Conjugation of two parallel lines

Given two parallel lines and one of them has a conjugate point M(Fig. 2.19, A). You need to build a pairing.

• 1) find the center of mate and the radius of the arc (Fig. 2.19, b). To do this from the point M restore the perpendicular to the intersection with the line at the point N. Line segment MN divided in half (see Fig. 2.7);
• 2) from a point ABOUT– center of mate with radius OM = ON describe an arc from the connecting points M And N(Fig. 2.19, V).

Rice. 2.19.

Given a circle with center ABOUT and point A. It is required to draw from point A tangent to the circle.

1. Point A connect a straight line to a given center O of a circle.

Construct an auxiliary circle with a diameter equal to OA(Fig. 2.20, A). To find the center ABOUT 1, divide the segment OA in half (see Fig. 2.7).

2. Points M And N intersection of the auxiliary circle with the given one - the required points of tangency. Full stop A connect straight lines to points M or N(Fig. 2.20, b). Straight A.M. will be perpendicular to the line OM, since the angle AMO based on diameter.

Rice. 2.20.

Drawing a line tangent to two circles

Given two circles of radii R And R 1. It is required to construct a straight line tangent to them.

There are two cases of touch: external (Fig. 2.21, b) and internal (Fig. 2.21, V).

At external touch construction is performed as follows:

• 1) from the center ABOUT draw an auxiliary circle with a radius equal to the difference between the radii of the given circles, i.e. R–R 1 (Fig. 2.21, A). A tangent line is drawn to this circle from center O1 Ο 1Ν. The construction of the tangent is shown in Fig. 2.20;
• 2) radius drawn from point O to point Ν, continue until they intersect at the point M with a given circle radius R. Parallel to radius OM draw radius Ο 1Ρ smaller circumference. Straight line connecting junction points M And R,– tangent to given circles (Fig. 2.21, b).

Rice. 2.21.

At inner touch the construction is carried out in a similar way, but the auxiliary circle is drawn with a radius equal to the sum of the radii R+R 1 (Fig. 2.21, V). Then from the center ABOUT 1 draw a tangent to the auxiliary circle (see Fig. 2.20). Full stop N connect with a radius to the center ABOUT. Parallel to radius ON draw radius O1 R smaller circumference. The required tangent passes through the connecting points M And R.

Conjugation of an arc and a straight arc of a given radius

Given an arc of a circle of radius R and straight. It is required to connect them with an arc of radius R 1.

• 1. Find the center of mating (Fig. 2.22, A), which should be at a distance R 1 from the arc and from the straight line. Therefore, an auxiliary straight line is drawn parallel to the given straight line at a distance equal to the radius of the mating arc R1) (Fig. 2.22, A). Compass opening equal to the sum of the given radii R+R 1 describe an arc from center O until it intersects with the auxiliary line. The resulting point O1 is the center of mate.
• 2. According to the general rule, the connecting points are found (Fig. 2.22, b): connect the straight centers of the mating arcs O1 and O and lower them from the center of the mating Ο 1 perpendicular to a given line.
• 3. From the mate center Οχ between junction points Μ And Ν draw an arc whose radius R 1 (Fig. 2.22, b).

Rice. 2.22.

Conjugation of two arcs with an arc of a given radius

Given two arcs whose radii are R 1 and R 2. It is required to construct a mate with an arc whose radius is specified.

There are three cases of touch: external (Fig. 2.23, a, b), internal (Fig. 2.23, V) and mixed (see Fig. 2.25). In all cases, the centers of mates must be located from the given arcs at a distance from the radius of the mate arc.

Rice. 2.23.

The construction is carried out as follows:

For external touch:

• 1) from centers Ο 1 and O2, using a compass solution equal to the sum of the radii of the given and mating arcs, draw auxiliary arcs (Fig. 2.23, A); radius of an arc drawn from the center Ο 1, equal R 1 + R 3; and the radius of the arc drawn from the center O2 is equal to R 2 + R 3. At the intersection of the auxiliary arcs, the center of mate is located – point O3;
• 2) connecting point Ο1 with point 03 and point O2 with point O3 by straight lines, find the connecting points M And N(Fig. 2.23, b);
• 3) from point 03 with a compass solution equal to R 3, between points Μ And Ν describe the conjugate arc.

For inner touch perform the same constructions, but the radii of the arcs are taken equal to the difference between the radii of the given and mating arcs, i.e. R 4 –R 1 and R 4 – R 2. Connection points R And TO lie on the continuation of the lines connecting point O4 with points O1 and O2 (Fig. 2.23, V).

For mixed (external and internal) touch(1st case):

• 1) a compass solution equal to the sum of the radii R 1 and R 3, an arc is drawn from point O2, as from the center (Fig. 2.24, a);
• 2) a compass solution equal to the difference in radii R 2 and R 3, from point O2 a second arc is drawn, intersecting with the first at point O3 (Fig. 2.24, b);
• 3) from point O1 draw a straight line to point O3, from the second center (point O2) draw a straight line through point O3 until it intersects with the arc at point M(Fig. 2.24, c).

Point O3 is the center of the mate, the point M And N – interface points;

4) placing the leg of the compass at point O3, with radius R 3 draw an arc between the connecting points Μ And Ν (Fig. 2.24, G).

