If the lines are parallel, then their projections of the same name are parallel.

If straight lines intersect, then their projections of the same name intersect with each other at points that are projections of the point of intersection of these lines.

Crossing straight lines do not intersect and not parallel among themselves, although their projections may intersect or be parallel.

The intersection points of these projections do not lie on the same communication line. one point 1 v match two points 1 n and 1" n. These points lie on the same perpendicular to the plane V(Fig.2.9a, b, c).

Rice. 2.9. Mutual position of segments on the plot:

A) parallel b) intersecting; c) crossing

2.3.1. Competing points

Points lying on the same perpendicular to the projection plane are called competing relative to this plane (Fig. 2.10a, b).

Competing points determine the visibility of geometric images on the diagram. Visible on a given projection will always be one of the competing points that lies farther away from this projection plane, hence closer to the viewer. points BUT and AT are frontally competitive. A point will be visible on the frontal projection plane BUT, because it is further from the plane V and closer to the observer. points BUT and FROM are horizontally competitive. A point will also be visible on the horizontal projection plane BUT, because it is off the plane H further than point FROM.

Rice. 2.10. Competing points: a) in dimetry; b) on the diagram

2.4. Plane Angle Projections

Two intersecting lines form a flat angle.

If the angle is located in a plane parallel to the plane of projections, then it is projected onto it in full size.

In general, a flat angle whose sides are not parallel to the projection plane is projected onto this plane with distortion.

2.4.1. Right angle projection theorem

In order for a right angle to be projected orthogonally in the form right angle, it is necessary and sufficient that at least one of its sides be parallel to the projection plane, and the second is not perpendicular to this plane(Fig.2.11a, b).

Rice. 2.11. Projections of a right angle on the plot:

A) on the frontal projection plane; b) on the horizontal projection plane

Proof: Let we have a right angle in space YOU. Project it onto a plane H orthogonally. Let's assume that the side AB given angle is parallel to the plane H. Then we have:  YOU= 90˚; AB || H; AA n H. Let us prove that  AT n BUT n FROM n= 90º (Fig.2.12).  BUT n AB= 90°, because figure AA n BB n- rectangle. Therefore, a straight line AB perpendicular to the projecting plane Q as perpendicular to two lines of this plane ( ABAC; ABAA n). That's why ABQ, but BUT n AT n || AB from here and BUT n AT n Q, which means that  AT n BUT n FROM n= 90º.

Figure 2.12 Right Angle Projection

A task: Determine distance from point BUT to the front (Fig.2.13).

Solution. The right angle between the desired perpendicular and the front sun projected in full size onto a plane V. Natural size of the perpendicular AK can be found using the right triangle method.

Rice. 2.13. Determining the distance from point A to the front BC

If two lines lie on a plane, then three different cases of their mutual arrangement are possible: 1) the lines intersect (that is, they have one common point), 2) the lines are parallel and do not coincide, 3) the lines coincide.

Let us find out how to find out which of these cases takes place if the lines are given by their equations

If the lines intersect, i.e., have one common point, then the coordinates of this point must satisfy both equations (15). Therefore, to find the coordinates of the point of intersection of the lines, it is necessary to solve their equations together. To this end, we first eliminate the unknown x, for which we multiply the first equation by , and the second by A, and subtract the first from the second. Will have:

To eliminate the unknown y from equations (15), we multiply the first of them by and the second by and subtract the second from the first. We get:

If then from equations (15) and (15") we obtain the solution of system (15):

Formulas (16) give the x, y coordinates of the point of intersection of two lines.

Thus, if then the lines intersect. If then formulas (16) do not make sense. How are the lines arranged in this case? It is easy to see that in this case the lines are parallel. Indeed, it follows from the condition that (if , then the lines are parallel to the Oy axis and, therefore, are parallel to each other).

So, if then the lines are parallel. The condition under consideration can be written in the form we can say that if in the equations of lines the corresponding coefficients at the current coordinates are proportional, then the lines are parallel.

In particular, parallel lines can coincide. Let us find out what is the analytical criterion for the coincidence of lines. To do this, consider equations (15) and ). If the free terms of these equations are both equal to zero, i.e.

i.e., the coefficients of the unknowns and the free terms of equations (15) are proportional. In this case, one of the equations of the system is obtained from the other by multiplying all its terms by some common factor, i.e., equations (15) are equivalent. Therefore, the parallel lines under consideration coincide.

If at least one of the free terms of equations (15) and ) is different from zero (or or

then equations (15) and (15"), and hence equations (15), will have no solutions (at least one of the equalities (15) or (15") will be impossible). In this case, the parallel lines will not coincide.

