Points M and M 1 are called symmetric about a given straight line Lif this straight line is the perpendicular to the segment MM 1 (Figure 1). Each point of a straight line L symmetrical to itself. Plane transformation, in which each point is mapped to a point symmetrical to it relative to a given line Lis called axially symmetrical with the L-axis and denoted S L : S L (M) \u003d M 1 .

Points M and M 1 are mutually symmetric with respect to L, so S L (M 1 ) \u003d M... Therefore, the transformation inverse to axial symmetry is the same axial symmetry: S L -1 \u003d S L , S L ° S L \u003d E... In other words, the axial symmetry of the plane is involutive transformation.

The image of a given point in axial symmetry can be simply built using only one compass. Let be L - axis of symmetry, Aand B - arbitrary points of this axis (Fig. 2). If S L (M) \u003d M 1, then, by the property of the points of the perpendicular to the segment, we have: AM \u003d AM 1 and BM \u003d BM 1 . So the point M 1 belongs to two circles: a circle with a center A radius AM and circles with center B radius BM (M - given point). Figure F and her image F 1 with axial symmetry are called symmetrical figures relative to a straight line L (Figure 3).

Theorem. Axial symmetry of a plane is motion.

If a AND and IN - any points of the plane and S L (A) \u003d A 1 , S L (B) \u003d B 1, then it is necessary to prove that A 1 B 1 \u003d AB... To do this, we introduce a rectangular coordinate system OXY so that the axis OX coincided with the axis of symmetry. Points AND and IN have coordinates A (x 1 , -y 1 ) and B (x 1 , -y 2 ) .Dots AND 1 and IN 1 have coordinates A 1 (x 1 , y 1 ) and B 1 (x 1 , y 2 ) (Figure 4 - 8). Using the formula for the distance between two points, we find:

It is clear from these relations that AB \u003d A 1 IN 1, as required.

Comparing the orientations of the triangle and its image, we find that the axial symmetry of the plane is movement of the second kind.

Axial symmetry maps each line to a line. In particular, each of the straight lines perpendicular to the axis of symmetry is mapped by this symmetry onto itself.

Theorem. A straight line other than the perpendicular to the axis of symmetry and its image at this symmetry intersect on the axis of symmetry or are parallel to it.

Evidence. Let a straight line not perpendicular to the axis be given L symmetry. If a m? L \u003d P and S L (m) \u003d m 1, then m 1 ? m and S L (P) \u003d P, so Pm1 (Figure 9). If m || L then m 1 || Lsince otherwise the straight lines m and m 1 would intersect at a point on a straight line L, which contradicts the condition m || L (Figure 10).

By virtue of the definition of equal figures, straight lines symmetrical about a straight line L, form with a straight line L equal angles (Figure 9).

Straight L called the axis of symmetry of the figure Fif symmetry with the axis L figure Fdisplayed on itself: S L (F) \u003d F... They say that the figure F symmetrical about a straight line L.

For example, any straight line containing the center of a circle is the axis of symmetry of this circle. Indeed, let M - arbitrary point of the circle u centered ABOUT, OL, S L (M) \u003d M 1 . Then S L (O) \u003d O and OM 1 \u003d OM, i.e. M 1 є u... So, the image of any point of the circle belongs to this circle. Hence, S L (u) \u003d u.

The symmetry axes of a pair of nonparallel straight lines are two perpendicular straight lines containing the bisectors of the angles between these straight lines. The axis of symmetry of a line segment is the line containing it, as well as the perpendicular to this segment.

Axial symmetry properties

• 1. With axial symmetry, the image of a straight line is a straight line, the image of parallel lines is parallel straight lines
• 3. Axial symmetry preserves a simple ratio of three points.
• 3. In case of axial symmetry, the segment turns into a segment, a ray - into a ray, a half-plane - into a half-plane.
• 4. With axial symmetry, the angle transforms into an equal angle.
• 5. In case of axial symmetry with the d axis, any straight line perpendicular to the d axis remains in place.
• 6. Under axial symmetry, the orthonormal frame is transformed into an orthonormal frame. In this case, point M with coordinates x and y relative to the frame R goes to point M` with the same coordinates x and y, but relative to the frame R`.
• 7. Axial symmetry of the plane transforms the right orthonormal frame to the left and, conversely, the left orthonormal frame to the right.
• 8. Composition of two axial symmetries of a plane with parallel axes is a parallel transfer to a vector perpendicular to these straight lines, the length of which is twice the distance between these straight lines

You will need

• - properties of symmetric points;
• - properties of symmetrical figures;
• - ruler;
• - square;
• - compasses;
• - pencil;
• - paper;
• - a computer with a graphic editor.