Rice. 2.24.

For mixed touch(2nd case):

• 1) two conjugate arcs of circles of radii R 1 and R 2 (Fig. 2.25);
• 2) distance between centers About i and O2 of these two arcs;
• 3) radius R 3 mating arcs;

required:

• 1) determine the position of the center O3 of the mating arc;
• 2) find the connecting points on the mating arcs;
• 3) draw a mating arc

Construction sequence

Set aside specified distances between centers Ο 1 and O2. From the center ABOUT 1 draw an auxiliary arc with a radius equal to the sum of the radii of the mating arc of radius R 1 and conjugate arc radius R 3, and from the center O2 a second auxiliary arc is drawn with a radius equal to the difference in radii R 3 and R 2, until it intersects with the first auxiliary arc at point O3, which will be the desired center of the mating arc (Fig. 2.25).

Rice. 2.25.

Conjugation points are found according to the general rule, connecting the centers of arcs O3 and O1 with straight lines , O 3 and O2. At the intersection of these lines with the arcs of the corresponding circles, points are found M And N.

Pattern curves

In technology there are parts whose surfaces are limited by flat curves: an ellipse, an involute circle, an Archimedes spiral, etc. Such curved lines cannot be drawn with a compass.

They are built along points that are connected by smooth lines using patterns. Hence the name pattern curves.

Shown in Fig. 2.26. Each point of a straight line, if rolled without sliding along a circle, describes an involute.

Rice. 2.26.

The working surfaces of the teeth of most gears have involute gearing (Fig. 2.27).

Rice. 2.27.

Archimedes spiral shown in Fig. 2.28. This is a flat curve described by a point moving uniformly from the center ABOUT along a rotating radius.

Rice. 2.28.

A groove is cut along the Archimedes spiral, into which the protrusions of the cams of a self-centering three-jaw chuck of a lathe enter (Fig. 2.29). When the bevel gear, on the back of which has a spiral groove, rotates, the cams are compressed.

When making these (and other) pattern curves in the drawing, you can use the reference book to make your work easier.

The dimensions of the ellipse are determined by the size of its major AB and small CD axes (Fig. 2.30). Describe two concentric circles. The larger diameter is equal to the length of the ellipse (major axis AB), the diameter of the smaller one is the width of the ellipse (minor axis CD). Divide a large circle into equal parts, for example 12. The division points are connected by straight lines passing through the center of the circles. From the points of intersection of straight lines with circles, lines are drawn parallel to the axes of the ellipse, as shown in the figure. When these lines intersect each other, points belonging to the ellipse are obtained, which, having previously been connected by hand with a thin smooth curve, are outlined using a pattern.

Rice. 2.29.

Rice. 2.30.

Practical application of geometric constructions

Given the task: make a drawing of the key shown in Fig. 2.31. How to do it?

Before starting to draw, an analysis of the graphic composition of the image is carried out to determine which cases of geometric constructions need to be applied. In Fig. Figure 2.31 shows these constructions.

Rice. 2.31.

To draw a key, you need to draw mutually perpendicular straight lines, describe circles, build hexagons by connecting their upper and lower vertices with straight lines, and connect arcs and straight lines with arcs of a given radius.

What is the sequence of this work?

First, draw those lines whose position is determined by the given dimensions and do not require additional construction (Fig. 2.32, A), i.e. draw axial and center lines, describe four circles according to given dimensions and connect the ends of the vertical diameters of smaller circles with straight lines.

Rice. 2.32.

Further work on the execution of the drawing requires the use of the geometric constructions set out in paragraphs 2.2 and 2.3.

In this case, you need to build hexagons and pair arcs with straight lines (Fig. 2.32, b). This will be the second stage of work.

Lesson No. 23.

Mates

Show multiple parts that have fillets.

Looking at the details, we see that in their design one surface often merges into another. Usually these transitions are made smooth, which increases the strength of the parts and makes them more convenient to use.

In the drawing, surfaces are depicted as lines that also smoothly transition into one another.

Such a smooth transition from one line (surface) to another line (surface) is called pairing.

When constructing a mate, it is necessary to determine the boundary where one line ends and another begins, i.e. find the transition point in the drawing, which is called mate point or point of contact .

Conjugation problems can be divided into 3 groups.

First group of tasks includes tasks on constructing conjugations where straight lines are involved. This can be a direct contact between a straight line and a circle, the conjugation of two straight lines with an arc of a given radius, as well as drawing a tangent line to two circles.

Let's construct a circle tangent to the line.

Constructing a circle tangent to a line , is associated with finding the point of tangency and the center of the circle.

A horizontal line is given AB , you need to construct a circle with radius R , tangent to this line (Fig. 1).

The touch point is chosen arbitrarily.

Since the point of tangency is not specified, the circle of radius R can touch a given line at any point. There are many such circles that can be drawn. The centers of these circles ( ABOUT 1 , ABOUT 2 etc.) will be at the same distance from the given straight line, i.e. on a line parallel to a given straight line AB at a distance equal to the radius of a given circle (Fig. 1). Let's call this line line of centers .

