Sine of the angle between a straight line and a plane. Angle between a straight line and a plane

The angle a between straight line l and plane 6 can be determined through the additional angle p between a given straight line l and a perpendicular n to a given plane drawn from any point on the straight line (Fig. 144). Angle P complements the desired angle a to 90°. Having determined the true value of the angle P by rotating around the straight line the level of the plane of the angle formed by the straight line l and the perpendicular and, it remains to complement it to a right angle. This additional angle will give the true value of the angle a between straight line l and plane 0.

27. Determining the angle between two planes.

The true value of the dihedral angle is between two planes Q and l. - can be determined either by replacing the projection plane in order to transform the edge of a dihedral angle into a projecting line (problems 1 and 2), or if the edge is not specified, as the angle between two perpendiculars n1 and n2 drawn to these planes from an arbitrary point M of space B plane of these perpendiculars at point M we obtain two plane angles a and P, which are respectively equal to the linear angles of two adjacent angles (dihedral) formed by the planes q and l. Having determined the true value of the angles between the perpendicular n1 and n2 by rotating around the straight line of the level, we will thereby determine the linear angle of the dihedral angle formed by the planes q and l.

Curved lines. Special points of curved lines.

In a complex drawing of a curve, its special points, which include points of inflection, return, break, and nodal points, are also special points on its projection. This is explained by the fact that the singular points of the curves are connected to the tangents at these points.

If the plane of the curve occupies a projecting position (Fig. A), then one projection of this curve has the shape of a straight line.

For a spatial curve, all its projections are curved lines (Fig. b).

To determine from the drawing which curve is given (plane or spatial), it is necessary to find out whether all points of the curve belong to the same plane. Specified in Fig. b the curve is spatial, since the point D the curve does not belong to the plane defined by three other points A, B And E this curve.

Circle - a plane curve of the second order, the orthogonal projection of which can be a circle and an ellipse

A cylindrical helical line (helix) is a spatial curve representing the trajectory of a point performing a helical movement.

29.Flat and spatial curved lines.

See question 28

30. Complex surface drawing. Basic provisions.

A surface is a set of sequential positions of lines moving in space. This line can be straight or curved and is called generatrix surfaces. If the generatrix is a curve, it can have a constant or variable appearance. The generatrix moves along guides, representing lines of a different direction than the generators. The guide lines set the law of movement for the generators. When moving the generatrix along the guides, a frame surface (Fig. 84), which is a set of several successive positions of the generatrices and guides. Examining the frame, one can be convinced that the generators l and guides T can be swapped, but the surface remains the same.

Any surface can be obtained in various ways.

Depending on the shape of the generatrix, all surfaces can be divided into ruled, which have a generative straight line, and non-ruled, which have a forming curved line.

Developable surfaces include the surfaces of all polyhedra, cylindrical, conical and torso surfaces. All other surfaces are non-developable. Non-ruled surfaces can have a generatrix of a constant shape (surfaces of revolution and tubular surfaces) and a generatrix of variable shape (channel and frame surfaces).

A surface in a complex drawing is specified by projections of the geometric part of its determinant, indicating the method of constructing its generators. In a drawing of a surface, for any point in space the question of whether it belongs to a given surface is unambiguously resolved. Graphically specifying the elements of the surface determinant ensures the reversibility of the drawing, but does not make it visual. For clarity, they resort to constructing projections of a fairly dense frame of generatrices and to constructing outline lines of the surface (Fig. 86). When projecting surface Q onto the projection plane, the projecting rays touch this surface at points forming a certain line on it l, which is called contour line. The projection of the contour line is called essay surfaces. In a complex drawing, any surface has: P 1 - horizontal outline, on P 2 - frontal outline, on P 3 - profile outline of the surface. The sketch includes, in addition to projections of the contour line, also projections of the cut lines.

Let some rectangular coordinate system and a straight line be given . Let And - two different planes intersecting in a straight line and given accordingly by equations. These two equations jointly define the straight line if and only if they are not parallel and do not coincide with each other, i.e. normal vectors
And
these planes are not collinear.

Definition. If the coefficients of the equations

are not proportional, then these equations are called general equations straight line, defined as the line of intersection of planes.

Definition. Any non-zero vector parallel to a line is called guide vector this straight line.

Let us derive the equation of the straight line passing through a given point
space and having a given direction vector
.

Let the point
- arbitrary point on a straight line . This point lies on a line if and only if the vector
, having coordinates
, collinear to the direction vector
straight. According to (2.28), the condition for collinearity of vectors
And looks like

. (3.18)

Equations (3.18) are called canonical equations straight line passing through a point
and having a direction vector
.