Instructions

Draw a straight line a, which will be the axis of symmetry. If its coordinates are not specified, draw it freely. On one side of this line, put an arbitrary point A. You need to find a symmetrical point.

Useful advice

Symmetry properties are constantly used in AutoCAD. For this, the Mirror option is used. To build an isosceles triangle or an isosceles trapezoid, it is enough to draw the lower base and the angle between it and the side. Flip them with the command indicated and extend the sides as needed. In the case of a triangle, this will be the point of their intersection, and for a trapezoid, a given value.

You constantly encounter symmetry in graphics editors when you use the "flip vertically / horizontally" option. In this case, the axis of symmetry is taken as a straight line corresponding to one of the vertical or horizontal sides of the picture frame.

Sources:

• how to draw central symmetry

The construction of the section of the cone is not such a difficult task. The main thing is to follow a strict sequence of actions. Then this task will be easily accomplished and will not require much labor from you.

You will need

• - paper;
• - a pen;
• - circus;
• - ruler.

Instructions

When answering this question, you first need to decide what parameters the section is given.
Let it be the line of intersection of the plane l with the plane and point O, which is the point of intersection with its section.

The construction is illustrated in Fig. 1. The first step in constructing a section is through the center of the section of its diameter, extended to l perpendicular to this line. As a result, point L is obtained. Then, through T.O, draw a straight line LW, and construct two guide cones lying in the main section O2M and O2C. At the intersection of these guides lies the point Q, as well as the already shown point W. These are the first two points of the desired section.

Now draw at the base of the cone BB1 \u200b\u200bperpendicular to the MC and construct the generators of the perpendicular section О2В and О2В1. In this section, through T.O, draw a straight line RG parallel to BB1. T.R and T.G - two more points of the desired section. If the cross-section of the ball is known, then it could be built already at this stage. However, this is not an ellipse at all, but something elliptical, with symmetry about the segment QW. Therefore, you should build as many points of the section as possible in order to connect them in the future with a smooth curve to obtain the most reliable sketch.

Draw an arbitrary section point. To do this, draw an arbitrary diameter AN at the base of the cone and draw the corresponding guides O2A and O2N. Through T.O, draw a straight line passing through PQ and WG, until it intersects with the just drawn guides at points P and E. These are two more points of the desired section. Continuing the same way and further, you can arbitrarily desired points.

True, the procedure for obtaining them can be slightly simplified using the symmetry with respect to QW. To do this, you can draw straight lines SS 'in the plane of the desired section, parallel to RG until they intersect with the surface of the cone. The construction is completed by rounding the constructed polyline from chords. It suffices to construct half of the desired section due to the already mentioned symmetry with respect to QW.

Related Videos

Tip 3: How to graph a trigonometric function

You need to draw schedule trigonometric function? Master the algorithm of actions using the example of building a sinusoid. To solve this problem, use the research method.

You will need

• - ruler;
• - pencil;
• - knowledge of the basics of trigonometry.

Instructions

Related Videos

note

If two semiaxes of a single-strip hyperboloid are equal, then the figure can be obtained by rotating the hyperbola with semiaxes, one of which is the above, and the other, different from two equal, around the imaginary axis.

Useful advice

When considering this figure with respect to the Oxz and Oyz axes, it is seen that its main sections are hyperbolas. And when a given spatial figure of rotation is cut by the Oxy plane, its section is an ellipse. The throat ellipse of a single-strip hyperboloid passes through the origin, since z \u003d 0.