Let's draw a line of centers parallel to the straight line AB on distance R . Since the center of the tangent circle is not specified, take any point on the line of centers, for example, the point ABOUT.

Before drawing a tangent circle, you must determine the point of tangency. The point of tangency will lie on the perpendicular drawn from the point ABOUT directly AB . At the intersection of a perpendicular with a line AB we get a point TO, which will be the point of contact. From the center ABOUT radius R from point TO Let's draw a circle. The problem is solved.

Write down the following rules in your notebooks:

If a straight line is involved in the pairing, then:

1)

the center of a circle tangent to a straight line lies on a straight line (line of centers) drawn parallel to a given straight line, at a distance equal to the radius of the given circle;

2) the point of tangency lies on a perpendicular drawn from the center of the circle to a given straight line.

Conjugation of two straight lines.

On a plane, two straight lines can be parallel or at an angle to each other.

To construct a conjugation of two lines, it is necessary to draw a circle tangent to these two lines.

Open your workbooks to page 31.

Consider the conjugation of two non-parallel lines.

Two non-parallel lines are located at an angle to each other, which can be straight, obtuse or acute. When making drawings of parts, such corners often need to be rounded with an arc of a given radius (Fig. 1). Rounding corners in a drawing is nothing more than the conjugation of two non-parallel straight lines with a circular arc of a given radius. To perform a mate, you need to find the center of the mate arc and the mate points.

It is known that if a straight line is involved in the conjugation, then the center of the conjugation arc is located on the line of centers, which is drawn parallel to a given straight line at a distance equal to the radius R mating arcs.

Since the angle is formed by two straight lines, draw two lines of centers parallel to each straight line at a distance equal to the radius R mating arcs. The point of their intersection will be the center of the mating arc.

To find connecting points from a point ABOUT lower perpendiculars to given lines and obtain connecting points TO And TO 1 . Knowing the points and the center of mate, from the point ABOUT radius R draw a mating arc. When tracing a drawing, you should first trace the arc, and then the tangent lines.

When constructing the conjugation of a right angle, drawing a line of centers is simplified, since the sides of the angle are mutually perpendicular. Segments equal to the radius are laid off from the vertex of the angle R arcs of conjugation, and through the resulting points TO And TO 1 , which will be the points of tangency, draw two lines of centers parallel to the sides of the angle. They will be both center lines and perpendiculars defining the connecting points TO And TO 1 (p. 31, fig. 1).

Page 31, task 4. Conjugation of two parallel lines.

To construct a conjugation of two parallel lines, it is necessary to draw an arc of a circle tangent to these lines (Fig. 3).

Fig.3

The radius of this circle will be equal to half the distance between the given straight lines. Since the point of tangency is not specified, many similar circles can be drawn. Their centers will be located on a straight line drawn parallel to the given straight lines at a distance equal to half the distance between them. This straight line will be the line of centers.

Touch points ( TO 1 And TO 2 ) lie on a perpendicular dropped from the center of the tangent circle onto given straight lines (Fig. 3a). Since the center of the tangent circle is not specified, the perpendicular is drawn arbitrarily. Line segment QC 1 divide in half (Fig. 3b), draw a straight line through the intersection points of the serifs parallel to the given straight lines, on which the centers of the circles tangent to the given parallel straight lines will be located, i.e. this line will be the line of centers. By placing the leg of the compass at the point ABOUT , draw a conjugation arc (Fig. 3c) from the point TO to the point TO 1 .

Construction of straight lines tangent to circles

(R.T. p.33).

Exercise 1. Draw a line tangent to the circle through a point A , lying on a circle.

From point ABOUT we conduct a direct O.B. through the point A . From point A We draw a circle with any radius. When crossing a straight line we got points 1 And 2. From these points we draw arcs of any radius until they intersect each other at points C And D . From point C or D draw a straight line through a point A .

It will be tangent to the circle, since a tangent is always perpendicular to the radius drawn to the point of contact.

Task 2.

This construction is similar to constructing a perpendicular to a line through a given point, which can be done using two squares.

First the square 1 placed so that its hypotenuse coincides with the points O And A . Then to square 1 a square is applied 2 , which will be the guide, i.e. along which the square will move 1 . Then the square 1 we put the other leg to the square 2. Then we roll the square 1 along the square 2 until the hypotenuse coincides with the point A . And draw a straight line tangent to the circle through the point A .

Task 3. Draw a line tangent to a circle through a point not lying on the circle.

Given a circle with radiusR and period A , not lying on the circle, must be drawn from the pointA a straight line tangent to a given circle in its upper part. To do this, you need to find the point of contact. We know that the point of tangency lies on the perpendicular drawn from the center of the circle to the tangent line. Therefore, a tangent and a perpendicular form a right angle.

Knowing that every angle inscribed in a circle and based on its diameter is a right angle, connecting the pointsA And ABOUT , take the segmentJSC for the diameter of the circumscribed circle. At the intersection of the circumcircle and the circle of radiusR there will be a vertex of a right angle (pointTO ). Line segment JSC divide in half using a compass, we get a pointABOUT 1 (Fig. 4, b).

From the center ABOUT 1 radius equal to the segmentJSC 1 , draw a circle, get pointsTO And TO 1 at the intersection with a circle of radiusR (Fig. 4,c).

