For two lines in space, four cases are possible:

The straight lines coincide;

The lines are parallel (but do not coincide);

Lines intersect;

Straight lines cross, i.e. have no common points and are not parallel.

Let's consider two ways to describe straight lines: canonical equations and general equations. Let the lines L 1 and L 2 be given by canonical equations:

L 1: (x - x 1)/l 1 = (y - y 1)/m 1 = (z - z 1)/n 1, L 2: (x - x 2)/l 2 = (y - y 2)/m 2 = (z - z 2)/n 2 (6.9)

For each line from its canonical equations we immediately determine the point on it M 1 (x 1 ; y 1 ; z 1) ∈ L 1, M 2 (x 2 ; y 2 ​​; z 2) ∈ L 2 and the coordinates of the direction vectors s 1 = (l 1; m 1; n 1) for L 1, s 2 = (l 2; m 2; n 2) for L 2.

If the lines coincide or are parallel, then their direction vectors s 1 and s 2 are collinear, which is equivalent to the equality of the ratios of the coordinates of these vectors:

l 1 /l 2 = m 1 /m 2 = n 1 /n 2. (6.10)

If the lines coincide, then the vector M 1 M 2 is collinear to the direction vectors:

(x 2 - x 1)/l 1 = (y 2 - y 1)/m 1 = (z 2 - z 1)/n 1. (6.11)

This double equality also means that the point M 2 belongs to the line L 1. Consequently, the condition for the lines to coincide is to satisfy equalities (6.10) and (6.11) simultaneously.

If the lines intersect or cross, then their direction vectors are non-collinear, i.e. condition (6.10) is violated. Intersecting lines lie in the same plane and, therefore, vectors s 1 , s 2 and M 1 M 2 are coplanarthird order determinant, composed of their coordinates (see 3.2):

Condition (6.12) is satisfied in three out of four cases, since for Δ ≠ 0 the lines do not belong to the same plane and therefore intersect.

Let's put all the conditions together:


Mutual position straight lines is characterized by the number of solutions of the system (6.13). If the lines coincide, then the system has infinitely many solutions. If the lines intersect, then this system has the only solution. In the case of parallel or crossing, there are no direct solutions. The last two cases can be separated by finding the direction vectors of the lines. To do this, it is enough to calculate two vector artwork n 1 × n 2 and n 3 × n 4, where n i = (A i; B i; C i), i = 1, 2, 3,4. If the resulting vectors are collinear, then the given lines are parallel. Otherwise they are interbreeding.

Example 6.4.


The direction vector s 1 of straight line L 1 is found by canonical equations this line: s 1 = (1; 3; -2). The direction vector s 2 of straight line L 2 is calculated using vector product normal vectors of planes, the intersection of which it is:

Since s 1 = -s 2, then the lines are parallel or coincide. Let us find out which of these situations is realized for these lines. To do this, we substitute the coordinates of the point M 0 (1; 2; -1) ∈ L 1 into general equations straight line L 2. For the first of them we obtain 1 = 0. Consequently, the point M 0 does not belong to the line L 2 and the lines under consideration are parallel.

Angle between straight lines. The angle between two straight lines can be found using direction vectors straight Acute angle between straight lines equal to angle between their direction vectors (Fig. 6.5) or is additional to it if the angle between the direction vectors is obtuse. Thus, if for lines L 1 and L 2 their direction vectors s x and s 2 are known, then acute angleφ between these lines is determined through the scalar product:

cosφ = |S 1 S 2 |/|S 1 ||S 2 |

For example, let s i = (l i ; m i ; n i ), i = 1, 2. Using formulas (2.9) and (2.14) to calculate vector length And dot product in coordinates, we get

If two lines lie on a plane, then three different cases of their relative position 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's find out how to find out which of these cases occurs if the lines are given by their own equations

If the lines intersect, that is, they have one common point, then the coordinates of this point must satisfy both equations (15). Consequently, to find the coordinates of the point of intersection of the lines, it is necessary to solve their equations together. To this end, let us 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. We will have:

To exclude 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 a solution to system (15):

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

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

In particular, parallel lines can coincide. Let us find out what is the analytical sign of 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. Consequently, the parallel lines under consideration coincide.

If at least one of free members equations (15) and ) will be different from zero (or or

then equations (15) and (15"), and therefore equations (15), will not have solutions (at least one of equalities (15) or (15") will be impossible). In this case, parallel lines will not coincide.

So, the condition (necessary and sufficient) for the coincidence of two straight 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.

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Slide captions:

Presentation on fine arts on the topic: “Straight lines and organization of space” Performed by: art teacher of Secondary School No. 1 named after I.D. Buvaltsev Krasnodar region Korenovsk Popovich Galina Ivanovna

The combination of various rectangles and lines gives the composition more variety and entertainment.

Straight lines are a simple but very expressive element.

Before you start working, determine the role of the line in the composition. First of all, the line divides the plane into separate parts.

The line divides the space and at the same time strengthens the interconnection of all elements of the composition. Lines help to unite them into a pictorial whole.

