with the previous problem, to find the line of intersection of these planes, we draw auxiliary secant planes g and d. Let us find the lines of intersection of these planes with the planes under consideration. Plane g intersects plane a along a straight line (12), and plane b - along a straight line (34). Point K - the point of intersection of these lines simultaneously belongs to three planes a, b and g, being thus a point belonging to the line of intersection of planes a and b. Plane d intersects planes a and b along lines (56) and (7C), respectively, their intersection point M is located simultaneously in three planes a, b, d and belongs to the straight line of intersection of planes a and b. Thus, two points are found belonging to the line of intersection of planes a and b - a straight line (KM).
Some simplification in constructing the line of intersection of the planes can be achieved if the auxiliary secant planes are drawn through the straight lines that define the plane.
Mutually perpendicular planes. It is known from stereometry that two planes are mutually perpendicular if one of them passes through a perpendicular to the other. Through the point A, you can draw a set of planes perpendicular to the given plane a (f, h). These planes form a bundle of planes in space, the axis of which is the perpendicular dropped from the point A to the plane a. In order to draw a plane perpendicular to the plane given by two intersecting lines hf from point A, it is necessary to draw a straight line n perpendicular to the plane hf from point A (the horizontal projection n is perpendicular to the horizontal projection of the horizontal h, the frontal projection n is perpendicular to the frontal projection of the frontal f). Any plane passing through the line n will be perpendicular to the plane hf, therefore, to set the plane through points A, we draw an arbitrary line m. The plane given by two intersecting straight lines mn will be perpendicular to the plane hf (Fig.64).
Figure 64. Mutually perpendicular planes
MUTUAL POSITION OF TWO PLANES.
| Parameter name | Meaning |
| Article subject: | MUTUAL POSITION OF TWO PLANES. |
| Rubric (thematic category) | Geology |
Two planes in space can be either parallel to each other or intersect.
Parallel planes. In projections with numerical marks, the parallelism of the planes on the plan is the parallelism of their horizontal lines, the equality of the foundations and the coincidence of the directions of incidence of the planes: pl. S || sq. L- h S || h L l S= l L , pad. I. (Fig. 3.11).
In geology, a flat homogeneous body composed of any rock is called a layer. The layer is bounded by two surfaces, the upper of which is called the roof, and the lower one is called the sole. If the layer is considered over a relatively small extent, then the roof and sole are equated to planes, obtaining in space a geometric model of two parallel inclined planes.
Plane S is the roof, and plane L is the bottom of the layer (Fig. 3.12, but). In geology shortest distance between the roof and the sole is called true power (in Fig. 3.12, but true power is indicated by the letter H). In addition to the true thickness, other parameters of the rock layer are also used in geology: vertical thickness - H in, horizontal thickness - L, apparent thickness - H view. vertical power in geology, they call the distance from the roof to the bottom of the layer, measured vertically. Horizontal Power layer is the shortest distance between the roof and the sole, measured in the horizontal direction. Apparent power - the shortest distance between the visible fall of the roof and the sole (the visible fall is called the rectilinear direction on the structural plane, that is, the straight line belonging to the plane). Τᴀᴋᴎᴍ ᴏϬᴩᴀᴈᴏᴍ, the apparent power is always greater than the true power. It should be noted that in horizontal layers, the true thickness, vertical and apparent thickness are the same.
Consider the method of constructing parallel planes S and L, spaced from each other at a given distance (Fig. 3.12, b).
On the plan, intersecting lines m And n the plane S is given. It is necessary to construct a plane L parallel to the plane S and spaced from it at a distance of 12 m (i.e., the true thickness is H = 12 m). Plane L is located under plane S (plane S is the top of the layer, plane L is the base).
1) Plane S is set on the plan by projections of contour lines.
2) On the scale of the foundations, a line of incidence of the plane S is built - u S. Perpendicular to the line u S plot a given distance of 12 m (true layer thickness H). Below the line of incidence of the plane S and parallel to it, a line of incidence of the plane L is drawn - u L. The distance between the lines of incidence of both planes in the horizontal direction is determined, i.e. the horizontal thickness of the layer L.