Since only one tangent needs to be drawn to the top of the circle, the desired point of tangency is selected. This point will be the pointTO . Full stop TO connect with dotsA And ABOUT , we get a right angle that rests on the diameterJSC circumscribed circle with radiusR 1 . Dot TO – vertex of this angle (Fig. 4, d), segmentsOK And AK – sides of a right angle, therefore, a pointTO will be the desired tangent point, and the straight lineAK – the desired tangent.

Fig.4

Drawing a straight line tangent to two circles.

Given two circles with radii R And R 1 , you need to construct a tangent to them. There are two possible cases of contact: external and internal.

With an external tangency, the tangent line is located on one side of the circles and does not intersect the segment connecting the centers of these circles.

In an internal tangency, the tangent line is located on different sides of the circles and intersects the segment connecting the centers of the circles.

Page 33. Task 5. Draw a straight line tangent to the two circles. External touch.

First of all, you need to find the touch points. It is known that they must lie on perpendiculars drawn from the centers of the circles ( ABOUT And ABOUT 1 ) to the tangent.

From point ABOUT draw a circle with radius R - R 1 , since the touch is external.

Divide the distance OO 1 in half and draw a circle with radius R =OO 2 =O 1 ABOUT 2

This circle intersects a circle with radius R - R 1 at the point TO. Connect this point with ABOUT 1 .

From point ABOUT through the point TO draw a straight line until it intersects with a circle of radius R . Got a point TO 1 – the first point of contact.

From point ABOUT 1 draw a straight line parallel QC 1 , until it intersects with a circle of radius R 1 . Got a second point of contact TO 2 . Connecting the dots TO 1 And TO 2 . This is the tangent to the two circles.

Task 6. Draw a straight line tangent to the two circles. The touch is internal.

The construction is similar, only with an internal touch the radius of the auxiliary circle drawn from the point ABOUT equal to the sum of the radii of the circles R + R 1 .

The second group of pairing problems includes problems that involve only circles and arcs. A smooth transition from one circle to another can occur either directly by touching, or through a third element - the arc of a circle.

The tangency of two circles can be external (RT: p. 32, Fig. 3) or internal (RT: p. 32, Fig. 4).

Task 3 (page 32)

When two circles touch externally, the distance between the centers of these circles will be equal to the sum of their radii.

From point ABOUT radius R + R C let's draw an arc. From point ABOUT 1 radius R 1 + R C ABOUT WITH - center of conjugation.

Connecting the dots ABOUT And ABOUT 1 with the center of mate ABOUT WITH . Points of tangency (conjugation) were obtained on the circles.

From point ABOUT WITH mating radius R C 30 connect the touch points.

Task 4 (page 32)

When two circles touch internally, one of the tangent circles is inside the other circle, and the distance between the centers of these circles will be equal to the difference in their radii.

From point ABOUT radius ( R C – R ) let's draw an arc. From point ABOUT 1 radius ( R C – R 1 ) draw an arc until it intersects with the first arc. Got a point ABOUT WITH - center of conjugation.

Pairing Center ABOUT WITH connect with dots ABOUT And ABOUT 1 s and extend the straight line further.

Points of tangency (conjugation) were obtained on the circles.

From point ABOUT WITH mating radius R C 60 connect the touch points.

The third group of problems on pairings includes tasks on connecting a straight line and a circular arc with an arc of a given radius.

When performing such a task, they solve two problems: drawing a tangent arc to a straight line and a tangent arc to a circle. Touch in this case can be both external and internal.

RT: page 32. Task 1. Conjugation of a circle and a straight line. External touch. R C 20 .

Given a straight line and a circle with radius R , it is required to construct a mate with an arc of radius R C 20 .

Since a straight line is involved in the conjugation, the center of the conjugation arc is located on a straight line drawn parallel to a given straight line at a distance equal to the conjugation radius R C 20 . Therefore, we draw another straight line parallel to the given straight line at a distance of 20 mm.

And the center of the conjugation arc when the two circles touch externally is located on a circle of radius equal to the sum of the radii R And R C . Therefore from the point ABOUT radius ( R + R C ABOUT WITH

Then we find the points of contact. The first point of tangency is a perpendicular dropped from the center of the mate to a given straight line. We find the second mate point by connecting the mate center ABOUT WITH and the center of the circle R . The point of tangency will lie at the first intersection with the circle, since the tangency is external.

Then from the point ABOUT WITH radius R C 20 connect the connecting points.

RT: page 32. Task 2. Conjugation of a circle and a straight line. The touch is internal. R C 60 .

Parallel to the given straight line, draw a line of centers at a distance of 60 mm. From point ABOUT radius ( R With - R ) draw an arc until it intersects with a new straight line (line of centers). Let's get a point ABOUT WITH , which is the center of conjugation.

From ABOUT WITH draw a straight line through the center of the circle ABOUT and perpendicular to a given line. We get two points of contact. And then from the center of the mate with a radius of 60 mm we connect the tangent points.

In this short article, the main types of conjugations will be discussed and you will learn how to construct a conjugation of angles, straight lines, circles and arcs, circles with a straight line.

Pairing is called smooth transition from one line to another. In order to build a mate, you need to find the center of the mate and the mate points.