The throat ellipse is described by the equation x² / a² + y² / b² \u003d 1, and the other ellipses are made up by the equation x² / a² + y² / b² \u003d 1 + h² / c².

Sources:

• Ellipsoids, paraboloids, hyperboloids. Straight generators

The shape of the five-pointed star has been widely used by humans since ancient times. We consider its form to be beautiful, since we unconsciously distinguish the ratio of the golden section in it, i.e. the beauty of the five-pointed star is mathematically based. Euclid was the first to describe the construction of the five-pointed star in his "Elements". Let's share his experience.

You will need

• ruler;
• pencil;
• compass;
• protractor.

Instructions

Building a star is reduced to building with the subsequent connection of its vertices with each other sequentially through one. In order to build the correct one, you need to break the circle into five.
Construct an arbitrary circle using a compass. Mark its center with O.

Mark point A and use the ruler to draw line segment OA. Now it is necessary to divide the segment OA in half, for this, draw an arc from point A with the radius OA until it intersects with the circle at two points M and N. Construct segment MN. Point E, at which MN intersects OA, will divide the segment OA in half.

Restore the OD perpendicular to radius OA and connect point D and E. Resection B at OA from point E with radius ED.

Now, using line segment DB, mark the circle into five equal parts. Designate the vertices of the regular pentagon sequentially with numbers from 1 to 5. Connect the points in the following sequence: 1 with 3, 2 with 4, 3 with 5, 4 with 1, 5 with 2. Here is a regular five-pointed star, in a regular pentagon. This is how he built

I ... Symmetry in mathematics :

Basic concepts and definitions.

Axial symmetry (definitions, construction plan, examples)

Central symmetry (definitions, construction plan, formeasures)

Summary table (all properties, features)

II ... Symmetry Applications:

1) in mathematics

2) in chemistry

3) in biology, botany and zoology

4) in art, literature and architecture

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1. Basic concepts of symmetry and its types.

Symmetry concept n rgoes through the entire history of mankind. It is found already at the origins of human knowledge. It arose in connection with the study of a living organism, namely a person. And it was used by sculptors as early as the 5th century BC. e. The word "symmetry" is Greek, it means "proportionality, proportionality, uniformity in the arrangement of parts." It is widely used by all areas of modern science without exception. Many great people thought about this pattern. For example, LN Tolstoy said: “Standing in front of a black board and drawing different figures on it with chalk, I was suddenly struck by the thought: why is symmetry clear to the eye? What is symmetry? This is an innate feeling, I answered myself. What is it based on? " The symmetry is indeed pleasing to the eye. Who has not admired the symmetry of nature's creations: leaves, flowers, birds, animals; or human creations: buildings, technology, - everything that surrounds us from childhood, those that strive for beauty and harmony. Hermann Weil said: "Symmetry is the idea through which man has been trying for centuries to comprehend and create order, beauty and perfection." Hermann Weil is a German mathematician. His activity falls on the first half of the twentieth century. It was he who formulated the definition of symmetry, established by what criteria to perceive the presence or, conversely, the absence of symmetry in one or another case. Thus, a mathematically rigorous concept was formed relatively recently - at the beginning of the 20th century. It's quite complicated. We will turn and once again remember the definitions given to us in the textbook.

2. Axial symmetry.

2.1 Basic definitions

Definition. Two points A and A 1 are called symmetric with respect to the straight line a if this straight line passes through the middle of the segment AA 1 and is perpendicular to it. Each point of the straight line a is considered symmetrical to itself.

Definition. The figure is called symmetrical about a straight line. andif for each point of the figure a point symmetrical to it with respect to a straight line and also belongs to this figure. Straight and called the axis of symmetry of the figure. The figure is also said to have axial symmetry.

2.2 Building plan

And so, to build a symmetrical figure relative to a straight line from each point, we draw a perpendicular to this straight line and extend it by the same distance, mark the resulting point. We do this with each point, we get symmetrical vertices of the new figure. Then we connect them in series and get a symmetrical figure of the given relative axis.

2.3 Examples of axially symmetric figures.

3. Central symmetry

3.1 Basic definitions

Definition. Two points A and A 1 are called symmetric with respect to point O if O is the midpoint of the segment AA 1. Point O is considered symmetrical to itself.

