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 the 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 the 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. By general rule find the connecting points (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 draw a second arc intersecting 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

Postpone 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 arcs corresponding circles find points 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 rotates, back side which a spiral groove is cut, 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 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 use 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 according to given dimensions four circles and connect the ends of the vertical diameters of the 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 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.

Average level

Circle and inscribed angle. Visual guide (2019)

Basic terms.

How well do you remember all the names associated with the circle? Just in case, let us remind you - look at the pictures - refresh your knowledge.

Firstly - The center of a circle is a point from which the distances from all points on the circle are the same.

Secondly - radius - a line segment connecting the center and a point on the circle.

There are a lot of radii (as many as there are points on the circle), but All radii have the same length.

Sometimes for short radius they call it exactly length of the segment“the center is a point on the circle,” and not the segment itself.

And here's what happens if you connect two points on a circle? Also a segment?

So, this segment is called "chord".

Just as in the case of radius, diameter is often the length of a segment connecting two points on a circle and passing through the center. By the way, how are diameter and radius related? Look carefully. Of course, radius equal to half diameter

In addition to chords, there are also secants.

Remember the simplest thing?

Central angle is the angle between two radii.

And now - the inscribed angle

Inscribed angle - the angle between two chords that intersect at a point on a circle.

In this case, they say that the inscribed angle rests on an arc (or on a chord).

Look at the picture:

Measurements of arcs and angles.

Circumference. Arcs and angles are measured in degrees and radians. First, about degrees. There are no problems for angles - you need to learn how to measure the arc in degrees.

The degree measure (arc size) is the value (in degrees) of the corresponding central angle

What does the word “appropriate” mean here? Let's look carefully:

Do you see two arcs and two central angles? Well, a larger arc corresponds to a larger angle (and it’s okay that it’s larger), and a smaller arc corresponds to a smaller angle.

So, we agreed: the arc contains the same number of degrees as the corresponding central angle.

And now about the scary thing - about radians!

What kind of beast is this “radian”?

Imagine this: Radians are a way of measuring angles... in radii!

An angle measuring radians is like this central angle, the arc length of which is equal to the radius of the circle.

Then the question arises - how many radians are there in a straight angle?

In other words: how many radii “fit” in half a circle? Or in another way: how many times is the length of half a circle? greater than radius?

Scientists asked this question back in Ancient Greece.

And so, after long search they discovered that the ratio of the circumference to the radius does not want to be expressed in “human” numbers like, etc.

And it’s not even possible to express this attitude through roots. That is, it turns out that it is impossible to say that half a circle is times or times larger than the radius! Can you imagine how amazing it was for people to discover this for the first time?! For the ratio of the length of half a circle to the radius, “normal” numbers were not enough. I had to enter a letter.

So, - this is a number expressing the ratio of the length of the semicircle to the radius.

Now we can answer the question: how many radians are there in a straight angle? It contains radians. Precisely because half the circle is times larger than the radius.

Ancient (and not so ancient) people throughout the centuries (!) tried to more accurately calculate this mysterious number, to better express it (at least approximately) through “ordinary” numbers. And now we are incredibly lazy - two signs after a busy day are enough for us, we are used to

Think about it, this means, for example, that the length of a circle with a radius of one is approximately equal, but this exact length is simply impossible to write down with a “human” number - you need a letter. And then this circumference will be equal. And of course, the circumference of the radius is equal.

Let's go back to radians.

We have already found out that a straight angle contains radians.

What we have:

That means I'm glad, that is, I'm glad. In the same way, a plate with the most popular angles is obtained.

The relationship between the values ​​of the inscribed and central angles.

There is an amazing fact:

The inscribed angle is half the size of the corresponding central angle.

Look how this statement looks in the picture. A “corresponding” central angle is one whose ends coincide with the ends of the inscribed angle, and the vertex is at the center. And at the same time, the “corresponding” central angle must “look” at the same chord () as the inscribed angle.

Why is this so? Let's figure it out first simple case. Let one of the chords pass through the center. It happens like that sometimes, right?

What happens here? Let's consider. It is isosceles - after all, and - radii. So, (labeled them).

Now let's look at. This is the outer corner for! Remember that the outer corner equal to the sums two internal ones, not adjacent to it, and write:

That is! Unexpected effect. But there is also a central angle for the inscribed.

This means that for this case they proved that the central angle is twice the inscribed angle. But it hurts too much special case: Isn’t it true that the chord doesn’t always go straight through the center? But it’s okay, now this particular case will help us a lot. Look: second case: let the center lie inside.

Let's do this: draw the diameter. And then... we see two pictures that were already analyzed in the first case. Therefore we already have that

This means (in the drawing, a)

Well, I stayed last case: center outside the corner.