Mating point– this is the common point for the mating lines. The mate point is also called the transition point.

Below we will discuss the main mate types.

Conjugation of corners (Conjugation of intersecting lines)

Right angle conjugation (Conjugation of intersecting lines at right angles)

In this example we will consider the construction right angle mate with a given conjugation radius R. First of all, let’s find the conjugation points. To find the connecting points, you need to place a compass at the vertex of a right angle and draw an arc of radius R until it intersects with the sides of the angle. The resulting points will be the connecting points. Next you need to find the center of the mate. The center of the mate will be the point equidistant from the sides of the angle. Let's draw two arcs with a conjugation radius R from points a and b until they intersect with each other. The point O obtained at the intersection will be the center of conjugation. Now, from the center of the conjugation of point O, we describe an arc with a conjugation radius R from point a to point b. The right angle conjugation is constructed.

Conjugation of an acute angle (Conjugation of intersecting lines at an acute angle)

Another example of conjugating an angle. This example will build pairing
acute angle. To construct the conjugation of an acute angle with a compass opening equal to the conjugation radius R, we draw two arcs from two arbitrary points on each side of the angle. Then we draw tangents to the arcs until they intersect at point O, the center of the conjugation. From the resulting mate center we lower a perpendicular to each side of the angle. This way we get the connecting points a and b. Then, from the center of the mate, point O, we draw an arc with a mate radius R, connecting the mate points a
and b. The conjugation of an acute angle is constructed.

Conjugation of an obtuse angle (Conjugation of intersecting lines at an obtuse angle)

It is constructed by analogy with the conjugation of an acute angle. We also first draw two arcs with a conjugation radius R from two arbitrarily chosen points on each side, and then draw tangents to these arcs until they intersect at point O, the center of the conjugation. Then we lower the perpendiculars from the center of the conjugation to each of the sides and connect the resulting points a and b with an arc equal to the conjugation radius of the obtuse angle R.

Pairing Parallel Straight Lines

Let's build conjugation of two parallel lines. We are given a conjugation point a lying on the same line. From point a we draw a perpendicular until it intersects with another line at point b. Points a and b are the connecting points of straight lines. Drawing an arc from each point with a radius greater than the segment ab, we find the center of conjugation - point O. From the center of conjugation we draw an arc of a given conjugation radius R.

Pairing circles (arcs) with a straight line

External conjugation of an arc and a straight line

In this example, a conjugation of a straight line defined by segment AB and a circular arc of radius R will be constructed with a given radius r.

First, let's find the center of conjugation. To do this, draw a straight line parallel to the segment AB and spaced from it by a distance of the conjugation radius r, and an arc from the center of the circle OR with radius R+r. The point of intersection of the arc and the line will be the center of conjugation - the point Or.

From the center of conjugation, point Or, we lower a perpendicular to line AB. Point D, obtained at the intersection of the perpendicular and segment AB, will be the conjugation point. Let's find the second conjugation point on the arc of a circle. To do this, connect the center of the circle OR and the conjugation center Or with a line. We obtain the second conjugation point - point C. From the center of the conjugation we draw a conjugation arc of radius r, connecting the conjugation points.

Internal conjugation of a straight line with an arc

By analogy, the internal conjugation of a straight line with an arc is constructed. Let's consider an example of constructing a conjugation of a straight line with radius r, specified by segment AB, and a circular arc of radius R. Let's find the center of the conjugation. To do this, we will construct a straight line parallel to the segment AB and spaced from it by a distance of radius r, and an arc from the center of the circle OR with radius R-r. Point Or, obtained at the intersection of a straight line and an arc, will be the center of conjugation.

From the center of conjugation (point Or) we lower a perpendicular to straight line AB. Point D, obtained based on the perpendicular, will be the mating point.

To find the second conjugation point on the arc of a circle, connect the conjugation center Or and the center of the circle OR with a straight line. At the intersection of the line with the arc of the circle, we obtain the second conjugation point - point C. From point Or, the center of conjugation, we draw an arc of radius r, connecting the conjugation points.

Conjugate circles (arcs)

External pairing a conjugation is considered in which the centers of the mating circles (arcs) O1 (radius R1) and O2 (radius R2) are located behind the conjugating arc of radius R. The example considers the external conjugation of arcs. First we find the center of conjugation. The center of conjugation is the point of intersection of arcs of circles with radii R+R1 and R+R2, constructed from the centers of circles O1(R1) and O2(R2), respectively. Then we connect the centers of circles O1 and O2 with straight lines to the center of the junction, point O, and at the intersection of the lines with the circles O1 and O2 we obtain the junction points A and B. After this, from the junction center we construct an arc of a given junction radius R and connect points A and B with it .

Internal pairing called a conjugation in which the centers of the mating arcs O1, radius R1, and O2, radius R2, are located inside the conjugate arc of a given radius R. The picture below shows an example of constructing an internal conjugation of circles (arcs). First, we find the center of conjugation, which is point O, the intersection point of circular arcs with radii R-R1 and R-R2 drawn from the centers of circles O1 and O2, respectively. Then we connect the centers of circles O1 and O2 with straight lines to the mate center and at the intersection of the lines with circles O1 and O2 we obtain the mate points A and B. Then from the mate center we construct a mate arc of radius R and construct a mate.