The line introduces dynamics and adds rhythmic expressiveness to the composition.

Emotional imagery

Golden planks

The composition consists not only of visual elements, but also of the spaces between them. The alternation of pictorial elements and free spaces, their frequency, condensation and sparseness is RHYTHM. The rhythm is influenced by the degree of brightness of the elements and their shape.

The main thing is to achieve a harmonious arrangement of lines and rectangles, to create a holistic, rhythmic, balanced composition.

Lines more than rectangles. They influence the rhythmic structure of the composition. By their direction, density, and intersections, they determine the movement and expression of the entire image.

Achieve differences in the close-up of plans - this creates visual polyphony, intonation richness and, accordingly, greater expressiveness of the composition.

Rhythm and emphasis of plans

TASKS: Straight lines are an element of organizing a planar composition. 1. By placing and mutually intersecting 3-4 straight lines of different thicknesses, achieve a harmonious division of spaces (use extending lines). 2. Create a composition of 2-3 rectangles and 3-4 straight lines, which by their arrangement connect the elements into a single compositional whole. Create: a) frontal composition; b) deep composition. 3. Make an interesting composition from an arbitrary number of elements. By rhythmically arranging the elements on the plane, achieve an emotional and figurative impression (for example, “flight”, “narrowing”, “slowing down”, etc.)

A line is not a “thinner rectangle”, but an independent graphic element. In works where the line is on the fly, it seems to take the pictorial action beyond the framework and makes the composition open, open-ended and more interesting.

Straight lines are an element of the organization of a planar composition. 1. By placing and mutually intersecting 3-4 straight lines of different thicknesses, achieve a harmonious division of spaces (use lines that extend). 2. Create a composition of 2-3 rectangles and 3-4 straight lines, which by their arrangement connect the elements into a single compositional whole. Create: a) frontal composition; b) deep composition. 3. Make an interesting composition from an arbitrary number of elements. By rhythmically arranging the elements on the plane, achieve an emotional and figurative impression (for example, “flight”, “narrowing”, “slowing down”, etc.)

Literature used: Textbook for grades 7-8 educational institutions edited by B.M. Nemensky, Moscow “Enlightenment” 2008, teacher’s work.

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Slide captions:

Presentation on fine arts on the topic: “Fundamentals of composition in the constructive arts. Harmony, contrast and emotional expressiveness of a planar composition" Performed by: art teacher of Secondary School No. 1 named after I.D. Buvaltsev, Krasnodar Territory, Korenovsk Popovich Galina Ivanovna

elements of composition Don't be confused by the fact that all exercises are done using rectangles. Firstly, they are quite expressive and, without distracting with the variety of forms, make it easier to master compositional techniques. Secondly, they are a prototype of future layouts of text masses and illustrations. Book cover design

All rectangular elements of the composition should be cut out of black or white paper (depending on the chosen background). Before finally gluing them, you need to move them along the sheet in search of best option layout, reduce or increase their size, achieving a balanced composition.

Create a conflict between the white field and the black spot. The plot, if you like - the intrigue, the constructive composition consists precisely in opposition, contrast, the ratio of masses (in this case - rectangles).

PRACTICAL WORK Let's do exercises to study the principles of balance and movement in a planar composition. We will select rectangles as the elements of the composition. Fold the A4 sheet in half and in half again - we get four rectangles for four exercises. These exercises can also be done on a computer. Exercise 1. Mass balance. Consider a white rectangle, evaluate the white space and select a black rectangle for it of such a size that the black and white colors are balanced, balanced

Exercise 2. Mass dynamics. Let's complicate the task and place the black rectangle at an angle to the white plane. What's more interesting? More expressive? The black rectangle, due to its location, creates a feeling of “movement”. By introducing additional elements into the composition, you can enhance the feeling of movement, or, on the contrary, you can “stop” it

Symmetry Balance in a composition is often associated with symmetry. Since ancient times, symmetry has been considered one of the conditions of beauty. The ancient Greeks believed that the universe was symmetrical simply because symmetry is beautiful. The idea of ​​symmetry was often the starting point in the hypotheses and theories of scientists of past centuries who believed in the mathematical harmony of the universe. The concept of symmetry is not limited to the symmetry of objects. It also applies to physical phenomena and their managers physical laws. It is symmetry that allows us to embrace a wide variety of bodies from a unified position. "Symmetry" translated from Greek means "proportionality"

Asymmetry The method of harmony, in which the image on the left is similar to the right, the top is similar to the bottom diagonally, horizontally, vertically or along another broken axis, is called symmetry, and the composition itself is symmetrical. Symmetry achieves harmony through the disappearance of pictorial conflict, and the composition itself turns into an ornament. The result is uniformity and monotony. Let us remember from Pushkin in “The Queen of Spades”: “The old woman’s furniture stood in sad symmetry.” Asymmetry allows you to achieve dynamism and tension in the composition without losing the harmony of the whole. When using asymmetry, the composition becomes more expressive and interesting. In case of asymmetry, there is no axis or plane of symmetry (Gaudi table) 14 years