We do the same thing: draw the diameter through the point. Everything is the same, but instead of a sum there is a difference.

That's all!

Let's now form two main and very important consequences from the statement that the inscribed angle is half the central angle.

Corollary 1

All inscribed angles based on one arc are equal to each other.

We illustrate:

There are countless inscribed angles based on the same arc (we have this arc), they may look completely different, but they all have the same central angle (), which means that all these inscribed angles are equal between themselves.

Corollary 2

The angle subtended by the diameter is a right angle.

Look: what angle is central to?

Certainly, . But he is equal! Well, therefore (as well as many more inscribed angles resting on) and is equal.

Angle between two chords and secants

But what if the angle we are interested in is NOT inscribed and NOT central, but, for example, like this:

or like this?

Is it possible to somehow express it through some central angles? It turns out that it is possible. Look: we are interested.

a) (as an external corner for). But - inscribed, rests on the arc -. - inscribed, rests on the arc - .

For beauty they say:

The angle between the chords is equal to half the sum angular values arcs enclosed in this angle.

They write this for brevity, but of course, when using this formula you need to keep in mind the central angles

b) And now - “outside”! How to be? Yes, almost the same! Only now (we apply the property again external corner For). That is now.

And that means... Let’s bring beauty and brevity to the notes and wording:

The angle between the secants is equal to half the difference in the angular values ​​of the arcs enclosed in this angle.

Well, now you are armed with all the basic knowledge about angles related to a circle. Go ahead, take on the challenges!

CIRCLE AND INSINALED ANGLE. AVERAGE LEVEL

Even a five-year-old child knows what a circle is, right? Mathematicians, as always, have an abstruse definition on this subject, but we will not give it (see), but rather let us remember what the points, lines and angles associated with a circle are called.

Important Terms

Firstly:

center of the circle- a point from which all points on the circle are the same distance.

Secondly:

There is another accepted expression: “the chord contracts the arc.” Here in the figure, for example, the chord subtends the arc. And if a chord suddenly passes through the center, then it has a special name: “diameter”.

By the way, how are diameter and radius related? Look carefully. Of course,

And now - the names for the corners.

Natural, isn't it? The sides of the angle extend from the center - which means the angle is central.

This is where difficulties sometimes arise. Pay attention - NOT ANY angle inside a circle is inscribed, but only one whose vertex “sits” on the circle itself.

Let's see the difference in the pictures:

Another way they say:

There is one tricky point here. What is the “corresponding” or “own” central angle? Just an angle with the vertex at the center of the circle and the ends at the ends of the arc? Not certainly in that way. Look at the drawing.

One of them, however, doesn’t even look like a corner - it’s bigger. But a triangle cannot have more angles, but a circle may well! So: the smaller arc AB corresponds to a smaller angle (orange), and the larger arc corresponds to a larger one. Just like that, isn't it?

The relationship between the magnitudes of the inscribed and central angles

Remember this very important statement:

In textbooks they like to write this same fact like this:

Isn’t it true that the formulation is simpler with a central angle?

But still, let’s find a correspondence between the two formulations, and at the same time learn to find in the drawings the “corresponding” central angle and the arc on which the inscribed angle “rests”.

Look: here is a circle and an inscribed angle:

Where is its “corresponding” central angle?

Let's look again:

What is the rule?

But! In this case, it is important that the inscribed and central angles “look” at the arc from one side. For example:

Oddly enough, blue! Because the arc is long, longer than half the circle! So don’t ever get confused!

What consequence can be deduced from the “halfness” of the inscribed angle?

But, for example:

Angle subtended by diameter

You have already noticed that mathematicians love to talk about the same things. in different words? Why do they need this? You see, the language of mathematics, although formal, is alive, and therefore, as in ordinary language, every time I want to say it in a way that is more convenient. Well, we have already seen what “an angle rests on an arc” means. And imagine, the same picture is called “an angle rests on a chord.” On what? Yes, of course, to the one that tightens this arc!

When is it more convenient to rely on a chord than on an arc?

Well, in particular, when this chord is a diameter.

There is a surprisingly simple, beautiful and useful statement for such a situation!

Look: here is the circle, the diameter and the angle that rests on it.

CIRCLE AND INSINALED ANGLE. BRIEFLY ABOUT THE MAIN THINGS

1. Basic concepts.

3. Measurements of arcs and angles.

An angle of radians is a central angle whose arc length is equal to the radius of the circle.

This is a number that expresses the ratio of the length of a semicircle to its radius.

The circumference of the radius is equal to.

4. The relationship between the values ​​of the inscribed and central angles.

Introduction. Let us consider sequentially the conjugation of two straight lines, a straight line and an arc, and two arcs at given radius R.