Mixed arc mate is a conjugation in which the center of one of the mating arcs (O1) lies outside the conjugate arc of radius R, and the center of the other circle (O2) lies inside it. The illustration below shows an example of a mixed conjugation of circles. First, we find the center of the mate, point O. To find the center of the mate, we build arcs of circles with radii R+R1, from the center of a circle of radius R1 of point O1, and R-R2, from the center of a circle of radius R2 of point O2. Then we connect the center of the conjugation point O with the centers of the circles O1 and O2 by straight lines and at the intersection with the lines of the corresponding circles we obtain the conjugation points A and B. Then we build the conjugation.

Module: Graphic design of drawings.

Result 1: Be able to draw up formats of standard sheets in accordance with GOST 2.303 - 68. Have the skills to draw the contours of parts, be able to apply dimensions, be able to make inscriptions in accordance with GOST 2.303 - 68.

Result 2: Know the construction rules and have the skills to construct a pairing. Be able to explain the rules of construction.

1. Rules for formatting, rules for filling out the title block in accordance with the standard.
2. Rules for applying dimensions, types of lines.
3. Rules for making inscriptions in fonts in accordance with GOST 2.303 – 68.
4. Rules for drawing the contours of technical parts. Geometric constructions.
5. Rules for drawing and constructing connections.

Lesson topic: Rules for constructing mates.

Goals:

• Know the definition of a mate, types of mates.
• Be able to build connections and explain the construction process.
• Develop technical literacy.
• Develop skills in group work and independent work.
• Cultivate a respectful attitude towards the speaker and the ability to listen.

DURING THE CLASSES

1. Organizational and motivational stage –10 minutes.

1.1. Student motivation:

• connection with other objects;
• consideration of parts, geometric bodies from which parts are composed and connections between them (smooth transitions from one line to another);

1.2. Dividing the group into subgroups of 5-6 people (into four subgroups).

All students in the group are asked to choose one from four types of geometric shapes; after the choice is made, the students are united into subgroups to work independently in subgroups.
Students are told what topic they have to study, get acquainted with the rules for constructing conjugations, which will help them understand how smooth transitions (conjugations) are constructed. Each group is invited to study and present one of the types of pairing (the teacher distributes material on the topic of the lesson to each section in sections).

2. Organization of independent activities of students on the topic of the lesson – 25 minutes.

2.1. The concept of pairing.
2.2. General algorithm for constructing mates.
2.3. Types of pairing. Rules for their construction.
2.3.1. Conjugation between two straight lines.
2.3.2. Internal and external conjugation between a straight line and an arc of a circle.
2.3.3. Conjugation internally and externally between two arcs of circles.
2.3.4. Mixed pairing.
3. Summing up, group reports on the topic after independent work in subgroups - 25 minutes.
4. Checking the degree of mastery of the material – 10 minutes.
5. Filling out diaries (about the lesson) – 5 minutes.
6. Evaluation of student activities.

Conjugation is a smooth transition from one line to another.

3. Construct a conjugation (smooth transition from one line to another)
2. 3.1. Constructing a conjugation of two sides of an angle of a circle of a given radius.

The conjugation of two sides of an angle (acute and obtuse) with an arc of a given radius R is performed as follows:

Two auxiliary straight lines are drawn parallel to the sides of the angle at a distance equal to the radius of the arc R. The intersection point of these lines (point O) will be the center of an arc of radius R, that is, the center of conjugation. From point O they describe an arc that smoothly turns into straight lines - the sides of the angle. The arc ends at the connecting points n and n1, which are the bases of the perpendiculars drawn from the center O to the sides of the angle. When constructing a mating of the sides of a right angle, it is easier to find the center of the mating arc using a compass. From the vertex of angle A, an arc of radius R is drawn until mutual intersection at point O, which is the center of conjugation. From the center O, describe the conjugation arc. The construction of the pairing of two sides of the angle is shown in Fig. 1.

General algorithm for constructing a pairing:

1. It is necessary to find the junction point.
2. It is necessary to find the connecting points.
3. Construction of a conjugation (smooth transition from one line to another).
2.3.2 Construction of internal and external connections between a straight line and a circular arc.

The conjugation of a straight line with a circular arc can be performed using an arc with an internal tangency of the arc and an external tangency. Figure 2(a, b) shows the conjugation of a circular arc of radius R and a straight line AB by a circular arc of radius r with an external tangency. To construct such a conjugation, draw a circle of radius R and a straight line AB. A straight line ab is drawn parallel to a given straight line at a distance equal to the radius r (radius of the conjugate arc). From the center O, draw an arc of a circle with a radius equal to the sum of the radii R and r until it intersects the straight line ab at point O1. Point O1 is the center of the mating arc. The conjugation point c is found at the intersection of straight line OO1 with a circular arc of radius R. Conjugation point O1 to this straight line AB. Using similar constructions, points O2, c2, c3 can be found. Figure 2(a, b) shows a bracket, when drawing it it is necessary to carry out the construction described above.

