Dividing a circle into equal parts using a compass and straightedge. Dividing a circle into equal parts

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DIVISION OF A CIRCLE INTO EQUAL PARTS

Some parts of machines and devices have elements evenly spaced around the circumference, for example, the parts in Fig. 52-59. When making drawings of such parts, you need to know the rules for dividing a circle into an equal number of parts.

Division of a circle into four and eight equal parts. On fig. 52, a shows a cover that has eight holes evenly spaced around the circumference. When constructing a drawing of the contour of the cover (Fig. 52 G) Divide the circle into eight equal parts. This can be done using a square with angles of 45 ° (Fig. 52, c), the hypotenuse of the square must pass through the center of the circle, or by construction.

Two mutually perpendicular diameters of a circle divide it into four equal parts (points 7, 3, 5, 7 in Fig. 52, b). To divide a circle into eight equal parts, the well-known technique of dividing a right angle with a compass into two equal parts is used. Get points 2, 4, 6, 8.

Division of a circle into three, six and twelve equal parts. In the flange (Fig. 53, a) there are three holes evenly spaced around the circumference. When drawing the contour of the flange (Fig. 53, d), it is necessary to divide the circle into three equal parts.

To find points dividing a circle of radius R into three equal parts, enough from any point on the circle, for example, a point BUT, draw an arc with a radius R . The intersections of the arc with the circle give the two desired points 2 and 3; the third point of division will be located at the intersection of the axis of the circle drawn from the point L with the circle (Fig. 53, b).

You can also divide the circle into three equal parts with a square with angles of 30 and 60 ° (Fig. 53, c), the hypotenuse of the square must pass through the center of the circle.

On fig. 54, b shows the division of the circle by a compass into six equal parts. In this case, the same construction is performed as in Fig. 53, b but the arc is described not once, but twice, from points and radius R equal to the radius of the circle.

You can also divide the circle into six equal parts with a square with angles of 30 and 60 ° (Fig. 54, c). On fig. 54, a a cover is shown, when drawing which it is necessary to divide the circle into six parts.

To make a drawing of a part (Fig. 55, a), which has 12 holes evenly spaced along the circles, you need to divide the axial circle into 12 equal parts (Fig. 55, d).

When dividing a circle into 12 equal parts using a compass, you can use the same technique as when dividing a circle into six equal parts (Fig. 54, b), but arcs with a radius R describe four times from points 1, 7, 4 and 10 (Fig. 55, b).

Using a square with angles of 30 and 60 °, followed by turning it by 180 °, divide the circle into 12 equal parts (Fig. 55, in).

Division of a circle into five, ten and seven equal parts. In the die (Fig. 56, a) there are five holes evenly spaced around the circumference. When drawing a die (Fig. 56, c), it is necessary to divide the circle into five equal parts. Through the intended center O (Fig. 56, b)

with the help of a T-square and a square, axial lines are drawn and from point O they describe a circle of a given diameter with a compass. From point A with radius R equal to the radius of the given circle, an arc is drawn that intersects the circle at point n. From point n, a perpendicular is lowered to the horizontal center line, point C is obtained. From point C with radius R 1 equal to the distance from point C to point 1, an arc is drawn that intersects the horizontal center line at point t. From point 1 with radius R equal to the distance from point 1 to point m, draw an arc that intersects the circle at point 2. Arc 12 is 1/5 of the circumference. Points 3,4 and 5 are found by setting aside segments equal to m1 with a compass.

Detail "asterisk" (Fig. 57, a) has 10 identical elements evenly spaced around the circumference. To draw an asterisk (Fig. 57, i), the circle should be divided into 10 equal parts. In this case, the same construction should be applied as when dividing the circle into five parts (see Fig. 56, b). Line segment p 1 will be equal to the chord that divides the circle into 10 equal parts.

On fig. 58, a a pulley is shown, and in fig. 58, in- a drawing of a pulley, where the circle is divided into seven equal parts.

