Various cases of mates when drawing drawings. Line fillet How to make external fillet in drawing
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.
Conjugation is a smooth transition from one line to another. A smooth transition can be made using circular lines
(circular arcs) and with the help of pattern curves (ellipse, parabola or hyperbola arcs). We will only consider cases of conjugations using circular arcs. From the whole variety of conjugations of various lines, the following main types of conjugations can be distinguished: conjugation of two differently located straight lines using a circular arc, conjugation of a straight line with a circular arc, construction of a common tangent to two circles, conjugation of two circles with a third. Any type of pairing should be performed in the following sequence:
– find the center of the mating arc,
- find connecting points,
– a conjugation arc is drawn with a given radius.
Various types of interfaces are shown in Table 2:
table 2
| Graphic construction of mates | Brief explanation of the construction |
| Conjugation of intersecting straight lines with an arc of a given radius | |
| Draw straight lines parallel to the sides of the angle at a distance R. From point O, the mutual intersection of these lines, dropping perpendiculars to the sides of the angle, we obtain conjugation points 1 and 2. With radius R, draw a conjugation arc between points 1 and 2. | |
| Conjugate a circle and a straight line using an arc of a given radius | |
 | At a distance R, draw a straight line parallel to the given straight line, and from the center O 1 with radius R + R 1 - an arc of a circle. Point O is the center of the mating arc. We get point 2 on the perpendicular lowered from point O to a given straight line, and point 1 at the intersection of straight line OO 1 and a circle of radius R. |
Continuation of table 2
| Conjugation of arcs of two circles with a straight line | |
 | From point O, draw an auxiliary circle with radius R-R 1. Divide segment OO 1 in half and from point O 2 draw a circle with a radius of 0.5 OO 1. This circle intersects the auxiliary circle at point K 0. By connecting point K 0 with point O 1 we obtain the direction of the common tangent. We find the tangent points K and K 1 at the intersection of perpendiculars from points O and O 1 with given circles. |
| Conjugation of arcs of two circles with an arc of a given radius (external conjugation) | |
 | From centers O 1 and O 2, draw arcs of radii R+R 1 and R+R 2. When these arcs intersect, we obtain point O - the center of the mating arc. Connect points O 1 and O 2 with point O. Points K and K 1 are conjugation points. Between points K and K1, draw a conjugation arc of radius R. |
Continuation of table 2
| Conjugation of arcs of two circles with an arc of a given radius (internal conjugation) | |
 | From the centers O 1 and O 2, draw arcs of radii R-R 1 and R-R 2. When these arcs intersect, we obtain point O - the center of the conjugation arc. Connect points O 1 and O 2 with point O until they intersect with the given circles. Points K and K 1 are conjugation points. Between points K and K 1 with radius R we draw a conjugation arc. |
| Conjugation of arcs of two circles with an arc of a given radius (mixed conjugation) | |
 | From the centers O 1 and O 2, draw arcs of radii R-R 1 and R+R 2. When these arcs intersect, we obtain point O - the center of the conjugation arc. We connect points O 1 and O 2 with point O until they intersect with the given circles. Points 1 and 2 are junction points. Between points 1 and 2 with radius R we draw a conjugation arc. |
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 the 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, or 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 |
>>Drawing: Mates
A smooth transition from one line to another is called pairing. The point common to the mating lines is called the junction point, or transition point. To construct mates, you need to find the center of the mate and the mate points. Let's look at the different types of mates. Right angle conjugation.
Let it be necessary to mate a right angle with a mate radius equal to the segment AB (H=AB). Let's find the connecting points. To do this, we place the leg of the compass at the top of the angle and with a compass opening equal to the segment AB, we make notches on the sides of the angle. The resulting points a and b are conjugation points. Let's find the center of the junction - a point equidistant from the sides of the angle. Using a compass opening equal to the conjugation radius, from points a and b we draw two arcs inside the corner until they intersect with each other. The resulting point O is the center of mate. From the center of the junction we describe an arc of a given radius from point a to point b. First we draw an arc, and then straight lines (Fig. 70).
Conjugation of acute and obtuse angles. To construct the conjugation of an acute angle, we take a compass opening equal to the given radius H = AB. Let us alternately place the leg of the compass at two arbitrary points on each side of the acute angle. Let's draw four arcs inside the corner, as shown in Fig. 71, a.
We draw two tangents to them until they intersect at point O - the center of the conjugation (Fig. 71, b). From the center of the junction we lower the perpendiculars to the sides of the angle.