So, the condition (necessary and sufficient) for the coincidence of two lines is the proportionality of the corresponding coefficients of their equations:

Example 1. Find the point of intersection of straight lines

Solving the equations together, multiply the second by 3.

Straight lines and organization of space

Straight lines - simple but very
expressive element:
a line divides the plane into
individual
parts;
-line helps unite
composition
into a whole;
line, more than
rectangle
affects the rhythm
compositions.

Frontal and deep compositions from lines
and rectangles

even by the simplest means
can achieve emotional
imagery

The line is not "lost weight
rectangle", and an independent
pictorial element line attached
expressiveness of the whole composition. AT
works where the line is right through (from edge to edge
sheet), she seems to endure
pictorial action beyond the scope and
makes the composition open, open
and more interesting.
thin, long and
straight lines are cut
by ruler

working
above
their
compositions,
seek differences in the size of plans,
because it creates a pictorial
polyphony, intonation richness and,
accordingly, greater expressiveness
compositions.

TASKS
Straight lines - an element of planar organization
compositions.
1. Location and mutual intersection of 3-4 straight lines
different thickness achieve harmonious articulation
spaces (use lines through).
2. Create a composition with 2-3 rectangles and 3-4 straight lines
lines that, by their arrangement, connect elements in
single compositional whole. Create: a) frontal
composition; b) deep composition.
3. From an arbitrary number of elements, make an interesting
composition.
Rhythmically arranging the elements on the plane, achieve
emotional-figurative impression (for example, “flight”, narrowing, “slowing down”, etc.).
Tasks can be completed on a computer.

RELATIONSHIP OF THE RIGHTS.

The angle between two lines, the conditions for parallelism and perpendicularity of two lines, the intersection of lines, the distance from a given point to a given line.

The angle between straight lines in a plane is understood to be the smaller (sharp) of the two adjacent corners formed by these lines.

If the lines l 1 and l 2 are given by equations with slope factors y \u003d k 1 x + b 1 and y \u003d k 2 x + b 2, then the angle φ between them is calculated by the formula

The parallelism condition for lines l 1 and l 2 has the form

and the condition of their perpendicularity

k 1 = - (or k 1 k 2 = - 1)

If lines l 1 and l 2 are given general equations A 1 x + B 1 y + C 1 \u003d 0 and A 2 x + B 2 y + C 2 \u003d 0,

then the value φ of the angle between them is calculated by the formula

tg φ=

their angular parallelisms

(or A 1 B 2 -A 2 B 1 \u003d 0)

The condition for their perpendicularity

A 1 A 2 + B 1 B 2 \u003d 0

To find common points of lines l 1 and l 2, it is necessary to solve the system

equations

A 1 x + B 1 y + C 1 \u003d 0, y \u003d k 1 x + b 1

or

A 2 x + B 2 y + C 2 \u003d 0, y \u003d k 2 x + b 2

Wherein:

If a
, then there is a single point of intersection of the lines;

If a
- lines l 1 and l 2 do not have a common point, i.e. parallel;

If a
-lines have an infinite number of points, i.e. they coincide

The distance d from the point M 0 (x 0; y 0) to the straight line Ax + Vy + C \u003d 0 is the length of the perpendicular dropped from this point to the straight line.

The distance d is determined by the formula

d=

Distance from point M 0 (x 0; y 0) to the straight line x cos + y sin - p=0 is calculated by the formula

d=

EXAMPLE: find the angle between lines:

1) y=2x-3 and y=
;

2) 2x-3y+10=0 and 5x – y+4=0;

3) y=
and 8x+6y+5=0;

4) y=5x+1 and y=5x-2;

=arctg
);

Tasks for practical exercises:

1. Find the angle between the lines:

1) y=0.5x-3 and y=2x-2;

2) 2x-3y-7=0 and 2x-y+5=0;

3) y=x+6 and 3x-2y-8=0;

4) y= 7x -1 and y=7x+1;

1) 3x+5y-9=0 and 10x-6y+4=0

2) 2x+5y-2=0 and x+y+4=0;

3) 2y=x-1 and 4y-2x+2=0;

4) x+8=0 and 2x-3=0;

5)
=1 and y=x+2;

6) x+y=0 and x-y=0

7) y+3=0 and 2x+y-1=0;

8) y=3-6x and 12x+2y-5=0;

9) 2x+3y=8 and x-y-3=0

10) x-y-1=0 and x+y+2=0

3. At what values the following pairs of lines are: a) parallel; b) are perpendicular.

1) 2x-3y+4=0 and x-6y+7=0;

2) x-4y+1=0 and -2x+y+2=0;

3) 4x+y-6=0 and 3x+ y-2=0;

4) x- y+5=0 and 2x+3y+3=0;

4. Through the point of intersection of lines 3x-2y + 5 \u003d 0; x+2y-9=0 a straight line is drawn parallel to the straight line 2x+y+6=0. Write its equation.