When drawing a flywheel, an arc of radius R is paired with a straight arc AB of radius r with an internal tangency. The center of the conjugation arc O1 is located at the intersection of an auxiliary line drawn parallel to this line at a distance r with the arc of an auxiliary circle described from the center O with a radius equal to the difference R-r. The point of conjugation with 1 is the base of the perpendicular dropped from point O1 to this line. The mating point c is found at the intersection of straight line OO1 with the mating arc. An example of constructing a connection between a straight line and a circular arc is shown in Figure 3.

Conjugation is a smooth transition from one line to another.

General algorithm for constructing a pairing:

1. It is necessary to find the center of the mate.
2. It is necessary to find the connecting points.
3. Construction of a conjugation line (smooth transition from one line to another).

2.3.3. Constructing a conjugation between two arcs of circles.

The conjugation of two arcs of circles can be internal or external.
With internal conjugation, the centers O and O1 of the mating arcs are located inside the mating arc of radius R. With external conjugation, the centers O and O1 of the mating arcs of radii R1 and R2 are located outside the mating arc of radius R.
Constructing an external interface:

a) radii of mating circles R and R1;

Required:

Shown in Figure 4(b). According to the given distances between the centers, centers O and O1 are marked in the drawing, from which conjugate arcs of radii R and R1 are described. From the center O1, draw an auxiliary arc of a circle with a radius equal to the difference between the radii of the mating arc R and the mating arc R2, and from the center O - with a radius equal to the difference in the radii of the mating arc R and the mating arc R1. The auxiliary arcs will intersect at point O2, which will be the desired center of the connecting arc. To find the points of intersection of the continuation of straight lines O2O and O2O1 with the mating arcs, the required conjugation points (points s and s1) are used.

Construction of internal interface:

a) radii R and R1 of mating circular arcs;
b) the distances between the centers of these arcs;
c) radius R of the mating arc;

Required:

a) determine the position O2 of the mating arc;
b) find the connecting points s and s1;
c) draw a mating arc;

The construction of the external interface is shown in Figure 4(c). Using given distances in the drawing, points O and O1 are found, from which conjugate arcs of radii R1 and R2 are described. From center O, draw an auxiliary arc of a circle with a radius equal to the sum of the radii of the mating arc R2 and the mating arc R. The auxiliary arcs will intersect at point O2, which will be the desired center of the mating arc. To find the connecting points, the centers of the arcs are connected by straight lines OO2 and O1O2. These two lines intersect the conjugate arcs at the conjugation points s and s1. From the center O2 with radius R, a conjugate arc is drawn, limiting it to points S and S1.

2.3.4. Construction of mixed conjugation.

An example of mixed pairing is shown in Figure 5.

a) The radii R and R1 of the mating mating arcs are specified;
b) the distances between the centers of these arcs;
c) radius R of the mating arc;

Required:

a) determine the position of the center O2 of the mating arc;
b) find the connecting points s and s1;
c) draw a mating arc;

According to the given distances between the centers, centers O and O1 are marked in the drawing, from which conjugate arcs of radii R1 and R2 are described. From the center O, an auxiliary arc of a circle is drawn with a radius equal to the sum of the radii of the mating arc R1 and the mating arc R, and from the center O1 - with a radius equal to the difference between the radii R and R2. The auxiliary arcs will intersect at point O2, which will be the desired center of the connecting arc. By connecting points O and O2 with a straight line, we obtain the conjugation point s1; connecting points O1 and O2, find the conjugation point s. From the center O2, a conjugation arc is drawn from s to s1. Figure 5 shows an example of constructing a mixed mate.

3. Summing up the results of students’ independent work in groups. Students' reports on each section of the lesson topic at the blackboard.
4. Checking the degree of student knowledge acquisition. Students from each group ask questions from students from the other group.
5. Filling out diaries. Each student is asked to fill out a diary at the end of the lesson.

In order to gain a good amount of knowledge, it is important to record how successfully the lesson went. This journal allows you to record every detail of your work during the lesson during the module. If you are satisfied, satisfied, disappointed with how your lesson went, then indicate your attitude towards the elements of the lesson in the appropriate cell of the questionnaire.

Lesson elements

Satisfied

Satisfied

Disappointed

Pairing.

Conjugation is a smooth transition from one line to another.

Conjugation of intersecting straight lines with a circular arc of a given radius.

The problem boils down to drawing a circle tangent to both given straight lines.

Option 1.

We draw auxiliary lines parallel to the given ones at a distance R from the given ones.

The point of intersection of these lines will be the center ABOUT mating arcs. Perpendiculars dropped from center O to

given straight lines will determine the tangent points K and K 1.

Option 2.

The construction is the same.

Pairings. Constructing line conjugation.

Option 3.

If you want to draw a circle so that it touches three intersecting straight lines, then in this case

The radius cannot be specified by the problem conditions. Center ABOUT the circle is at the intersection bisectors corners

IN And WITH. The radius of the circle is the perpendicular dropped from the center O to any of the 3 given lines

Lines.

Pairings. Constructing line connections.

Construction of an external conjugation of a given circle with a given straight arc of a given radius R 1.

From the center ABOUT given a circle, draw an arc of an auxiliary circle with a radius R+R 1.

We draw a straight line parallel to the given one at a distance R1.