3) Putting on the plan the horizontal power from the horizontal h S, a horizontal line of the plane L is drawn parallel to it with the same numerical mark h L. Note that if the L plane is located under the S plane, then the horizontal power should be deposited in the direction of the rise of the S plane.
4) Based on the condition of parallelism of two planes, horizontal lines of the plane L are drawn on the plan.
Intersecting planes. A sign of the intersection of two planes is usually the parallelism on the plan of projections of their contour lines. The line of intersection of two planes in this case is determined by the intersection points of two pairs of similar (having the same numerical marks) contour lines (Fig. 3.13): ; . Connecting the obtained points N and M with a straight line m, determine the projection of the required line of intersection. If the plane S (A, B, C) and L(mn) are not given horizontally on the plan, then to construct their intersection line t it is extremely important to build two pairs of contour lines with the same numerical marks, which at the intersection will determine the projections of the points R and F of the desired line t(fig.3.14). Figure 3.15 shows the case when two intersecting
planes S and L, the horizontals are parallel. The line of intersection of such planes will be a horizontal line h. It is worth saying that to find the point A belonging to this line, an arbitrary auxiliary plane T is drawn, which intersects the planes S and L. The plane T intersects the plane S along the straight line but(C 1 D 2), and the plane L - in a straight line b(K 1 L 2).
Point of intersection of lines but And b, belonging respectively to the planes S and L, will be common for these planes: =A. The elevation of point A can be determined by interpolating the lines a And b. It remains to draw a horizontal line through A h 2.9, which is the line of intersection of the S and L planes.
Consider another example (Fig. 3.16) of constructing a line of intersection of an inclined plane S with a vertical plane T. The desired line m is determined by points A and B, where the horizontals h 3 and h 4 planes S intersect the vertical plane T. It can be seen from the drawing that the projection of the intersection line coincides with the projection of the vertical plane: mº T. In solving geological exploration problems, a section of one or a group of planes (surfaces) by a vertical plane is usually called a section. The additional vertical projection of the straight line constructed in the example under consideration m is called the profile of a cut made by the plane T in a given direction.
MUTUAL POSITION OF TWO PLANES. - concept and types. Classification and features of the category "MUTUAL ARRANGEMENT OF TWO PLANES." 2017, 2018.
Angle between two planes. Terms parallelism and perpendicularity two planes:
let two planes Q 1 and Q 2 be given:
A 1 x + B 1 y + C 1 z + D 1 \u003d 0
A 2 x + B 2 y + C 2 z + D 2 =0
An angle between planes is understood as one of the dihedral angles formed by these planes.
If the planes are perpendicular, then their normals are the same, i.e. . But then, i.e.
A 1 A 2 + B 1 B 2 + C 1 C 2 = 0. The resulting equality is condition of perpendicularity of two planes.
If the planes are parallel, then their normals will also be parallel. But then, as you know, the coordinates of the vectors are proportional: 
. That's what it is condition of parallelism of two planes.
Mutual arrangement of lines.
Angle between lines. Conditions for parallelism and perpendicularity of lines.
Half the angle between these lines understand the angle between the direction vectors S 1 and S 2 .
For finding acute angle between the lines L 1 and L 2 the numerator of the right side of the formula should be taken modulo.
If the lines L 1 and L 2 perpendicular, then in this and only this case we have cos =0. therefore, the numerator of the fraction = 0, i.e. 
=0.
If the lines L 1 and L 2 parallel, then their direction vectors S 1 and S 2 are parallel. therefore, the coordinates of these vectors are proportional: .
The condition under which two lines lie in the same plane:
=0.
When this condition is met, the lines either lie in the same plane, that is, they either intersect.
Mutual arrangement of a straight line and a plane.
