By doing graphic works many construction problems have to be solved. The most common tasks in this case are dividing line segments, angles and circles into equal parts, constructing 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.
Dividing a circle into four equal parts.
Dot-dash 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 Dividing a circle into 4 equal parts.
Dividing a circle into eight equal parts.
To divide a circle into eight equal parts, arcs equal to a quarter 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 a circle, notches are made beyond its boundaries. The resulting points 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. Dividing a circle into 8 equal parts.
Dividing a circle into sixteen equal parts.
Using a compass, dividing an arc equal to 1/8 into two equal parts, apply notches to the circle. By connecting all the serifs with straight segments, we get a regular hexagon.
Fig.3. Dividing a circle into 16 equal parts.
Dividing 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. Dividing a circle into 3 equal parts.
Dividing a circle into six equal parts. Side regular hexagon, inscribed in a circle, is equal to the radius of the circle (Fig. 5.).
To divide a circle into six equal parts, you need points 1 And 4 intersection of the center line with the circle, make two notches with a radius on the circle R, equal to the radius of the circle. By connecting the resulting points with straight line segments, we obtain a regular hexagon.
Rice. 5. Dividing a circle into 6 equal parts
Dividing a circle into twelve equal parts.
To divide a circle into twelve equal parts, the circle must be divided 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 of the same radius are drawn until they intersect with the circle. Received points 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 and dots A , IN, WITH, D divide the circle into twelve equal parts (Fig. 6).
Rice. 6. Dividing a circle into 12 equal parts
Dividing a circle into five equal parts
From point A draw an arc with the same radius as the radius of the circle until it intersects with the circle - we get a point IN. Dropping the perpendicular from this point, we get the point WITH.From point WITH- the middle of the radius of a circle, as from the center, an arc of radius CD make a notch on the diameter, we get a point E. Line segment DE equal to length sides of the inscribed regular pentagon. Making it a radius DE serifs on the circle, we get the points of dividing the circle into five equal parts.
Rice. 7. Dividing a circle into 5 equal parts
Dividing a circle into ten equal parts
By dividing a 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 circle - we get 5 more points.
Rice. 8. Dividing a 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) are described as an additional arc from the center the same radius R- get a point IN. Dropping a perpendicular from a point IN- we get a point WITH.Line segment Sun equal to the length of the side of the inscribed regular heptagon.
Rice. 9. Dividing a circle into 7 equal parts
Dividing a circle into four equal parts and constructing a regular inscribed quadrilateral(Fig. 6).
Two mutually perpendicular center lines divide the circle into four equal parts. By connecting the points of intersection of these lines with the circle with straight lines, a regular inscribed quadrilateral is obtained.
Dividing a circle into eight equal parts and constructing a regular inscribed octagon(Fig. 7).
The circle is divided into eight equal parts using a compass as follows.
From points 1 and 3 (points of intersection of the center lines with the circle) arbitrary radius R draw arcs until they intersect each other, and with the same radius from point 5 make a notch on the arc drawn from point 3.
Straight lines are drawn through the intersection points of the serifs and the center of the circle until they intersect with the circle at points 2, 4, 6, 8.
If the resulting eight points are connected sequentially by straight lines, you will get a regular inscribed octagon.
Dividing a circle into three equal parts and constructing a regular inscribed triangle(Fig. 8).
Option 1.
When dividing a circle with a compass into three equal parts, from any point on the circle, for example, point A of the intersection of the center lines with the circle, draw an arc of radius R equal to the radius of the circle, obtaining points 2 and 3. The third point of division (point 1) will be located at the opposite end of the diameter passing through point A. By sequentially connecting points 1, 2 and 3, a regular inscribed triangle is obtained.
Option 2.
When constructing a regular inscribed triangle, if one of its vertices is given, for example point 1, find point A. To do this, through given point carry out the diameter (Fig. 8). Point A will be located at the opposite end of this diameter. Then an arc of radius R equal to the radius of the given circle is drawn, points 2 and 3 are obtained.
Dividing a circle into six equal parts and constructing a regular inscribed hexagon(Fig.9).
When dividing a circle into six equal parts using a compass, arcs are drawn from two ends of the same diameter with a radius equal to the radius of the given circle until they intersect with the circle at points 2, 6 and 3, 5. By sequentially connecting the resulting points, a regular inscribed hexagon is obtained.
Dividing a circle into twelve equal parts and constructing a regular inscribed dodecagon(Fig. 10).