If a symmetrical equilibrium form is perceived easily and immediately, then an asymmetrical dynamic form is read gradually. The balanced, balanced composition of V. Lebedev can be contrasted with the dynamic, asymmetrical composition of D. Shterenberg

Peter Cornelis Mondrian is an abstract artist who dedicated his life to the search for equilibrium and balance, creating and leading the “Style” group, which left a bright mark on history contemporary art. In his works he “destroyed” dynamics. His compositions are completely balanced and impeccably balanced. In addition, Mondrian was also the founder of “neoplasticism” - a strict abstract movement based on the use of a lattice of intersecting horizontal and vertical lines as the main compositional motif. For thirty years of his life, he performed sacred acts on canvases, painted them into rectangles and squares and painted the resulting geometric fields either with intense bright colors, or (later) with lightweight and transparent shades of white, gray, beige or bluish

Exercise 3. Symmetry. The white plane is already defined. Cut out several black or colored rectangles and create a symmetrical composition.

Rhythm Among compositional patterns, one should highlight a group of means united by the concept of rhythm. The word “rhythm” itself, translated from ancient Greek, means “beat” or “proportionality”. We live in a world of changing rhythms. Place your hand on your chest, listen to the rhythm of your heart - uniform and calm. Listen to the rhythms of the city - the sound of cars, footsteps, gusts of wind, the sound of raindrops. Rhythm can be perceived not only auditorily, but also visually. Observe the alternation of light and shadow as you move. However, rhythm is characteristic not only of movement, but also of a static object. Look at the rows of desks in the classroom, at the alternation of window openings in the school corridors. Rhythm, thanks to the repetition of elements, creates the impression of conditioned movement. The alternation of pictorial elements and free spaces, their frequency, condensation and sparseness is called rhythm. The rhythm can be calm and restless, directed in one direction or converging towards the center, directed both horizontally and vertically. You can alternate elements, volumes, color spots, some details, etc.

Contrasts are the influencing force of a composition and determine its expressiveness. Contrast is a sharply expressed opposite: long - short, thick - thin, large - small. Contrast is one of the main means of composition. There are contrasts in size, volume and plane, light and shadow (tonal contrasts), warm and cold colors, different textures, etc. Contrasting comparisons help sharpen the perception of the whole. Contrast enhances and emphasizes the difference in the properties of the form, making their unity more intense and impressive. A very strong contrast can visually destroy the compositional structure, so the degree of contrast used is limited by the requirement to maintain the integrity of the impression. In form, proportions, color, contrast emphasizes a clearly expressed opposite, and nuance carries within itself a barely noticeable transition, a shade. Nuance, like contrast, is a way of expressiveness in a composition. Expressiveness in a composition is closely related to harmony, the main task of which is to create the impression of balance, grace and precision of the work (El Lissitzky. poster “Beat the Whites with a Red Wedge,” 1920)

Exercise 4. Rhythm. Let's create a rhythmic composition using lines and rectangles, circles and dots. You can complete the task by cutting rhythmically alternating lines. It is advisable to cut out all elements of the composition not with scissors, but with a breadboard knife.

Static frontal composition A static frontal composition or a more dynamic deep composition should be built on the difference in the sizes of rectangles. The dominant is the center of attention in the composition (Fig. 2). The dominant is not always the largest element of the composition; it can be the smallest isolated form that creates a plastic conflict. To achieve mass balance, you can “push” rectangles onto each other in a composition. The figure within the boundaries of the “overlap” should be white if the rectangles are black, and vice versa

Pay attention to such a banal moment of work as a signature. Make sure that the signature is written on the reverse side of the sheet and in pencil. In the future, after becoming familiar with the font, you can create your own mark, your own sign, with which everyone will mark their works, including layouts.


Straight lines and organization of space

Straight lines - simple, but very
expressive element:
-line divides the plane into
separate
parts;
-line helps to unite
composition
into a single whole;
-line, to a greater extent than
rectangle
influences rhythmic structure
compositions.

Frontal and deep compositions of lines
and rectangles

even by the simplest means
you can achieve emotional
imagery

The line is not "thinner"
rectangle", and independent
figurative element Line attached
expressiveness of the entire composition. IN
works where the line runs right through (from edge to edge
sheet), she seems to take out
figurative action beyond the boundaries and
makes the composition open, open
and more interesting.
Thin, long and
straight lines are cut
along the line

Working
over
their
compositions,
achieve differences in the size of plans,
because it creates a visual
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 thicknesses achieve harmonious division
space (use straight lines).
2. Create a composition of 2-3 rectangles and 3-4 straight lines
lines that, by their location, connect elements in
a single compositional whole. Create: a) frontal
composition; b) deep composition.
3. Make an interesting one from an arbitrary number of elements
composition.
By rhythmically arranging the elements on the plane, achieve
emotional-figurative impression (for example, “flight”, narrowing”, “slowing down”, etc.).
Assignments can be completed on a computer.