If straight is given by general equations (3.17), then the direction vector this line is orthogonal to the normal vectors
And
planes specified by equations. Vector
according to the vector product property, it is orthogonal to each of the vectors And . According to the definition, as a direction vector straight you can take a vector
, i.e.
.

To find a point
consider the system of equations
. Since the planes defined by the equations are not parallel and do not coincide, then at least one of the equalities does not hold
. This leads to the fact that at least one of the determinants ,
,
different from zero. For definiteness, we will assume that
. Then, taking an arbitrary value , we obtain a system of equations for the unknowns And :

According to Cramer's theorem, this system has a unique solution defined by the formulas

,
. (3.19)

If you take
, then the straight line given by equations (3.17) passes through the point
.

Thus, for the case when
, the canonical equations of the line (3.17) have the form

The canonical equations of the straight line (3.17) are written similarly for the case when the determinant is nonzero
or
.

If a line passes through two different points
And
, then its canonical equations have the form

. (3.20)

This follows from the fact that the straight line passes through the point
and has a direction vector.

Let us consider the canonical equations (3.18) of the straight line. Let us take each of the relations as a parameter , i.e.
. One of the denominators of these fractions is non-zero, and the corresponding numerator can take any value, so the parameter can take on any real values. Considering that each of the ratios is equal , we get parametric equations straight:

,
,
. (3.21)

Let the plane is given by a general equation, and the straight line - parametric equations
,
,
. Dot
intersection of a straight line and planes must simultaneously belong to a plane and a line. This is only possible if the parameter satisfies the equation, i.e.
. Thus, the point of intersection of a straight line and a plane has coordinates

Example 32. Write parametric equations for a line passing through points
And
.

Solution. For the directing vector of the straight line we take the vector

. A straight line passes through a point , therefore, according to formula (3.21), the required straight line equations have the form
,
,
.

Example 33. Vertices of the triangle
have coordinates
,
And
respectively. Compose parametric equations for the median drawn from the vertex .

Solution. Let
- middle of the side
, Then
,
,
. As the guide vector of the median, we take the vector
. Then the parametric equations of the median have the form
,
,
.

Example 34. Compose the canonical equations of a line passing through a point
parallel to the line
.

Solution. The straight line is defined as the line of intersection of planes with normal vectors
And
. As a guide vector take the vector of this line
, i.e.
. According to (3.18), the required equation has the form
or
.

3.8. The angle between straight lines in space. Angle between a straight line and a plane

Let two straight lines And in space are given by their canonical equations
And
. Then one of the corners between these lines is equal to the angle between their direction vectors
And
. Using formula (2.22), to determine the angle we get the formula

. (3.22)

Second corner between these lines is equal
And
.

Condition for parallel lines And is equivalent to the condition of collinearity of vectors
And
and lies in the proportionality of their coordinates, i.e. the condition for parallel lines has the form

. (3.23)

If straight And are perpendicular, then their direction vectors are orthogonal, i.e. the perpendicularity condition is determined by the equality

. (3.24)

Consider a plane , given by the general equation, and the straight line , given by the canonical equations
.

Corner between the straight line and plane is complementary to the angle between the directing vector of the straight line and the normal vector of the plane, i.e.
And
, or

. (3.24)

Condition for parallelism of a line and planes is equivalent to the condition that the direction vector of the line and the normal vector of the plane are perpendicular, i.e., the scalar product of these vectors must be equal to zero:

If the line is perpendicular to the plane, then the direction vector of the line and the normal vector of the plane must be collinear. In this case, the coordinates of the vectors are proportional, i.e.

. (3.26)

Example 35. Find an obtuse angle between straight lines
,
,
And
,
,
.

Solution. The direction vectors of these lines have coordinates
And
. Therefore one corner between straight lines is determined by the ratio, i.e.
. Therefore, the condition of the problem is satisfied by the second angle between the lines, equal to
.

3.9. Distance from a point to a line in space

Let
 point in space with coordinates
, straight line given by canonical equations
. Let's find the distance from point
to a straight line .

Let's apply a guide vector
to the point
. Distance from point
to a straight line is the height of a parallelogram built on vectors And
. Let's find the area of a parallelogram using the cross product:

On the other side, . From the equality of the right-hand sides of the last two relations it follows that

. (3.27)

3.10. Ellipsoid

Definition. Ellipsoid is a second-order surface, which in some coordinate system is defined by the equation

. (3.28)

Equation (3.28) is called the canonical equation of the ellipsoid.

From equation (3.28) it follows that the coordinate planes are planes of symmetry of the ellipsoid, and the origin of coordinates is the center of symmetry. Numbers
are called semi-axes of the ellipsoid and represent the lengths of segments from the origin to the intersection of the ellipsoid with the coordinate axes. An ellipsoid is a bounded surface enclosed in a parallelepiped
,
,
.