Let us consider sequentially the conjugation of two straight lines, a straight line and an arc, and two arcs for a given radius R.

To construct a conjugation of two intersecting lines l 1 or l 2 at a distance of a given radius R, draw two auxiliary straight lines, respectively, parallel to the given ones l 1 And l 2 ( Figure 32). The intersection point of these lines is the conjugation center O. From the resulting center we lower the perpendiculars to the given lines - we obtain the conjugation points M and N . From the center O with the size of a given radius R draw an arc within the limits between the found points M and N.

To construct a conjugation of a straight line l with an arc of radiusR 1 , carried out from the center O 1 (Figure 33), draw an auxiliary line parallel to the line l , at a distance of a given conjugation radius R, and from the center O 1 draw an auxiliary arc with a radius R 1 + R. At the point of intersection of these auxiliary lines we obtain the center of mate ABOUT. From this center we lower the perpendicular to the straight line - we get a conjugation point on the straight line M, then connect the center ABOUT with arc center O 1 - at the intersection of a line OO 1 with a given arc we obtain a conjugation point on the arc - a point N. Between found points M And N radius R draw a mating arc.

Figure 32 Figure 33

To construct a conjugation of two arcs: arcs R 1 from the center O 1 and arcs R 2 from the center O2(Figure 34), draw two auxiliary arcs with radii respectively equal R 1 + R And R2+R . The intersection point of the auxiliary arcs determines the center of the mate - the point ABOUT. To define mate points M And N connect the center of mating ABOUT with the centers of given arcs O 1 And O2. Radius R draw a conjugation arc within MN.

Figure 34

Conjugation of two arcs at a given radius R possible with following condition: O 1 O 2 ≤ R 1 + 2R + R 2

Having considered the most typical cases of mates for a given radius, it is possible to identify a general rule for constructing mates for such cases. The center of the mate is determined at the intersection of two auxiliary lines parallel to the given arcs and spaced from the given lines at a distance of the mate radius.

Mating points are determined: on straight lines- perpendicular, lowered from the center of the mates to the straight line; on arcs- a straight line connecting the center of the mates with the center of a given arc (Figures 32 – 34).

7.2.2 Specified mate point

Let's consider several typical cases of conjugation of two straight lines, a line and an arc, and two arcs when one conjugation point is given M.

To construct a conjugation of two intersecting linesl 1 and l 2 (Figure 35) mate center ABOUT determined at the point of intersection of the perpendicular to the line l 1 , restored from a given point M, and the bisector of the angle formed by straight lines l 1 And l 2 . Second mate point N on a straight line l 2 determined using a perpendicular dropped from the center O directly l 2 . The mate radius is determined graphically: R X =| OM|= |ON| .

Figure 35

Constructing a straight line mate l c arc of radius R 1, carried out from the center O 1 . This problem can be solved in two ways, period M can be specified on an arc and on a straight line. Let's consider both options sequentially.

First option. Dot M specified on an arc. At the point M draw a tangent to the arc. The point of intersection of the bisector of the angle formed by a tangent and a given line l , with radius extension O 1 M determine the center of the mating arc ABOUT(Figure 36).

Second mate point N on a line is determined by the perpendicular dropped from the point ABOUT directly l . The mate radius was determined graphically: R X =| OM|= |ON| .

Figure 36 Figure 37

Second option. Dot M given on a straight line. From a given point M restore the perpendicular to the line l and put on it a distance equal to R 1(Figure 37). The resulting point TO connect to the center O 1 and divide the segment O 1 TO O determined at the intersection point of the perpendicular restored from the middle of the segment O 1 TO and a line passing through the points M And TO.

Second mate point N on an arc we determine at the point of intersection of the line O O 1 with a given arc. Blend radius R X =| OM| = |ON| .

Construct a conjugation of two arcs R 1 from the center O 1 and R 2 from the center O 2. Mating point M defined on an arc drawn from the center O 1 . Connecting a given point M with center O 1 and set aside on the continuation of the radius O 1 M distance equal to R 2(Figure 38). The further construction is similar to the previous case; received point TO connect to the center O2 and divide the segment KO 2 in half. Mating arc center ABOUT determined at the intersection point of the perpendicular restored from the middle of the segment KO 2, and a line passing through the points M And O 1 . The second conjugation point on the second arc is determined at the point of intersection of the arc with the straight line OO 2. Blend radius R X =| OM|= |ON| .

Figure 38

When tracing conjugate lines, you should first trace the arcs to the conjugation points, and then the straight sections.

7.3 Pattern curves

Pattern curves have great application in technology. Let's consider the most common methods of constructing plane curves: ellipse, parabola, cycloid, sinusoid, involute. These curves are usually outlined using patterns, which is why they are called pattern curves.