When dividing a circle with a compass, from the four ends of two mutually perpendicular diameters of the circle, an arc with a radius equal to the radius of the given circle is drawn until it intersects with the circle (Fig. 10). By connecting sequentially obtained intersection points, a regular inscribed dodecagon is obtained.
Dividing a circle into five equal parts and constructing a regular inscribed pentagon ( Fig. 11).
When dividing a circle with a compass, half of any diameter (radius) is divided in half to obtain point A. From point A, as from the center, draw an arc with a radius equal to the distance from point A to point 1, to the intersection with the second half of this diameter at point B. Segment 1B equal to chord subtending an arc whose length is equal to 1/5 of the circumference. Making notches on a circle of radius R1, equal to the segment 1B, divide the circle into five equal parts. The starting point A is chosen depending on the location of the pentagon.
From point 1, construct points 2 and 5, then from point 2, construct point 3, and from point 5, construct point 4. The distance from point 3 to point 4 is checked with a compass; if the distance between points 3 and 4 is equal to segment 1B, then the construction was carried out accurately.
It is impossible to make notches sequentially, in one direction, since measurement errors accumulate and the last side of the pentagon turns out to be skewed. By sequentially connecting the found points, a regular inscribed pentagon is obtained.
Dividing a circle into ten equal parts and constructing a regular inscribed decagon(Fig. 12).
Dividing a circle into ten equal parts is carried out similarly to dividing a circle into five equal parts (Fig. 11), but first divide the circle into five equal parts, starting construction from point 1, and then from point 6, located at the opposite end of the diameter. By connecting all the points in series, a regular inscribed decagon is obtained.
Dividing a circle into seven equal parts and constructing a regular inscribed heptagon(Fig. 13).
From any point on a circle, for example point A, an arc is drawn with the radius of a given circle until it intersects with the circle at points B and D of the straight line.
Half of the resulting segment (in in this case segment BC) will be equal to the chord that subtends an arc constituting 1/7 of the circumference. With a radius equal to the segment BC, notches are made on the circle in the sequence shown when constructing a regular pentagon. By connecting all the points in sequence, a regular inscribed heptagon is obtained.
Dividing a circle into fourteen equal parts and constructing a regular inscribed quadrangle (Fig. 14).
Dividing a circle into fourteen equal parts is carried out similarly to dividing a circle into seven equal parts (Fig. 13), but first divide the circle into seven equal parts, starting construction from point 1, and then from point 8, located at the opposite end of the diameter. By connecting all the points in series, a regular inscribed quadrangle is obtained.
Dividing a circle into six equal parts and constructing a regular inscribed hexagon is done using a square with angles of 30, 60 and 90º and/or a compass. When dividing a circle into six equal parts with a compass, arcs are drawn from two ends of the same diameter with a radius equal to the radius of the given circle until they intersect with the circle at points 2, 6 and 3, 5 (Fig. 2.24). By sequentially connecting the resulting points, a regular inscribed hexagon is obtained.
Figure 2.24
When dividing a circle with a compass, from the four ends of two mutually perpendicular diameters of the circle, an arc with a radius equal to the radius of the given circle is drawn until it intersects with the circle (Fig. 2.25). By connecting the resulting points, a dodecagon is obtained.
Figure 2.25
2.2.5 Dividing a circle into five and ten equal parts
and construction of regular inscribed pentagon and decagon
The division of a circle into five and ten equal parts and the construction of a regular inscribed pentagon and decagon is shown in Fig. 2.26.
Figure 2.26
Half of any diameter (radius) is divided in half (Fig. 2.26 a), point A is obtained. From point A, as from the center, draw an arc with a radius equal to the distance from point A to point 1 to the intersection with the second half of this diameter, at point B( Fig. 2.26 b ). Segment 1 is equal to a chord subtending an arc whose length is equal to 1/5 of the circumference. Making notches on the circle (Fig. 2.26, in ) radius TO equal to segment 1B, divide the circle into five equal parts. Starting point 1 is chosen depending on the location of the pentagon. From point 1, build points 2 and 5 (Fig. 2.26, c), then from point 2, build point 3, and from point 5, build point 4. The distance from point 3 to point 4 is checked with a compass. If the distance between points 3 and 4 is equal to segment 1B, then the construction was carried out accurately. It is impossible to make serifs sequentially, in one direction, as errors occur and the last side of the pentagon turns out to be skewed. By sequentially connecting the found points, a pentagon is obtained (Fig. 2.26, d).
Dividing a circle into ten equal parts is carried out similarly to dividing a circle into five equal parts (Fig. 2.26), but first divide the circle into five parts, starting construction from point 1, and then from point 6, located at the opposite end of the diameter (Fig. 2.27, A). By connecting all the points in series, they get a regular inscribed decagon (Fig. 2.27, b).
