Geometry lesson in grade 10
In one of the previous lessons, you got acquainted with the concept of the projection of a point on a given plane parallel to a given line.
In this lesson, you will continue your study of lines and planes; Learn how to find the angle between a line and a plane. You will get acquainted with the concept of orthogonal projection onto a plane and consider its properties. The lesson will give definitions of the distance from a point to a plane and from a point to a line, the angle between a line and a plane. The famous theorem on three perpendiculars will be proved.
orthogonal projection
Orthogonal projection of a point and figure.
Orthogonal projection of the part.
Orthogonal projection of point A on a given plane is called the projection of a point on this plane parallel
straight line perpendicular to this plane. orthogonal projection
figure onto a given plane p consists of orthogonal projections onto the plane p of all points of this figure. Orthogonal projection is often used to depict spatial bodies on a plane, especially in technical drawings. It gives a more realistic image than an arbitrary parallel projection, especially of round bodies.
Perpendicular and oblique
Let a line be drawn through a point A not belonging to the plane p, perpendicular to this plane and intersecting it at point B. Then
segment AB is called
perpendicular, lowered from the point
And on this plane, and point B itself is the base of this perpendicular. Any segment AC, where C -
an arbitrary point of the plane p, other than B, is called inclined to
this plane.
Note that the point B in this definition is orthogonal
projection of point A, and segment AC - Perpendicular and oblique. orthogonal projection of oblique AB.
Orthographic projections have all the properties of ordinary parallel projections, but they also have a number of new properties.
Let a perpendicular and several inclined lines be drawn from one point to the plane. Then the following statements are true.
1. Any oblique is longer than both the perpendicular and the orthogonal projection of the oblique onto this plane.
2. Equal obliques have equal orthogonal projections, and vice versa, obliques having equal projections are also equal.
3. One oblique is longer than the other if and only if the orthogonal projection of the first oblique is longer than the orthogonal projection of the second oblique.
Properties of orthogonal projection
Proof.
Let a perpendicular AB and two inclined AC and AD be drawn from the point A to the plane p; then the segments BC and BD are orthogonal projections of these segments onto the plane p.
Let us prove the first assertion: any oblique is longer than both the perpendicular and the orthogonal projection of the oblique onto this plane. Consider, for example, an oblique AC and a triangle ABC formed by the perpendicular AB, this oblique AC, and its orthogonal projection BC. This triangle is right-angled with a right angle at vertex B and hypotenuse AC, which, as we know from planimetry, is longer than each of the legs, i.e. and the perpendicular AB, and the projection BC.
From point A to the plane pi, a perpendicular AB and two inclined AC and AD are drawn.
Properties of orthogonal projection
triangles
ABC and ABD
equal in length and hypotenuse.
Now let's prove the second statement, namely: equal oblique ones have equal orthogonal projections, and vice versa, oblique ones having equal projections are also equal.
Consider right triangles ABC and ABD. They
have a common leg AB. If oblique AC and AD are equal, then right triangles ABC and ABD are equal in leg and hypotenuse, and then BC=BD. Conversely, if the projections BC and BD are equal, then these same triangles are equal in two legs, and then their hypotenuses AC and AD are also equal.
contradicts the condition. If the sun< BD, как мы только что доказали, АС < AD, что опять противоречит условию.
There remains a third possibility: BC > BD. The theorem has been proven.
If BC is greater than BD,
then AC is greater than the side
AE equal to AD.
Orthogonal projection is a special case of parallel projection, when the projection direction S is perpendicular (orthogonal) to the projection plane S 1 (Fig. 1.11).
Rice. 1.11. Right Angle Orthographic Projection
Orthogonal projection is widely used in engineering practice for depicting geometric figures on a plane, since it has a number of advantages over central and parallel (oblique) projection, which include:
a) the simplicity of graphic constructions for determining orthogonal projections of points;
b) the possibility, under certain conditions, to preserve the shape and dimensions of the projected figure on the projections.
These advantages ensured the widespread use of orthogonal projection in engineering, in particular for the preparation of engineering drawings.
For orthogonal projection, all nine invariant properties considered above are true. In addition, it is necessary to note one more, tenth, invariant property, which is valid only for orthogonal projection.
10. If at least one side of the right angle is parallel to the projection plane, then the right angle is projected onto this projection plane without distortion (Fig. 1.11)
On fig. 1.11 shows a right angle ABD, both sides of which are parallel to the projection plane 1. According to the invariant property 9.2, this angle is projected onto the plane 1 without distortion, i.e. A 1 B 1 D 1 =90.
Let's take an arbitrary point C on the projecting beam DD 1, then the resulting ABC will be straight, because ABBB 1 DD 1 .
The projection of this right angle ABC, in which only one side AB is parallel to the plane of projections 1, will be a right angle A 1 B 1 D 1.
Speaking about geometric figures and their projections, it must be remembered that the projection of a figure is the set of projections of all its points.
1.6. System of three planes of projections. Epure Monge.
All spatial geometric figures can be oriented relative to the Cartesian rectangular system of coordinate axes - a system of three mutually perpendicular coordinate planes (Fig. 1.12).
Rice. 1.12. Image of a system of three projection planes
These coordinate planes are designated:
horizontal plane of projections - 1;
frontal plane of projections - 2;
profile plane of projections - 3 .
The lines of intersection of these planes form the coordinate axes: the abscissa axis is X; y-axis - Y; the applicate axis is Z. The point O of the intersection of the coordinate axes is taken as the origin of coordinates and is denoted by the letter O. The positive directions of the axes are: for the x axis - to the left of the origin, for the Y axis - towards the viewer from the plane 2, for the z axis - up from the plane 1; opposite directions are considered negative.
To simplify further reasoning, we will consider only the part of space located to the left of the profile plane of projections 3 .