The angle between a line and a plane. Conditions of parallelism and perpendicularity of a line and a plane.
Let the plane be given by the equation Ax + By + Cz + D=0, and the line L by the equations 
. An angle between a line and a plane is any of two adjacent angles formed by a line and its projection onto the plane. Denote by the angle between the plane and the line.
.
If the line L is parallel to the plane Q, then the vectors n and S are perpendicular, and therefore , i.e.
0 is parallelism condition straight and plane.
If the line L is perpendicular to the plane Q, then the vectors n and S are parallel. Therefore, the equalities
Are perpendicularity conditions straight and plane.
Intersection of a line with a plane. The condition of belonging to a straight plane:
Consider the line and plane Ax + By + Cz + D=0.
Simultaneous fulfillment of equalities:
Ax 0 + By 0 + Cz 0 + D=0 are the condition of belonging to a straight plane.
Ellipse.
The locus of points, the sum of the distances from which to two fixed points of the plane (usually called focal points) is constant, is called ellipse.
If the coordinate axes are located so that Ox passes through the foci F 1 (C,0) and F 2 (-C,0), and O(0,0) coincides with the middle of the segment F 1 F 2, then according to F 1 M + F 2 M we get:
canonical equation of the ellipse 
,
b 2 \u003d - (c 2 -a 2).
a and b are the semi-axes of the ellipse., a-large, b-smaller.
Eccentricity. , (if a>b)
(if a Eccentricity characterizes the convexity of the ellipse. The ellipse has an eccentricity of 0. The case =0 occurs only when c=0, and this is the case of a circle - it is an ellipse with zero eccentricity. Headmistresses (D) The locus of points, the ratio of the distances from which to the point of the ellipse to the distance from this point of the ellipse to the focus is constant and equal to the value, is called directors. . Note: The circle has no directrix. Hyperbola. The locus of points, the modulus of the difference in distances from which to two fixed points of the plane is constant, is called hyperbole. The canonical equation of a hyperbola is: A hyperbola is a line of the second order. A hyperbola has 2 asymptotes: and The hyperbole is called equilateral if its semiaxes are equal. (a=b). Canonical equation: Eccentricity is the ratio of the distance between the foci to the value of the real axis of the hyperbola: Since for a hyperbola c>a, then the eccentricity of the hyperbola >1. Eccentricity characterizes the shape of a hyperbola: . The eccentricity of an equilateral hyperbola is . Headmistresses- straight. Focal radii: There are hyperbolas that have common asymptotes. Such hyperbolas are called conjugated. Parabola. Parabola- the set of all points of the plane, each of which is equally distant from a given point, called the focus, and a given line, called the directrix. Distance from focus to directrix parabola parameter(p>0).- semifocal diameter. A parabola is a line of the second order. M(x,y) is an arbitrary point of the parabola. Connect the point M with F, draw the segment MN perpendicular to the directrix. According to the definition of the parabola MF=MN. According to the formula for the distance between 2 points, we find: Canonical parabola equation: Ellipsoid. Exploring the surface given by the equation: Consider sections of the surface with planes parallel to the xOy plane. Equations of such planes: z=h, where h is any number. The line obtained in the section is determined by two equations: Examining the surface: A) if then Line of intersection of the surface with the planes z=h does not exist. B) if , the line of intersection degenerates into two points (0,0,s), and (0,0,-s). The plane z = c, z = - c touches the given surface. C) if , then the equations can be rewritten as: The considered sections allow us to depict the surface as a closed oval surface. The surface is called ellipsoids. If any semiaxes are equal, the triaxial ellipsoid turns into an ellipsoid of revolution, and if a=b=c, then into a sphere. Hyperboloid and cone.
, where .
And 
.
=> 
= =>
=>
y 2 = 2px.
, as you can see, the line of intersection is an ellipse with semi-axes a1 = , b1 = . In this case, the smaller h, the larger the semiaxis. At n=0 they reach their highest values. a1=a, b1=b. The equations will take the form: 