Definition. A figure is called symmetric about point O if for each point of the figure the point symmetric to it about point O also belongs to this figure.

3.2 Building plan

Construction of a triangle symmetrical to a given one about the center O.

To draw a point symmetrical to a point ANDrelative to point ABOUT, it is enough to draw a straight line OA(fig. 46 ) and on the other side of the point ABOUTpostpone a segment equal to a segment OA. In other words , points A and ; In and ; With and are symmetric with respect to some point O. In Fig. 46 built a triangle symmetrical to a triangle ABC relative to point ABOUT.These triangles are equal.

Draws symmetrical points about the center.

In the figure, points M and M 1, N and N 1 are symmetric about point O, and points P and Q are not symmetric about this point.

In general, figures symmetrical about some point are equal .

3.3 Examples

Here are some examples of figures with central symmetry. The simplest figures with central symmetry are the circle and the parallelogram.

Point O is called the center of symmetry of the figure. In such cases, the figure has central symmetry. The center of symmetry of a circle is the center of the circle, and the center of symmetry of a parallelogram is the point of intersection of its diagonals.

The straight line also has central symmetry, however, unlike the circle and the parallelogram, which have only one center of symmetry (point O in the figure), the straight line has infinitely many of them - any point of the straight line is its center of symmetry.

The figures show an angle symmetrical about the vertex, a segment symmetrical to another segment about the center AND and a quadrilateral symmetric about its vertex M.

An example of a shape that does not have a center of symmetry is a triangle.

4. Lesson summary

Let's summarize the knowledge gained. Today in the lesson we got acquainted with two main types of symmetry: central and axial. Let's look at the screen and systematize the knowledge gained.

Summary table

	Axial symmetry	Central symmetry
Feature	All points of the figure must be symmetrical about some straight line.	All points of the figure must be symmetrical about the point selected as the center of symmetry.
Properties	1. Symmetrical points lie on perpendiculars to a straight line. 3. Straight lines turn into straight lines, angles into equal angles. 4. Sizes and shapes of figures are saved.	1. Symmetrical points lie on a straight line passing through the center and the given point of the figure. 2. The distance from a point to a straight line is equal to the distance from a straight line to a symmetrical point. 3. Sizes and shapes of figures are saved.

II. Applying symmetry

© Sukhacheva Elena Vladimirovna, 2008-2009.

How many different axes of symmetry any triangle can have depends on its geometric shape. If this is an equilateral triangle, then it will have as many as three axes of symmetry at once.

And if it is an isosceles triangle, it will have only one axis of symmetry.

The sister's son is just taking this topic at school in geometry lessons. The axis of symmetry is a straight line, when rotated around which by a specific angle, a symmetrical figure will take the same position in space that it occupied before the rotation, and the same others will replace some of its parts. In an isosceles triangle - three, in a right-angled one - one, in the rest - not, since their sides are not equal to each other.

It depends on which triangle. An equilateral triangle has three axes of symmetry that pass through three of its vertices. An isosceles triangle, respectively, has one axis of symmetry. The rest of the triangles, the axis of symmetry do not have.



The simplest thing to remember is that an equilateral triangle has three sides equal and it has three axes of symmetry

This makes it easier to remember the following

There are no equal sides, that is, all sides are different, which means there are no axes of symmetry

And there is only one axis in an isosceles triangle



You cannot simply answer how many axes of symmetry a triangle has without understanding which triangle we are talking about.

An equilateral triangle has three axes of symmetry, respectively.

An isosceles triangle has only one axis of symmetry.

Any other triangles with sides of different lengths do not have any axis of symmetry at all.

A triangle with all sides different in size has no axes of symmetry.

A right-angled triangle can have one axis of symmetry if its legs are equal.

In a triangle in which two sides are equal (isosceles), you can draw one axis, and in which all three sides are equal (equilateral) - three.

Before answering the question of how many axes of symmetry a triangle has, you first need to remember in general what an axis of symmetry is.

So, speaking simply, in geometry, the axis of symmetry is a line, along which a figure is bent, we get the same halves.

but it is worth remembering that triangles are also different.