Figure 2.27
2.2.6 Dividing a circle into seven and fourteen equal parts
parts and construction of a regular inscribed heptagon and
quadragon
The division of a circle into seven and fourteen equal parts and the construction of a regular inscribed heptagon and a fourteen-sided triangle are shown in Fig. 2.28 and 2.29.
From any point on the circle, for example point A , draw an arc with the radius of a given circle (Fig. 2.28, a ) until it intersects with the circle at points B and D . Let's connect the points Vi D with a straight line. Half of the resulting segment (in this case segment BC) will be equal to the chord that subtends an arc constituting 1/7 of the circumference. With a radius equal to the segment BC, notches are made on the circle in the sequence shown in Fig. 2.28, b . By connecting all the points in series, they get a regular inscribed heptagon (Fig. 2.28, c).
Dividing the circle into fourteen equal parts is done by dividing the circle into seven equal parts twice from two points (Fig. 2.29, a).
Figure 2.28
First, the circle is divided into seven equal parts from point 1, then the same construction is performed from point 8 . The constructed points are connected sequentially by straight lines and a regular inscribed quadrangle is obtained (Fig. 2.29, b).
Figure 2.29
Construction of an ellipse
Image of a circle in a rectangular isometric projection in all three projection planes it represents ellipses of the same shape.
The direction of the minor axis of the ellipse coincides with the direction of the axonometric axis, perpendicular to that projection plane in which the depicted circle lies.
When constructing an ellipse depicting a circle of small diameter, it is enough to construct eight points belonging to the ellipse (Fig. 2.30). Four of them are the ends of the ellipse axes (A, B, C, D), and the other four (N 1, N 2, N 3, N 4) are located on straight lines parallel to the axonometric axes, at a distance equal to the radius of the depicted circle from the center of the ellipse.
Sometimes, to make stencils, templates, drawings, patterns, and crafts, it is necessary to separate into 6 parts.
For example, we needed to make a template for a flower in the shape of a six-pointed star.
For those who have forgotten geometry, I remind you that you can divide a circle into 6 parts in two ways:
- By using protractor.
- By using compass.
1. How to divide a circle into 6 parts using a protractor
Dividing a circle using a protractor is very easy.
Draw a line connecting the center and any point (for example, point 1) on the circle. From this line, using a protractor, we plot an angle of 60, 120, 180 degrees. We put points on the circle (for example, points 2, 3, 4). We unfold the protractor and divide the other part of the circle in the same way.
2. How to divide a circle into 6 parts using a compass
It happens that you don’t have a protractor at hand. Then the circle can be divided into 6 equal parts using a compass.
Draw a circle, for example, with a radius of 5 cm (red circle). Without changing the radius, we move the leg of the compass to the circle (point 1) and draw another circle. We get two points of intersection of the black and red circles 6 and 2.
We move the leg of the compass to point 2 and again draw a circle. We get point 3.
We move the leg of the compass to point 3. Again we draw a circle.
Thus, we continue to divide the circle until we divide it into 6 equal parts.
Today in the post I am posting several pictures of ships and patterns for them for embroidery with isofilament (pictures are clickable).
Initially, the second sailboat was made on studs. And since the nails have a certain thickness, it turns out that two threads come off each one. Plus, layering one sail on top of the second. As a result, a certain split image effect appears in the eyes. If you embroider a 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 center point(on the underside of the sail), from which rays extend to points along the perimeter of the sail.
Joke:
- Do you have any threads?
- Eat.
- And the harsh ones?
- Yes, it’s just a nightmare! I'm afraid to approach!
Master class: Embroidering a peacock
This is my first debut Master Class. I hope not the last. We will embroider a peacock. Product diagram.When marking puncture sites, pay attention Special attention, so that there are them in closed loops 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), it’s better painted on both sides(for Kiev residents - I bought it from the stationery department at the Central Department Store on Khreshchatyk). Threads- floss (any manufacturer, I had DMC), in one thread, i.e. We unwind the bundles into individual fibers. How to transfer the diagram to the base. Embroidery consists of three layers thread At first Using the laying method, we embroider the first layer of feathers on the peacock’s head, the wing (light blue thread color), as well as the dark blue circles of the tail. The first layer of the body is embroidered in chords with variable pitches, trying to ensure that the threads run tangent to the contour of the wing. Then we embroider branches (snake stitch, mustard-colored threads), leaves (first dark green, then the rest...