The resulting points a and b will be the conjugation points (Fig. 71, b). Having placed the leg of the compass at the center of the conjugation (O), with a compass opening equal to the given radius of the conjugation (H = AB), we will draw a conjugation arc.
Similar to the construction of the conjugation of an acute angle, the conjugation (rounding) of an obtuse angle is constructed. Conjugation of two parallel lines. Given two parallel lines and a point<1, лежащая на одной из них (рис.72). Рассмотрим последовательность построения сопряжения двух прямых. В точке (1 восставим перпендикуляр до пересечения его с другой прямой. Точки d и е являются точками сопряжения. Разделив отрезок de пополам, найдем центр сопряжения. Из него радиусом сопряжения проводим дугу, сопрягающую прямые.
Conjugation of arcs of two circles with an arc of a given radius
There are several types of conjugation of the arcs of two circles with an arc of a given radius: external, internal and mixed. Consider an example of the external conjugation of the arcs of two circles with an arc of a given radius. The radii R1 and R2 of the arcs of two circles are given (the lengths of the radii are shown as line segments). It is necessary to construct their conjugation with a third arc of radius R (Fig. 73, a). To find the center of the mate, we draw two auxiliary arcs: one with a radius O 1 O = R 1 + R, and the other O 2O = R 2 + R. The intersection point of the auxiliary arcs is the center of the mate.
The conjugation points K lie at the intersection of the lines O 1 O and O 2O with the arcs of given circles. From the center of the mate, we draw an arc with the mate radius, connecting the mate points. When outlining constructions, first draw a conjugation arc, and then the arcs of conjugate circles (Fig. 73, b).
Internal conjugation of arcs of two circles with an arc of a given radius. With internal conjugation, the mating arcs of circles are located inside the conjugation arc (Fig. 74). Given two arcs of circles with centers O 1 and O 2, the radii of which are respectively equal to R 1 and R 2. It is necessary to construct a conjugation of these arcs with a third arc of radius R. Find the center of the conjugation. To do this, from the center O 1 with a radius equal to R-R 1 and from the center O 2 with a radius equal to R-R 2, describe auxiliary arcs until their mutual intersection at point O. Point O will be the center of the conjugate arc of radius R. Conjugation points K lie on the lines OO 1 and OO 2 connecting the centers of the circular arcs with the mate center.
Conclusion. When determining the size of the radii of auxiliary arcs, you should:
a) for external conjugation, take the sum of the radii of the given arcs and the conjugation radius, i.e. R 1 + R; R 2 + R (Fig. 73);
b) for internal conjugation, you need to use the difference between the conjugation radius R and the radii of the given circular arcs, i.e. R-R 1 and R-R 2 (Fig. 74). 
Questions and tasks
1. What is called pairing?
2. Which point is called the center of conjugation?
3. Which points are the connecting points?
Graphic work
Using a visual representation of the part, draw it using the rules for constructing mates (Fig. 75).
N.A. Gordeenko, V.V. Stepakova - Drawing., 9th grade
Submitted by readers from Internet sites
PRACTICAL LESSON No. 4
TOPIC: CONJUGING STRAIGHTS AND CIRCLES
MATCHES APPLIED IN THE CONTOURS OF TECHNICAL DETAILS
Conjugation is the smooth transition of one line to another.
The point at which one line passes into another is called mate point.
Arcs, with the help of which a smooth transition from one line to another is carried out, are called arcs of mates.
Tangent is a straight line that has only one common point with a closed curve. This is the limiting position of the secant, the points of intersection of which with the curve, tending towards each other, merge into one point - the point of tangency.
The construction of conjugations is based on the properties of tangents to curves and comes down to determining the position of the center of the conjugating arc and the conjugation (touching) points, i.e. points at which given lines transform into a connecting arc
CONJUNCTION OF ANGLES (CONJUNCTION OF INTERSECTING STRAIGHTS)
Right angle mate
(Conjugation of intersecting lines at right angles)
In this example, we will consider the construction of a right angle mating with a given mating radius R. First of all, we will find the mating 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.
Acute angle mate
(Conjugation of intersecting lines at an acute angle).
Another example of conjugating an angle. In this example, an acute angle mate will be created. 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 is how we get the connecting points a And b. Then we draw from the center of conjugation, points ABOUT, arc with conjugation radius R, connecting the connecting 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)
The conjugation of an obtuse angle 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 perpendiculars from the center of the mate to each of the sides and connect with an arc equal to the mate radius of the obtuse angle R, received points a And b.