The division of the circle into seven equal parts is shown in Fig. 58b. From a point BUT an auxiliary arc is drawn with a radius R, equal to the radius of the given circle that intersects the circle at a point. From a point n lower the perpendicular to the horizontal center line. From a point 1 radius equal to the segment nc, make seven serifs around the circumference and get seven desired points.

Division of a circle into any number of equal parts. With sufficient accuracy, you can divide the circle into any number of equal parts, using the table of coefficients for calculating the length of the chord (Table 9).

Knowing how many (n) it is necessary to divide the circle, find the coefficient from the table. When multiplying the coefficient k by the diameter of the circle D, the length of the chord l is obtained, which is plotted with a compass on the circle n once.

When constructing a drawing of a ring (Fig. 59, a) it is necessary to divide a circle with a diameter of D \u003d 142 mm into 32 equal parts. The number of parts of the circle n=32 corresponds to the coefficient k=0.098. Calculate the length of the chord l= Dk= 142x0.098 \u003d 13.9 mm, it is laid with a compass on a circle 32 times (Fig. 59, b and in).

And the construction of regular inscribed polygons

Dividing the circle into 3, 6 and 12 equal parts. Construction of a regular inscribed triangle, hexagon and dodecagon.

To construct a regular inscribed triangle, it is necessary from a point BUT the intersection of the center line with the circle set aside a size equal to the radius R, to one side and the other. We get vertices 1 and 2( rice. 26, a). Vertex 3 lies on the opposite point BUT end of diameter.

1/3 1/6 1/12

a B C)

Rice. 26

The side of the hexagon is equal to the radius of the circle. The division into 6 parts is shown in fig. 26, b.

In order to divide the circle into 12 parts, it is necessary to set aside a size equal to the radius on the circles in one direction and the other from four centers (Fig. 26, in).

Dividing the circle into 4 and 8

inscribed quadrilateral and octagon.

Rice. 27

The circle is divided into 4 parts by two mutually perpendicular center lines. To divide into 8 parts, an arc equal to a quarter of a circle must be divided in half ( Fig.27.)

Dividing the circle into 5 and 10 equal parts. Building the right

inscribed pentagon and decagon.

1/5 1/10

a) b)

Rice. 28

Half of any diameter (radius) is divided in half ( rice. 28, a), get a point N. From a point N, as from the center, draw an arc with a radius R1, equal to the distance from the point N to the point BUT, until it intersects with the second half of this diameter, at the point R. Line segment AR equal to a chord subtending an arc whose length is 1/5 of the circumference. Making serifs on a circle with a radius R2, equal to the segment AR, divide the circle into five equal parts. The starting point is chosen depending on the location of the pentagon. ( ! It is impossible to perform serifs in one direction, since errors occur and the last side of the pentagon turns out to be skewed.)

The division of a circle into 10 equal parts is performed similarly to the division of a circle into five equal parts ( rice. 28b), but first divide the circle into five parts, starting construction from point A, and then from point B, located at the opposite end of the diameter. Can be used to draw a segment OR- the length of which is equal to the chord 1/10 of the circumference.

Dividing the circle into 7 equal parts.

1/7

a B C)

Rice. 29

From anywhere (eg. BUT) circles, with a radius of a given circle, draw an arc until it intersects with a circle at points AT and D (Fig. 29, a). By connecting the dots AT and D straight, get a cut sun, equal to the chord that subtends an arc that is 1/7 of the circumference. Serifs are performed in the sequence indicated on rice. 29 b.

Pairings

Often in the design of parts, one surface passes into another. Usually these transitions are made smooth, which increases the strength of the parts and makes them more convenient to work with. Pairing is a smooth transition from one line to another. The construction of conjugations comes down to three points: 1) determining the center of conjugation; 2) finding junction points; 3) construction of an arc of conjugation of a given radius. To build a mate, the mate radius is most often specified. The center and junction point are defined graphically.

Division of a circle into 3 equal parts.

To divide a circle of radius R into 3 equal parts and inscribe an equilateral triangle into it, from the point of intersection of the diameter with the circle (for example, from point A), an additional arc of radius R is described as from the center. Points 2 and 3 are obtained. Points 1, 2, 3 divide circle into three equal parts. By connecting straight lines points 1, 2, 3 build an inscribed equilateral triangle.

Division of a circle into 6 equal parts.

To divide the circle into 6 equal parts, two arcs of radius R are drawn from two opposite points (1 and 4) of the intersection of the diameter with the circle. Points (2, 3, 5, 6) are obtained. Together with the points that were obtained at the intersection of the diameter with the circle, he divides the circle into 6 equal parts.

Dividing a circle into 12 equal parts.

To divide the circle into 12 equal parts from the four points of intersection of the axes of symmetry with the circle, 4 arcs of radius R are described. The points obtained, together with those obtained by crossing the axes of symmetry with the circle, divide the circle into 12 equal parts.

Types of section designations in drawings

To show the transverse shape of parts, use images called sections (Fig. 13). In order to obtain a section, the part is mentally dissected by an imaginary cutting plane in the place where its shape needs to be revealed. The figure obtained as a result of cutting the part with a cutting plane is depicted in the drawing. Hence a section is an image of a figure obtained by mentally dissecting an object by a plane or several planes.

The section shows only what is obtained directly in the cutting plane.

For clarity of the drawing, the sections are highlighted with hatching. Inclined parallel hatching lines are drawn at an angle of 45 ° to the lines of the drawing frame, and if they coincide in direction with the contour lines or center lines, then at an angle of 30 ° or 60 °.

Exposed section.

The contour of the rendered section is outlined with a solid thick line of the same thickness as the line adopted for the visible contour of the image. If the section is taken out, then, as a rule, an open line is drawn, two thickened strokes, and arrows indicating the direction of view. From the outside of the arrows, the same capital letters are applied. Above the section, the same letters are written through a dash with a thin line below. If the section is a symmetrical figure and is located on the continuation of the section line (dash-dotted line), then no designations are applied.

Superimposed section.

The contour of the superimposed section is a solid thin line (S/2 - S/3), and the contour of the view at the location of the superimposed section is not interrupted. The superimposed section is usually not indicated. But if the section is not a symmetrical figure, strokes of an open line and arrows are drawn, but letters are not applied.

Section designation

The position of the cutting plane is indicated in the drawing by a section line - an open line, which is drawn in the form of separate strokes that do not intersect the contour of the corresponding image. The thickness of the strokes is taken in the range from $ to 1 1/2 S, and their length is from 8 to 20 mm. On the initial and final strokes, perpendicular to them, at a distance of 2-3 mm from the end of the stroke, put arrows indicating the direction of view. At the beginning and end of the section line, they put the same capital letter of the Russian alphabet. The letters are applied near the arrows indicating the direction of view from the outside, fig. 12. Above the section, an inscription is made according to the type A-A. If the section is in a gap between parts of the same type, then with a symmetrical figure, the section line does not pass R4. The section can be rotated, then the symbol A-A must be added to the inscription

turned O, that is, A-AO.

With the help of a compass and straightedge, it is possible to divide a circle into more than any number of parts. Mathematicians have proven that it is possible to divide into 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, ..., 257, ... parts, but not into 7, 9, 11, 13, 14, ... parts .

Unfortunately, there is no single way to divide. Let's take a look at the most important ones.

1) Division of the circle into 6, 3, 12, 24, …, 3×2 k (k=0,1,2,3,…) equal parts.

Starting with dividing the circle into 6 parts. To do this, with the same solution of the compass with which the circle was drawn, from any point on the circle, as from the center, it is necessary to draw a circle. Then repeat the procedure, taking as the center the point of intersection of the initial and new circles.

To divide a circle into 3 parts, you need to divide it into 6 parts and take points through one (Fig. 5a). To divide a circle into 12 parts, you need to divide it into 6 parts and divide each arc in half, then the process of dividing the arcs in half can be continued indefinitely.

The length of the perpendicular dropped from the center of the circle to the side of the hexagon is a good approximation for the length of the side of the heptagon inscribed in the circle (shown in shading in Figure 5a). Perpendicular length ≈0.866R, heptagon side length ≈0.868R – accuracy ≈2%.

2) Division of the circle into 2, 4, 8, 16,…, 2 k (k=1,2,3,…) equal parts.

You can divide the circle into 2 parts using a ruler by drawing a straight line through the center of the circle. But it is possible to postpone the radius of the circle from any point of the circle 3 times. The start and end points bisect the circle (a diameter can be drawn through them - Fig. 5a). To divide the circle into 4 parts, it is necessary to divide the resulting arcs in half. Consistent execution of the division of the resulting arcs in half ensures the division of the circle into 8, 16, etc. parts.

3) Division of the circle into 5 parts.

The construction method adopted in drawing uses the ratio between the side of a regular decagon ( a 10) and a regular pentagon ( a 5)- a 5 2 = R 2 + a 10 2 . The construction is carried out as follows. Let's draw 2 perpendicular lines through the center of the circle O. A and B are the points of their intersection with the circle. From point A, as from the center, we draw a circle of the same radius (we find the middle of the segment AO - point C). From the middle of segment AO of point C, we draw another circle of radius CB. Segment BE is equal to the side of the pentagon, OE is equal to the decagon (Fig. 5b).

You can divide the circle into 5 and 10 parts in the way shown in Figure 5c. Segment BC is the side of the pentagon, AC is the side of the decagon. About the remarkable properties of the pentagon and decagon and why the construction method shown in Figure 5c is correct, we will tell in the next chapter.

Madrasah Kukeldash (XVI century, Tashkent)

Figure 5d demonstrates the reception of an approximate geometric solution to the problem of dividing a circle into any number of parts. Let, for example, it is required to divide the given circle into 7 equal parts. We construct an equilateral triangle ABC on the diameter of the circle AB and divide the diameter AB by the point D in relation to AD:AB=2:7 (generally 2:n). To do this, you need to draw an auxiliary line, set aside n + 2 identical segments on it, connect the extreme point with point B and draw a line parallel to line BF through the second point. Draw a line DC to the intersection with the circle. The arc AE will be the 7th part of the circle (in the general case, the nth). This method for n<11 дает погрешность не более 1%.

Algorithms for dividing a circle into equal parts can be used, for example, to construct reference points for spirals - the Archimedes spiral, named after the great ancient Greek scientist Archimedes (III century BC), who first studied this line, and the logarithmic spiral.

Today in the post I post several pictures of ships and diagrams for them for embroidery with isothread (pictures are clickable).

Initially, the second sailboat was made on carnations. And since the carnation has a certain thickness, it turns out that two threads depart from each. Plus, layering one sail on the second. As a result, a certain effect of splitting the image appears in the eyes. If you embroider the ship on cardboard, I think it will look more attractive.
The second and third boats are somewhat easier to embroider than the first. Each of the sails has a central point (on the underside of the sail) from which rays extend to points along the perimeter of the sail.
Joke:
- Do you have threads?
- There is.
- And the harsh ones?
- It's just a nightmare! I'm afraid to come!

My first debut Master Class. Hopefully not the last. We will embroider a peacock. Product diagram.When marking the places of punctures, pay special attention so that they are in closed contours even number.The basis of the picture is dense cardboard(I took brown with a density of 300 g / m2, you can try it on black, then the colors will look even brighter), better dyed on both sides(for the people of Kiev - I took it in the stationery department at the Central Department Store on Khreshchatyk). Threads- floss (of any manufacturer, I had DMC), in one thread, i.e. we unwind the bundles into individual fibers. Embroidery consists of three layers thread. At first we embroider the first layer in feathers on the peacock's head, the wing (light blue thread color), as well as dark blue circles of the tail using the flooring method. The first layer of the body is embroidered with chords with variable pitch, trying to make the threads run tangentially to the contour of the wing. Then we embroider twigs (serpentine seam, mustard-colored threads), leaves (first dark green, then the rest ...