For two straight lines in space, four cases are possible:
Straight lines match;
The lines are parallel (but not the same);
The lines intersect;
The lines intersect, i.e. do not have common points and are not parallel.
Consider two ways of describing lines: canonical equations and general equations. Let the lines L 1 and L 2 be given by the 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 \u003d (z - z 2) / n 2 (6.9)
For each straight line from its canonical equations, we immediately determine a 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 \u003d m 1 / m 2 \u003d n 1 / n 2. (6.10)
If the lines coincide, then the direction vectors are also collinear with the vector M 1 M 2 :
(x 2 - x 1) / l 1 \u003d (y 2 - y 1) / m 1 \u003d (z 2 - z 1) / n 1. (6.11)
This double equality also means that the point M 2 belongs to the line L 1 . Therefore, the condition for the lines to coincide is the fulfillment of 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. The 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 cases out of four, since for Δ ≠ 0 the lines do not belong to the same plane and therefore intersect.
Let's bring all the conditions together:
The mutual arrangement of the lines is characterized by the number of solutions for the system (6.13). If the lines coincide, then the system has infinitely many solutions. If the lines intersect, then this system has a unique solution. There are no direct solutions in the case of parallel or crossing direct solutions. The last two cases can be separated by finding the direction vectors of the lines. To do this, it suffices to calculate two vector works 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 directing vector s 1 of the straight line L 1 is found by the canonical equations of this straight line: s 1 = (1; 3; -2). The directing vector s 2 of the straight line L 2 is calculated using the vector product of the normal vectors of the planes, the intersection of which it is:
Since s 1 \u003d -s 2, then the lines are parallel or coincide. Let us find out which of these situations is realized for given lines. To do this, we substitute the coordinates of the point M 0 (1; 2; -1) ∈ L 1 into the general equations of the line L 2 . For the first of them, we obtain 1 = 0. Therefore, the point M 0 does not belong to the line L 2 and the lines under consideration are parallel.
Angle between lines. The angle between two lines can be found using direction vectors direct. The acute angle between the lines is equal to the angle between their direction vectors (Fig. 6.5) or is complementary to it if the angle between the direction vectors is obtuse. Thus, if for the lines L 1 and L 2 their direction vectors s x and s 2 are known, then the acute angle φ 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 the scalar product in coordinates, we get
Straight lines and organization of space
Straight lines - simple but veryexpressive 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. V
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.
"Sexually transmitted diseases" - Intended for students of medical, pediatric, military medical, dental faculties. The materials are intended for dermatovenereologists, clinical microbiologists, urologists, obstetricians and gynecologists. Addressed to students of all specialties of universities for self-preparation for classes.
"Sexually transmitted diseases" - Sexually transmitted diseases. A patient with syphilis at the 3rd stage of the disease. Chancre. Sexually transmitted diseases (STDs) are traditionally called sexually transmitted diseases. Prevention of sexually transmitted diseases. Symptoms of syphilis Symptoms of secondary syphilis make themselves felt after 6-8 weeks.
"The use of ICT in the educational process" - The main directions of the use of ICT in the educational process. 1) Confidently and regularly use ICT - 30% of teachers. 2) Can do lesson planning using ICT - 60% . 3) Prepare a lesson using ICT by students - 50%. 4) Select software for educational purposes - 60%. 5) Find study materials - 70%. 6) Use of ICT to monitor student development - 40%. 7) The use of ICT for explanation in the classroom - 40%.
"Use of resources" - Directions for improving the Catalog 1. Enlargement of the list of academic disciplines, further gradation into smaller subsections 2. Introduction of additional structuring criteria (for example, combining links to resources by type - simulators, games, etc.), 3. Increase number of references to methodological, technological and technical manuals 4. A more detailed description of teaching methods using educational resources.
"Use of technology" - Radio communication refers to the transmission of information using radio waves - electromagnetic waves, the frequencies of which cover a wide range from 30,000 to 3,000,000,000 Hz. Principles of radio communication. Demodulation is the process of inverse modulation. The use of modern educational technologies in the practice of teaching is a prerequisite for the intellectual, creative and moral development of students.
"Composition" - The main options for splitting the title. Unity. Large heading option. Unlike a line and a strip, a line makes sense, that is, it carries information. 1. The task can be completed in Word or Paint. Any letter or hieroglyph is primarily an image. Form. The dependence of the rhythmic structure on the size of the inter-letter spaces.
https://accounts.google.com
Slides captions:
Presentation on fine arts on the topic: “Straight lines and organization of space” Completed by: art teacher MOBU secondary school No. 1 named after I.D. Buvaltsev Krasnodar Territory 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 starting work, 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 enhances the interconnection of all elements of the composition. Lines help to unite them into a pictorial whole.
The line brings dynamics and adds rhythmic expressiveness to the composition.
Emotional Imagery
Golden bars
The composition consists not only of pictorial elements, but also of the spaces between them. The alternation of pictorial elements and free spaces, their frequency, density and sparseness - this 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. Influence the rhythmic construction of the composition. With their focus, density, intersections, they determine the movement and expression of the entire image.
Achieve differences in the size of plans - this creates pictorial polyphony, intonation richness and, accordingly, greater expressiveness of the composition.
Rhythm and accent plans
ASSIGNMENTS: Straight lines are an element of the organization of a planar composition. 1. Arrange and mutually intersect 3-4 straight lines of different thicknesses to achieve a harmonious division of spaces (use lines for departure). 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. From an arbitrary number of elements, make an interesting composition. Rhythmically arranging the elements on the plane, achieve an emotional-figurative impression (for example, “flight”, “narrowing”, “slowing down”, etc.)