The intersection of the direct and auxiliary arcs will give the center point of the mating arc O 1.

Point of tangency of the arcs TO lies on the line OO 1.

Point of tangency between arc and line K 1 lies at the intersection of the perpendicular from point O 1 to the straight line with the arc.

Pairings. Constructing an external connection between a circle and a straight line.

Construction of the internal conjugation of a given circle with a given straight arc of a given radius R 1.

From the center ABOUT given a circle, draw an auxiliary circle with a radius R-R 1.

Pairings. Construction of the internal conjugation of a circle with a straight line.

Constructing the conjugation of two given circles with an arc of a given radius R 3.

External touch.

From the center of the circle O 1 R 1 + R 3.

From the center of the circle O 2 describe the arc of the auxiliary circle with radius R 2 + R 3 .

Intersection arcs of auxiliary circles will give a point O 3, which is the center of the conjugation arc

Touch points K 1 And K 2 are on the lines O 1 O 3 And O 2 O 3.

Inner Touch

From the center of the circle O 1 describe the arc of the auxiliary circle with radius R 3 -R 1.

From the center of the circle O 2 describe the arc of the auxiliary circle with radius R 3 - R 2.

Intersection

(circles with radius R 3).

Pairings. Conjugation of two circles with an arc.

External and internal touch.

Two circles with centers O 1 and O 2 with radii r 1 and r 2 are given. It is necessary to draw a circle of a given

Radius R so as to provide internal contact with one circle, and external contact with the other.

From the center of the circle O 1 describe the arc of the auxiliary circle with radius R-r 1.

From the center of the circle O 2 describe the arc of the auxiliary circle with radius R+r 2 .

Intersectionarcs of auxiliary circles will give a point that is the center of the conjugation arc

(circles with radius R).

Pairings. Conjugation of two circles with an arc.

Constructing a circle passing through a given point A and tangent to the given circle

at a given point B.

Finding the middle of a straight line AB. Draw a perpendicular through the middle of line AB. Continuation intersection

Line OB and perpendicular gives a point O 1. O 1 - center of the desired circle with radius R = O 1 B = O 1 A.

Pairings. Internal tangency of circle and arc.

Constructing a conjugation of a circle with a straight line at a given point A on a straight line.

From a given point A of line LM we restore the perpendicular to the straight line LM. On continuation

We lay out a perpendicular segment AB. AB = R. We connect point B with the center of the circle O 1 with a straight line.

From point A we draw a straight line parallel to BO 1 until it intersects with the circle. Let's get a point TO- point

Touches. Let's connect point K to the center of circle O1. Let's extend lines O 1 K and AB until they intersect. Let's get a point

O 2, which is the center of the conjugate arc with the radius O 2 A = O 2 K.

Pairings. Conjugation of a circle with a straight line at a given point.

Constructing a conjugation of a circle with a straight line at point A specified on the circle.

External touch.

We carry out tangent to a circle through a point A. The intersection of the tangent with the straight line LM will give the point IN.

Divide the angle in half

O 1. O 1 O 1 A = O 1 K.

Inner touch.

We carry out tangent to a circle through a point A. The intersection of the tangent with the line LM will give the point IN.

Divide the angle, formed by the tangent and straight line LM, in half. The intersection of the angle bisector and

Continuation of the radius OA will give a point O 1. O 1 - O 1 A = O 1 K.

Pairings. Conjugation of a circle with a straight line at a given point on the circle.

Constructing the conjugation of two non-concentric circular arcs with an arc of a given radius.

Draw from the center of the arc O 1 auxiliary arc with radius R 1 -R 3 . Draw from the center of the arc ABOUT 2 auxiliary

Arc radius R 2 + R 3. The intersection of arcs will give a point O. O- center of the arc of conjugation with the radius R 3. Touch points

K 1 And K 2 lie on the lines OO 1 And OO 2.

Pairings. Conjugation of 2 non-concentric arcs of circles with an arc.

Construction of a pattern curve by selecting arcs.

By selecting the centers of arcs that coincide with sections of the curve, you can draw any pattern curve with a compass.

In order for the arcs to smoothly transition into one another, it is necessary that the points of their conjugation (touching)

They were located on straight lines connecting the centers of these arcs.

Sequence of constructions.

Selecting a center 1 arcs of an arbitrary section ab.

On continuation first radius, select the center 2 arc radius of the area bc.

On continuation second radius, select the center 3 arc radius of the area CD etc.

This is how we build the entire curve.

Pairings. Selection of arcs.

Constructing the conjugation of two parallel lines with two arcs.

Points defined on straight parallel lines A And IN connect with a line AB.

Choose on a straight line AB arbitrary point M.

Divide the segments AM And VM in half.

We restore perpendiculars in the middles of the segments.

At points A and B, given lines, we restore perpendiculars to the lines.

Intersection relevant perpendiculars will give points O 1 And O 2.

O 1 center of the arc of conjugation with the radius O 1 A = O 1 M.

O 2 center of the arc of conjugation with the radius O 2 B = O 2 M.

If the point M choose on middle lines AB, That radii arcs of conjugation will be are equal.

Arcs touching at a point M, located on the line O 1 O 2 .

Pairings. Conjugation of parallel lines with two arcs.