So, isosceles a triangle (a triangle with two equal sides) has one axis of symmetry.

Equilateral the triangle, respectively, has 3 axes of symmetry, since all sides of this triangle are equal.

And here versatile a triangle has no axes of symmetry at all. Even if you fold it and even where you draw straight lines, since the sides are different, then two identical halves will not work.



As far as I remember geometry, an equilateral triangle has three axes of symmetry passing through its vertices, these are its bisectors. A right-angled triangle, like a versatile, obtuse and acute-angled triangles, has no axes of symmetry at all, but an isosceles triangle has only one.

And it's easy to check - just imagine a line along which it can be cut in two so as to get two identical triangles.

Since triangles are different, their axes of symmetry are, respectively, in different quantities. For example, a triangle with different sides without any axes of symmetry at all. And the equilateral has three. There is another type of triangle that has one axis of symmetry. It has two sides equal, and one right angle.

An arbitrary triangle has no axes of symmetry. An isosceles triangle has one axis of symmetry - this is the median to the single side. An equilateral triangle has three axes of symmetry - these are its three medians.

Today we will talk about a phenomenon that each of us constantly encounters in life: about symmetry. What is symmetry?

Approximately we all understand the meaning of this term. The dictionary says: symmetry is proportionality and full correspondence of the arrangement of parts of something relative to a straight line or point. Symmetry is of two types: axial and radial. Let's consider axial first. This is, let's say, "mirror" symmetry, when one half of the object is completely identical to the second, but repeats it as a reflection. Look at the halves of the sheet. They are mirror-symmetrical. The halves of the human body (full face) are also symmetrical - the same arms and legs, the same eyes. But let's not be mistaken, in fact, in the organic (living) world, you cannot find absolute symmetry! The halves of the leaf copy each other far from perfect, the same applies to the human body (take a closer look); it is the same with other organisms! By the way, it should be added that any symmetrical body is symmetrical with respect to the viewer in only one position. It is worth, say, turning the sheet, or raising one hand, and what? - you see for yourself.

People achieve true symmetry in the works of their labor (things) - clothes, cars ... In nature, it is characteristic of inorganic formations, for example, crystals.

But let's get down to practice. It is not worth starting with complex objects like people and animals; as the first exercise in a new field, we will try to finish the mirror half of the sheet.

How to draw a symmetrical object - lesson 1

We make sure that it turns out as similar as possible. For this we will literally build our soul mate. Do not think that it is so easy, especially the first time, to draw a mirror-corresponding line with one stroke!

Let's mark some anchor points for the future symmetrical line. We proceed as follows: we draw several perpendiculars to the axis of symmetry - the midrib of the leaf with a pencil without pressing. Four or five is enough for now. And on these perpendiculars we measure to the right the same distance as on the left half to the line of the edge of the leaf. I advise you to use a ruler, don't rely on the eye too much. We, as a rule, tend to reduce the drawing - it has been noticed from experience. We do not recommend measuring distances with your fingers: the error is too large.

We connect the resulting points with a pencil line:

Now we are meticulously looking - are the halves really the same. If everything is correct, we will circle it with a felt-tip pen, we will clarify our line:

The poplar leaf has been completed, now you can swing at the oak one.

How to draw a symmetrical shape - lesson 2

In this case, the difficulty lies in the fact that the veins are indicated and they are not perpendicular to the axis of symmetry and you will have to not only the dimensions, but also the angle of inclination exactly. Well, we train the eye:

So a symmetrical oak leaf was drawn, or rather, we built it according to all the rules:

How to draw a symmetrical object - lesson 3

And let's fix the theme - draw a symmetrical lilac leaf.

He also has an interesting shape - heart-shaped and with ears at the base you will have to pant:

So they drew:

Take a look at the resulting work from a distance and see how accurately we managed to convey the required similarity. Here's a tip: take a look at your image in the mirror and it will tell you if there are any mistakes. Another way: bend the image exactly along the axis (we have already learned how to bend it correctly) and cut the leaf along the original line. Look at the figure itself and at the cut paper.