Let us establish the geometric form of the ellipsoid. To do this, let us find out the shape of the lines of intersection of its planes parallel to the coordinate axes.

To be specific, consider the lines of intersection of the ellipsoid with the planes
, parallel to the plane
. Equation for the projection of the intersection line onto a plane
is obtained from (3.28) if we put in it
. The equation of this projection is

. (3.29)

If
, then (3.29) is the equation of an imaginary ellipse and the points of intersection of the ellipsoid with the plane
No. It follows that
. If
, then line (3.29) degenerates into points, i.e. planes
touch the ellipsoid at points
And
. If
, That
and you can introduce the notation

,
. (3.30)

Then equation (3.29) takes the form

, (3.31)

i.e. projection onto a plane
lines of intersection of the ellipsoid and the plane
is an ellipse with semi-axes, which are determined by equalities (3.30). Since the line of intersection of the surface with planes parallel to the coordinate planes is a projection “raised” to a height , then the intersection line itself is an ellipse.

When decreasing the value axle shafts And increase and reach their greatest value at
, i.e. in the section of the ellipsoid by the coordinate plane
the largest ellipse with semi-axes is obtained
And
.

The idea of an ellipsoid can be obtained in another way. Consider on the plane
family of ellipses (3.31) with semi-axes And , defined by relations (3.30) and depending on . Each such ellipse is a level line, that is, a line at each point of which the value the same. “Raising” each such ellipse to a height , we obtain a spatial view of the ellipsoid.

A similar picture is obtained when a given surface is intersected by planes parallel to the coordinate planes
And
.

Thus, an ellipsoid is a closed elliptical surface. When
The ellipsoid is a sphere.

The line of intersection of an ellipsoid with any plane is an ellipse, since such a line is a limited line of the second order, and the only limited line of the second order is an ellipse.

Maintaining your privacy is important to us. For this reason, we have developed a Privacy Policy that describes how we use and store your information. Please review our privacy practices and let us know if you have any questions.

Collection and use of personal information

Personal information refers to data that can be used to identify or contact a specific person.

You may be asked to provide your personal information at any time when you contact us.

Below are some examples of the types of personal information we may collect and how we may use such information.

What personal information do we collect:

When you submit an application on the site, we may collect various information, including your name, telephone number, email address, etc.

How we use your personal information:

The personal information we collect allows us to contact you with unique offers, promotions and other events and upcoming events.
From time to time, we may use your personal information to send important notices and communications.
We may also use personal information for internal purposes, such as conducting audits, data analysis and various research in order to improve the services we provide and provide you with recommendations regarding our services.
If you participate in a prize draw, contest or similar promotion, we may use the information you provide to administer such programs.

Disclosure of information to third parties

We do not disclose the information received from you to third parties.

Exceptions:

If necessary - in accordance with the law, judicial procedure, in legal proceedings, and/or on the basis of public requests or requests from government bodies in the Russian Federation - to disclose your personal information. We may also disclose information about you if we determine that such disclosure is necessary or appropriate for security, law enforcement, or other public importance purposes.
In the event of a reorganization, merger, or sale, we may transfer the personal information we collect to the applicable successor third party.

Protection of personal information

We take precautions - including administrative, technical and physical - to protect your personal information from loss, theft, and misuse, as well as unauthorized access, disclosure, alteration and destruction.

Respecting your privacy at the company level

To ensure that your personal information is secure, we communicate privacy and security standards to our employees and strictly enforce privacy practices.

This means finding the angle between this line and its projection onto a given plane.

A spatial model illustrating the task is presented in the figure.

Problem solution plan:
1. From an arbitrary point A∈a lower the perpendicular to the plane α ;
2. Determine the meeting point of this perpendicular with the plane α . Dot A α- orthogonal projection A to the plane α ;
3. Find the point of intersection of the line a with plane α . Dot a α- straight trail a on surface α ;
4. We carry out ( A α a α) - projection of a straight line a to the plane α ;
5. Determine the real value ∠ Aa α A α, i.e. ∠ φ .

The solution of the problem find the angle between a line and a plane can be greatly simplified if we do not define ∠ φ between a straight line and a plane, and complementary to 90° ∠ γ . In this case, there is no need to determine the projection of the point A and straight line projections a to the plane α . Knowing the magnitude γ , calculated by the formula:

$ φ = 90° - γ $

a and plane α , defined by parallel lines m And n.

a α
By rotating around the horizontal line specified by points 5 and 6, we determine the natural size ∠ γ . Knowing the magnitude γ , calculated by the formula: