When performing graphic work, you have to solve many construction tasks. The most common tasks in this case are the division of line segments, angles and circles into equal parts, the construction of various conjugations.
Dividing a circle into equal parts using a compass
Using the radius, it is easy to divide the circle into 3, 5, 6, 7, 8, 12 equal sections.
Division of a circle into four equal parts.
Dash-dotted center lines drawn perpendicular to one another divide the circle into four equal parts. Consistently connecting their ends, we get a regular quadrilateral(Fig. 1) .
Fig.1 Division of a circle into 4 equal parts.
Division of a circle into eight equal parts.
To divide a circle into eight equal parts, arcs equal to the fourth part of the circle are divided in half. To do this, from two points limiting a quarter of the arc, as from the centers of the radii of the circle, notches are made outside it. The points obtained are connected to the center of the circles and at their intersection with the line of the circle, points are obtained that divide the quarter sections in half, i.e., eight equal sections of the circle are obtained (Fig. 2 ).
Fig.2. Division of a circle into 8 equal parts.
Division of a circle into sixteen equal parts.
Dividing an arc equal to 1/8 into two equal parts with a compass, we will put serifs on the circle. Connecting all serifs with straight line segments, we get a regular hexagon.
Fig.3. Division of a circle into 16 equal parts.
Division of a circle into three equal parts.
To divide a circle of radius R into 3 equal parts, from the point of intersection of the center line 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 the circle into three equal parts.
Rice. 4. Division of a circle into 3 equal parts.
Division of a circle into six equal parts. The side of a regular hexagon inscribed in a circle is equal to the radius of the circle (Fig. 5.).
To divide a circle into six equal parts, it is necessary from points 1 And 4 intersection of the center line with the circle, make two serifs on the circle with a radius R equal to the radius of the circle. Connecting the obtained points with line segments, we get a regular hexagon.
Rice. 5. Dividing the circle into 6 equal parts
Division of a circle into twelve equal parts.
To divide a circle into twelve equal parts, it is necessary to divide the circle into four parts with mutually perpendicular diameters. Taking the points of intersection of the diameters with the circle A , IN, WITH, D beyond the centers, four arcs are drawn by the radius to the intersection with the circle. Received points 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 and points A , IN, WITH, D divide the circle into twelve equal parts (Fig. 6).
Rice. 6. Dividing the circle into 12 equal parts
Dividing a circle into five equal parts
From a point A draw an arc with the same radius as the radius of the circle before it intersects with the circle - we get a point IN. Lowering the perpendicular from this point - we get the point WITH.From point WITH- the midpoint of the radius of the circle, as from the center, by an arc of radius CD make a notch on the diameter, get a point E. Line segment DE equal to the length of the side of the inscribed regular pentagon. By making a radius DE serifs on the circle, we get the points of dividing the circle into five equal parts.
Rice. 7. Dividing the circle into 5 equal parts
Dividing a circle into ten equal parts
By dividing the circle into five equal parts, you can easily divide the circle into 10 equal parts. Drawing straight lines from the resulting points through the center of the circle to opposite sides circles - we get 5 more points.
Rice. 8. Dividing the circle into 10 equal parts
Dividing a circle into seven equal parts
To divide a circle of radius R into 7 equal parts, from the point of intersection of the center line with the circle (for example, from the point A) describe how from the center an additional arc the same radius R- get a point IN. Dropping a perpendicular from a point IN- get a point WITH.Line segment sun equal to the length of the side of the inscribed regular heptagon.
Rice. 9. Dividing the circle into 7 equal parts
A circle is a closed curved line, each point of which is located on the same distance from one point O, called the center.
Straight lines connecting any point on a circle with its center are called radii R.
A line AB connecting two points of a circle and passing through its center O is called diameter D.
The parts of the circles are called arcs.
A line CD joining two points on a circle is called chord.
A line MN that has only one point in common with a circle is called tangent.
The part of a circle bounded by a chord CD and an arc is called segment.
The part of a circle bounded by two radii and an arc is called sector.
Two mutually perpendicular horizontal and vertical lines intersecting at the center of a circle are called circle axes.
The angle formed by two radii of KOA is called central corner.
Two mutually perpendicular radius make an angle of 90 0 and limit 1/4 of the circle.
Division of a circle into parts
We draw a circle with horizontal and vertical axes that divide it into 4 equal parts. Drawn with a compass or square at 45 0, two mutually perpendicular lines divide the circle into 8 equal parts.
Division of a circle into 3 and 6 equal parts (multiples of 3 by three)
To divide the circle into 3, 6 and a multiple of them, we draw a circle of a given radius and the corresponding axes. The division can be started from the point of intersection of the horizontal or vertical axis with the circle. The specified radius of the circle is successively postponed 6 times. Then the obtained points on the circle are successively connected by straight lines and form a regular inscribed hexagon. Connecting points through one gives an equilateral triangle, and dividing the circle into three equal parts.
The construction of a regular pentagon is performed as follows. We draw two mutually perpendicular axes of the circle equal to the diameter of the circle. Divide the right half of the horizontal diameter in half using the arc R1. From the obtained point "a" in the middle of this segment with radius R2, we draw an arc of a circle until it intersects with the horizontal diameter at point "b". Radius R3 from point "1" draw an arc of a circle to the intersection with a given circle (point 5) and get the side of a regular pentagon. The "b-O" distance gives the side of a regular decagon.
Dividing a circle into N-th number of identical parts (building a regular polygon with N sides)
It is performed as follows. We draw horizontal and vertical mutually perpendicular axes of the circle. From the top point "1" of the circle we draw under arbitrary angle a straight line to the vertical axis. On it we set aside equal segments of arbitrary length, the number of which is equal to the number of parts into which we divide the given circle, for example 9. We connect the end of the last segment with the lower point of the vertical diameter. We draw lines parallel to the obtained one from the ends of the segments to the intersection with the vertical diameter, thus dividing the vertical diameter of the given circle into a given number of parts. With a radius equal to the diameter of the circle, from the lower point of the vertical axis we draw an arc MN until it intersects with the continuation of the horizontal axis of the circle. From points M and N we draw rays through even (or odd) division points of the vertical diameter until they intersect with the circle. The resulting segments of the circle will be the desired ones, because points 1, 2, …. 9 divide the circle into 9 (N) equal parts.
Division of a circle into three equal parts. Install a square with angles of 30 and 60 ° with a large leg parallel to one of the center lines. Along the hypotenuse from a point 1 (first division) draw a chord (Fig. 2.11, A), getting the second division - point 2. Turning the square and drawing the second chord, get the third division - point 3 (Fig. 2.11, b). By connecting points 2 and 3; 3 And 1 straight lines form an equilateral triangle.
Rice. 2.11.
a, b - c using a square; V- using a circle
The same problem can be solved using a compass. By placing the support leg of the compass at the lower or upper end of the diameter (Fig. 2.11, V) describe an arc whose radius is equal to the radius of the circle. Get the first and second divisions. The third division is at the opposite end of the diameter.
Dividing a circle into six equal parts
The compass opening is set equal to the radius R circles. From the ends of one of the diameters of the circle (from the points 1, 4 ) describe arcs (Fig. 2.12, a, b). points 1, 2, 3, 4, 5, 6 divide the circle into six equal parts. By connecting them with straight lines, they get a regular hexagon (Fig. 2.12, b).
Rice. 2.12.
The same task can be performed using a ruler and a square with angles of 30 and 60 ° (Fig. 2.13). The hypotenuse of the square must pass through the center of the circle.
Rice. 2.13.
Dividing a circle into eight equal parts
points 1, 3, 5, 7 lie at the intersection of the center lines with the circle (Fig. 2.14). Four more points are found using a square with angles of 45 °. When receiving points 2, 4, 6, 8 the hypotenuse of a square passes through the center of the circle.
Rice. 2.14.
Dividing a circle into any number of equal parts
To divide a circle into any number of equal parts, use the coefficients given in Table. 2.1.
Length l chord, which is laid on a given circle, is determined by the formula l = dk, Where l- chord length; d is the diameter of the given circle; k- coefficient determined from Table. 1.2.
Table 2.1
Coefficients for dividing circles
To divide a circle of a given diameter of 90 mm, for example, into 14 parts, proceed as follows.
In the first column of Table. 2.1 find the number of divisions P, those. 14. From the second column write out the coefficient k, corresponding to the number of divisions P. In this case, it is equal to 0.22252. The diameter of a given circle is multiplied by a factor and the length of the chord is obtained l=dk= 90 0.22252 = 0.22 mm. The resulting length of the chord is set aside with a measuring compass 14 times on a given circle.
Finding the center of the arc and determining the size of the radius
An arc of a circle is given, the center and radius of which are unknown.
To determine them, you need to draw two non-parallel chords (Fig. 2.15, A) and set up perpendiculars to the midpoints of the chords (Fig. 2.15, b). Center ABOUT arc is at the intersection of these perpendiculars.
Rice. 2.15.
Pairings
When performing engineering drawings, as well as when marking workpieces in production, it is often necessary to smoothly connect straight lines with arcs of circles or an arc of a circle with arcs of other circles, i.e. perform pairing.
Pairing called a smooth transition of a straight line into an arc of a circle or one arc into another.
To build mates, you need to know the value of the radius of the mates, find the centers from which the arcs are drawn, i.e. interface centers(Fig. 2.16). Then you need to find the points at which one line passes into another, i.e. connection points. When constructing a drawing, mating lines must be brought exactly to these points. The point of conjugation of the arc of a circle and a straight line lies on a perpendicular lowered from the center of the arc to the mating line (Fig. 2.17, A), or on a line connecting the centers of mating arcs (Fig. 2.17, b). Therefore, to construct any conjugation by an arc of a given radius, you need to find interface center And point (points) conjugation.
Rice. 2.16.
Rice. 2.17.
The conjugation of two intersecting lines by an arc of a given radius. Given straight lines intersecting at right, acute and obtuse angles (Fig. 2.18, A). It is necessary to construct conjugations of these lines by an arc of a given radius R.
Rice. 2.18.
For all three cases, the following construction can be applied.
1. Find a point ABOUT- the center of the mate, which must lie at a distance R from the sides of the corner, i.e. at the point of intersection of lines passing parallel to the sides of the angle at a distance R from them (Fig. 2.18, b).
To draw straight lines parallel to the sides of an angle, from arbitrary points taken on straight lines, with a compass solution equal to R, make serifs and draw tangents to them (Fig. 2.18, b).
- 2. Find the junction points (Fig. 2.18, c). For this, from the point ABOUT drop perpendiculars to given lines.
- 3. From point O, as from the center, describe an arc of a given radius R between junction points (Fig. 2.18, c).
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?
- Eat.
- 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 ...
Learning task 1 is to find the center of the circle with the help of a center-finder square (Fig. 11, a). The square consists of two strips connected at an angle of 90°, and a rigidly reinforced ruler, the working edge of which divides the angle of 90° in half.
Rice. 11. Finding the center of the circle using the center finder:
a - hinged first risks; b - drawing the second risk; a - determining the position of the center
The markup is performed in the following sequence.
1. The part is installed on the marking plate so that the marked end is on top.
2. A center finder square is placed on the upper end of the part so that its two sides (slats) touch the cylindrical surface of the part.
3. With the left hand, firmly press the ruler of the square to the surface of the butt, and with the right hand draw the first diametrical risk with the scriber.
4. The center-detector square is rotated along the cylindrical surface of the part by about 90 ° and a second diametrical risk is drawn with a scriber (Fig. 11, b). The intersection point of the two marks will be the center of the marked circle (Fig. 11, c).
Rice. 12. Method for checking the accuracy of marking the center of a circle with a marking compass
The marking of the center of the part with a roughly machined cylindrical surface is carried out in the same sequence. In this case, to more accurately find the center of the circle, it is necessary to apply five to seven scratches, and the center will be the point at which the largest number of scratches intersect.
The accuracy of marking the center of the circle is checked with a marking compass (Fig. 12). The tip of one leg of the compass is set to the marked center, and the other leg is moved so that its tip slightly touches the cylindrical part of the part. If the tip of the leg of the compass touches the part along the entire circumference, then the center is marked correctly.
Rice. 13. An example of dividing a circle into four parts with the construction of an inscribed square
Learning task 2 is a division of the circle into four equal parts with the construction of an inscribed square (Fig. 13).
1. In the center of the marked plane, a circle R = 28 mm is drawn with a compass (the radius can be arbitrary).
2. A straight line is drawn through the center of the circle along the ruler so that it intersects the circle at two points A and B and divides it into two equal parts.
3. The supporting leg of the compass is set to point A and, having moved the compass to a distance slightly greater than half of the segment AB, draw an arc V.
4. The supporting leg of the compass is transferred to point B and, without changing the solution of the compass, draw an arc b so that it crosses the first completed arc at points 1 and 2 (Fig. 13, 14).
Rice. 14. Reception of marking a square
5. Through points 1 and 2, a line is drawn along the line, which forms points C and D on the circle.
6. Connecting the points AD, DB, BC and CA with direct risks, we get a square inscribed in a circle.
Learning task 3 consists in dividing the circle into three equal parts with the construction of an inscribed triangle (Fig. 15).
Rice. 15. Division of a circle into three parts with the construction of an inscribed triangle
1. In the center of the marked plane, using a compass, draw a circle R = 26 mm (the radius can be arbitrary).
2. A straight line is drawn through the center of the circle along the ruler with the intersection of the circle at points A and B.
3. The supporting leg of the compass is set to point A and with the compass opening equal to the radius of the drawn circle, two notches are made on the circle (points C and D), where the length of the arc between them will be equal to one third of the circumference.
4. By connecting the points with direct risks CD, CB and BD, an inscribed equilateral triangle is obtained.
5. The correctness of the construction is checked with a compass, setting the compass opening equal to the length of one of the sides of the triangle and determining the equality of the remaining sides of the triangle with the same size.
Learning task 4 (Fig. 16) is a division of the circle into six parts with the construction of an inscribed hexagon (Fig. 17).
Rice. 16. Division of a circle into six parts with the construction of an inscribed hexagon
Rice. 17. An example of marking a hexagon for the size of the throat of a wrench
1. In the center of the marked plane, a circle R = 27 mm is drawn with a compass (the radius can be arbitrary).
2. A line is applied along the line, passing through the center of the circle and crossing it at points A and B.
3. From point A, as from the center, an arc is drawn with a radius equal to the radius of the drawn circle, and points 1 and 2 are obtained.
A similar construction is made from point B, drawing points 3 and 4. The resulting intersection points and end points of the diameter will be the desired points for dividing the circle into six parts.
4. Connecting the points with direct risks A-2, 2-4, 4-B, B-3, 3-1 and 1-A, an inscribed hexagon is obtained.
When marking the faces of the hexagon to the size h of the wrench mouth (Fig. 17), the radius of the circumscribed circle of the inscribed hexagon is determined by the formula R = 0.577h.