The line is not a “thinner rectangle”, but an independent pictorial element. In works where the line goes out, it seems to take the pictorial action out of the frame and makes the composition open, open and more interesting.
Straight lines are an element of the organization of a planar composition. 1. Arrange and mutually intersect 3-4 straight lines of different thicknesses to achieve a harmonious division of spaces (use lines for departure). 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. From an arbitrary number of elements, make an interesting composition. Rhythmically arranging the elements on the plane, achieve an emotional-figurative impression (for example, “flight”, “narrowing”, “slowing down”, etc.)
Used literature: Textbook for grades 7-8 of educational institutions, edited by B.M. Nemensky, Moscow "Enlightenment" 2008, teacher's work.
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Slides captions:
Presentation on fine arts on the topic: “Fundamentals of composition in the constructive arts. Harmony, contrast and emotional expressiveness of a planar composition "Completed by: art teacher MOBU secondary school No. 1 named after I.D. Buvaltsev Krasnodar Territory Korenovsk Popovich Galina Ivanovna
composition elements Do not be confused by the fact that all exercises are done using rectangles. Firstly, they are quite expressive and, without being distracting by the variety of forms, make it easier to learn compositional techniques. Secondly, they are a prototype of future arrangements 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 selected background). Before finally gluing them, they need to be moved around the sheet in search of the best 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, is an intrigue, a constructive composition just consists in opposition, contrast, the ratio of masses (in this case, rectangles).
PRACTICAL WORK Let's perform exercises to study the principles of balance and movement in a planar composition. Let's choose rectangles as composition elements. Let's bend sheet A4 in half and again in half - 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 fit a black rectangle to it of such a size that 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, gives rise to a feeling of "movement". By introducing additional elements into the composition, you can enhance the feeling of movement, or, on the contrary, “stop” it.
Symmetry The balance of 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 is 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 extends to physical phenomena and the physical laws that govern them. It is symmetry that allows us to cover a wide variety of bodies from a unified position. "Symmetry" in 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 recall from Pushkin in The Queen of Spades: "The old woman's furniture stood in sad symmetry." Asymmetry allows you to achieve dynamism, tension of the composition, without losing the harmony of the whole. When using asymmetry, the composition becomes more expressive, more interesting. With asymmetry, the axis or plane of symmetry is absent (Gaudi table) 14 years
If the symmetrical balanced form is perceived easily and immediately, then the asymmetric dynamic one is read gradually. The balanced, balanced composition of V. Lebedev can be opposed to the dynamic, asymmetric composition of D. Shterenberg
Pieter Cornelis Mondrian is an abstract artist. He devoted his life to the search for balance and balance, having created and led the Style group, which left a bright mark on the history of contemporary art. In his works, he "destroyed" the dynamics. His compositions are perfectly balanced and impeccably balanced. In addition, Mondrian was also the founder of "neoplasticism" - a strict abstract direction 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 served as a priest over canvases, drawing them into rectangles and squares, and painting over the resulting geometric fields either with intense bright colors, or (later) with lighter and transparent shades of white, gray, beige or bluish.
Exercise 3. Symmetry. The white plane is already set. We cut out several black or colored rectangles and perform a symmetrical composition.
Rhythm Among the compositional patterns, one should single out a group of means united by the concept of rhythm. The very word "rhythm" in translation from ancient Greek means "tact" or "proportion". We live in a world of changing rhythms. Put your hand to your chest, listen to the rhythm of the 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 by ear, but also visually. Watch 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 corridors of the school. Rhythm, thanks to the repetition of elements, creates the impression of a conditional movement. The alternation of pictorial elements and free spaces, their frequency, density and sparseness are called rhythm. The rhythm can be calm and restless, directed in one direction or converging to the center, directed both horizontally and vertically. You can alternate elements, volumes, color spots, some details, etc.
Contrasts are the influencing force of the composition and determine its expressiveness. Contrast is a pronounced opposite: long - short, thick - thin, large - small. Contrast is one of the main means of composition. There are contrasts of values, volume and plane, light and shadow (tonal contrasts), warm and cold colors, different textures, etc. Contrasting comparisons sharpen the perception of the whole. The contrast enhances, emphasizes the difference in the properties of the form, makes their unity more intense, impressive. Very strong contrast can visually destroy the compositional structure, so the degree of contrast used is limited by the requirement to preserve the integrity of the impression. In form, proportions, color, contrast emphasizes a pronounced opposite, and nuance carries a barely noticeable transition, shade. Nuance, like contrast, is a way of expressiveness in a composition. Harmony is closely related to expressiveness in the composition, the main task of which is to create the impression of balance, grace and accuracy of the work (El Lissitzky, poster “Beat the Whites with the Red Wedge.” 1920)
Exercise 4. Rhythm. Let's make a rhythmic composition using lines and rectangles, circles and dots. You can complete the task by cutting out rhythmically alternating lines. It is advisable to cut out all the elements of the composition not with scissors, but with a breadboard knife.
Static frontal composition A static frontal composition or a more dynamic depth composition should be based on the difference in the sizes of the 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. As the balance of masses is achieved, it is possible to “run into” each other with rectangles in the composition. The shape within the "zoom" boundaries must 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 on the back of the sheet and in pencil. In the future, after becoming familiar with the font, you can create your own brand, your own sign, with which everyone will mark their work, including layouts.