With this assumption, three coordinate projection planes form four spatial angles - octant (in the general case - 8 octants).
From fig. 1.12 it can be seen that the x-axis X divides the horizontal projection plane 1 into two parts: the front floor 1 (X and Y axes) and the back floor 1 (X and - Y axes).
X-axis divides frontal projection plane 2 also into two parts: the upper floor 2 (X and Z axes) and the lower floor 2 (X and - Z axes).
The y-axis and the applicate Z divide the profile projection plane 3 into four parts:
upper front floor 3 (Y and Z axes)
upper rear floor 3 (-Y and Z axes)
lower front floor 3 (Y and –Z axes)
lower rear floor 3 i (axes - Y and -Z)
In order to obtain a flat (two-dimensional) model of the spatial coordinate planes of projections, the horizontal 1 and profile 3 planes are combined with the frontal 2 in the order shown by the arrows in Fig. 1.12.
P 
In this case, the horizontal projection plane 1 rotates around the X axis by 90, and the profile projection plane 3 also rotates around the Z axis by 90 (the direction of rotation is shown in Fig. 1.12).
The combination of three projection planes obtained in this way (Fig. 1.13) is a flat model of a system of three spatial
To
Rice. 1.13. Spatial model of point A
To build a flat model of a spatial geometric figure, each of its points is projected orthogonally onto the projection planes 1 , 2 and 3 , which are then combined into one plane. The flat model of a spatial geometric figure obtained in this way is called the Monge plot.
The order of plotting a point plot located in the first octant.
On fig. 1.13 shows a spatial point A, the coordinates of which (x, y, z) show the distances at which the point is removed from the projection planes.
D 
In order to obtain orthogonal projections of point A, it is necessary to lower the perpendiculars on the projection plane from this point.
The intersection points of these perpendiculars with the projection planes form the projections of point A:
A 1 - horizontal projection of the point;
A 2 - frontal projection of the point;
A
Rice. 1.14. Plot point A
On fig. 1.14 the projection planes 1 and 3 are aligned with the drawing plane (with the projection plane 2), and together with them are aligned with the drawing plane and the projection of point A (A 1, A 2, A 3) and thus a planar model of coordinate planes is obtained projections and a planar model of the spatial point A - its diagram.
The position of the projections of point A on the diagram is uniquely determined by its three coordinates (Fig. 1.14).
On fig. 1.13 and fig. 1.14 also shows that on the diagram the horizontal and frontal projections of the point lie on the same perpendicular to the X axis, as well as the frontal and profile projections - on the same perpendicular to the Z axis:
A 1 A 2 X, A 2 A 3 Z.
Figure 1.12 shows that points located in different octants have certain signs of coordinates.
The table shows the signs of the coordinates of points located in different octants
Table of coordinate signs
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Coordinate signs |
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Questions for self-control
What is the idea behind the projection method?
What is the essence of central projection and what are its main properties?
What is the essence of parallel projection and what are its main properties?
What is the essence of orthogonal (rectangular) projection?
How is the right angle projection theorem formulated?
Related tasks:
The side AB of the rhombus ABCD is a, one of the angles is 60 degrees. A plane alpha is drawn through side AB at a distance a/2 from point D.
a) find the distance from point C to plane alpha.
b) show in the figure the linear angle of the dihedral angle DABM. M belongs to alpha.
c) Find the sine of the angle between the rhombus plane and the alpha plane.
The side AB of the rhombus ABCD is a, one of the angles is 60 degrees. A plane alpha is drawn through side AB at a distance a/2 from point D. a) find the distance from point C to plane alpha. b) show in the figure the linear angle of the dihedral angle DABM. M belongs to alpha. c) Find the sine of the angle between the rhombus plane and the alpha plane.
The side AB of the rhombus ABCD is equal to a, and one of its angles is equal to 60°. A plane alpha is drawn through side AB at a distance a2 from point D.
a) Find the distance from point C to plane alpha.
b) Show in the figure the linear angle of the dihedral angle DABM, M belongs to the square. alpha.
c) Find the sine of the angle between the rhombus plane and the alpha plane.
Geometry lesson in grade 10
In this lesson, you will continue your study of lines and planes; Learn how to find the angle between a line and a plane. You will get acquainted with the concept of orthogonal projection onto a plane and consider its properties. The lesson will give definitions of the distance from a point to a plane and from a point to a line, the angle between a line and a plane. The famous three theorem will be proved. perpendiculars.
The orthogonal projection of a point A onto a given plane is the projection of a point onto this plane parallel to a straight line perpendicular to this plane. The orthogonal projection of a figure onto a given plane p consists of orthogonal projections onto the plane p of all points of this figure.
Orthogonal projection is often used to depict spatial bodies on a plane, especially in technical drawings. It gives a more realistic image than an arbitrary parallel projection, especially of round bodies.
Let a straight line be drawn through a point A that does not belong to the plane p, perpendicular to this plane and intersecting it at point B. Then the segment AB is called the perpendicular dropped from the point A to this plane, and the point B itself is the base of this perpendicular. Any segment AC, where C is an arbitrary point of the plane p, different from B, is called inclined to this plane.
Note that the point B in this definition is the orthogonal projection of the point A, and the segment AC is the orthogonal projection of the oblique AB. Orthographic projections have all the properties of ordinary parallel projections, but they also have a number of new properties.
Let a perpendicular and several inclined lines be drawn from one point to the plane. Then the following statements are true.
1. Any oblique is longer than both the perpendicular and the orthogonal projection of the oblique onto this plane.
2. Equal obliques have equal orthogonal projections, and vice versa, obliques having equal projections are also equal.
3. One oblique is longer than the other if and only if the orthogonal projection of the first oblique is longer than the orthogonal projection of the second oblique.