5. Find the equation of a straight line passing through point A (-1; 2):

a) parallel to the straight line y \u003d 2x-7;

b) perpendicular to the line x+3y-2=0.

6. Find the length of the height of the VD in a triangle with vertices A (4; -3); B (-2; 6) and C (5; 4).

7. The equations of the triangle sides are given: x+3y-3=0, 3x-11y-29=0 and 3x-y+11=0.

Find the vertices of this triangle.

Tasks for independent solution

1. Find sharp corner between lines:

1) y \u003d 3x and y \u003d - x

2) 2x-3y+6=0 and 3x-y-3=0

4) 3x+4y-12=0 and 15x-8y-45=0

2. Explore the relative position of the following pairs of lines:

1) 2x-3y+4=0 and 10x+3y-6=0

2) 3x-4y+12=0 and 4x+3y-6=0

3) 25x+20y-8=0 and 5x+4y+4=0

4) 4x+5y-8=0 and 3x-2y+4=0

5) y=3x+4 and y=-3x+2

3. Find the equation of a straight line passing through point B (2;-3)

a) parallel to the straight line connecting the points M 1 (-4; 0) and M 2 (2; 2);

b) perpendicular to the line x-y=0.

4. Write an equation of a straight line containing the height of the VD in a triangle with vertices

A (-3; 2), B (5; -2), C (0; 4)

5. Find the area of ​​the triangle formed by the lines 2x+y+4=0, x+7y-11=0 and 3x-5y-7=0.

6. Through the point of intersection of the lines 3x + 2y-4 \u003d 0 and x-5y + 8 \u003d 0, lines are drawn, one of which passes through the origin, and the other is parallel to the Ox axis. Write their equations.

7. Given a quadrilateral ABCD with vertices A (3; 5); In (6;6); C (5; 3); D (1; 1). Find:

a) coordinates of the point of intersection of the diagonals;

b) the angle between the diagonals .

8. Given the vertices of the triangle A (2; -2), B (3; 5), C (6; 1). Find:

1) the lengths of the sides AC and BC;

2) equations of lines on which the sides BC and AC lie;

3) the equation of the straight line on which lies the height drawn from B;

4) the length of this height;

5) the equation of a straight line on which lies the median drawn from point A;

6) the length of this median;

7) the equation of a straight line on which the bisector of angle C lies;

8) the center of gravity of the triangle;

9) area of ​​a triangle;

10) angle C;

Answers to tasks for independent solution:

1. 1) 63 0 ; 2) 37,9 0 ; 3) 31,3 0 ; 4) 81,2 0 . 2. 1) Parallel;

2) Perpendicular; 3) Parallel; 4) Intersect; 5) Intersect;

3. a) x-3y-11=0; b) x + y + 1 = 0; 4. 3x+2y-11=0; 5. 13; 6. 7x-y=0 and 17y-28=0; 7. a)(4;4);

b); 8. 1) -5;5 2) 4x+3y-27=0.3x-4y-14=0; 3) 4x+3y-27=0; 4) 5; 5) 2x-y-6=0; 6) ; 7) x+7y-13=0; 8) (;); 9); 10)

If we draw parallel lines AB and C through given D planes perpendicular to the horizontal plane of projections, then these two planes will be parallel, and at their intersection with the plane H, two mutually parallel lines will be obtained A"B" and C"D", which are orthogonal projections of the data of straight lines AB and CD on the horizontal plane of projections (Fig. 25).

Similarly, orthogonal projections of given lines onto the frontal plane V can be obtained.

In the complex drawing, the projections of the same name parallel lines are parallel: A"B"C"D" and A""B""C""D"" (Fig. 25).

intersecting lines

Mutually intersecting lines have a common point, for example, line segments AB and CD intersect at a point To. The projections of intersecting lines intersect, and their intersection points ( K" and K"") lie on the same line of communication - perpendicular to the axis x(Fig. 26).

Crossed lines

These are lines that are not parallel and do not intersect. On the complex drawing, the projections of intersecting lines (straight lines AB and CD) can intersect, but the intersection points ( 1 ,2 and 3 ,4 ) lie on different communication lines (Fig. 27). The intersection points of the same-named projections of skew lines correspond in space to two points: in one case - 1 and 2 , and in the other 3 and 4 located on straight lines. In the drawing, the intersection point of the horizontal projections of the lines corresponds to two frontal projections of points 1 "" and 2 "". Similarly - with dots 3 and 4 .