Ellipse(Figure 39). An ellipse is a closed plane curve for which the sum of the distances from any of its points to two points of the same plane - the foci of the ellipse - is a constant value equal to the major axis of the ellipse. The segment MN is called the major axis of the ellipse, and the segment DE is its minor axis. If you draw an arc with radius R=MN from point D or E: 2 , then on the major axis of the ellipse its foci (points F 1 And F 2).

Figure 39

To construct an ellipse, two concentric circles are drawn, the diameters of which are equal to the axes of the ellipse. These circles are divided into several parts (12...16). Draw through the division points on the great circle vertical lines, through the corresponding division points on the small circle - horizontal lines. The intersection of these lines will give the points of the ellipse I, II, III... (for other methods of constructing an ellipse, see the recommended literature).

Parabola(Figure 40). A parabola is a plane curve, each point of which is located at the same distance from a given straight line, called the directrix, and a point called the focus of the parabola, located in the same plane.

Let's consider one of the ways to construct a parabola. Given: vertex of parabola ABOUT, one of the points of the parabola D and the direction of the OS axis. A rectangle is constructed on segments OS and CD, the sides of this rectangle OB and BD are divided into arbitrary same number equal parts and number the division points. Vertex O is connected to the division points BD, and straight lines are drawn from the division points of the segment OB, parallel axes. The intersection of lines passing through points with the same numbers determines a number of points of the parabola (for other methods of constructing a parabola, see the recommended literature).

Figure 40

Cycloid(Figure 41). Point trajectory A belonging to a circle that rolls along a straight line without sliding is called a cycloid. To build it from starting position points A a segment is laid on the straight guide AA 1 , equal to length given circle 2πR . Circle and segment AA 1 divided into the same number of equal parts. Reconstructing perpendiculars from points dividing a line AA 1 until it intersects with a line passing parallel to the center of a given circle AA 1, mark a series of sequential positions of the center of the rolling circle O 1, O 2, O 3, ..., O 8. Describing circles of radius R from these centers, mark the points of intersection with them of straight lines running parallel AA 1, through the dividing points of the circle 1 ,2, 3, etc.

At the intersection of the horizontal line passing through point 1 with the circle described from the center O 1, one of the points of the cycloid is located; at the intersection of the straight line passing through point 2 with the circle drawn from the center O 2, there is another point of the cycloid, etc. By connecting the resulting points with a smooth curve, we obtain a cycloid.

Figure 41

Sine wave(Figure 42). To construct a sinusoid, divide a circle of a given radius into equal parts ( 6 , 8 , 12 etc.) and onwards center line from the conditional beginning - point A- draw a straight segment AB, equal 2πR . Then the straight line is divided into the same number of equal parts as the circle ( 6 , 8 , 12 etc.). From points on a circle 1, 2, 3, ..., 12 draw straight lines parallel to the selected line until they intersect with the corresponding perpendiculars restored or omitted from the points of division of the line. The resulting intersection points ( 1" , 2" , 3" , ... , 12" ) will be points of a sinusoid with an oscillation period equal to 2πR . Points 3" and 9" of the curve are the vertices of point A, 6 and B are inflection points.

Figure 42

Involute(circle scan, Figure 43). An involute is a trajectory described by each point of a straight line that rolls around a circle without sliding. In mechanical engineering, the profile of the teeth of gear wheels is outlined using an involute. To construct an involute, the circle is first divided into arbitrary number equal parts; At the division points, tangents to the circle are drawn, directed in one direction. On the tangent drawn through the last division point, lay a segment equal to the circumference 2πR, and divide it by the same number n equal parts. Laying down on the first tangent one division equal to πD/n, on the second - two, on the third - three, etc., get a series of points I, II, III etc., which are connected according to the pattern.

Figure 43

For the construction of hyperbolas, epicycloids, hypocycloids, Archimedes' spirals, strophoids, etc., see the recommended literature.

To trace a curve according to a pattern, it is recommended to connect the resulting points with a thin line by hand by eye, while trying to give the curved line the smoothest possible outlines, and only after that select a pattern corresponding to the curvature of one or another section of it (Figure 44), connecting at least three points at the same time.

Figure 44

7.4 Conjugates of a straight line with pattern curves (tangents to pattern curves)

Previously, various cases of conjugating straight lines, a straight line with an arc, and two arcs were considered. In practice, it is not uncommon to pair a straight line with pattern curves, in which the mating straight line must be directed tangent to the curve drawn through a given conjugation point.

Let's look at examples of construction mates of a straight line with an ellipse(Figure 45). Mating point specified D. The tangent to the ellipse at a given point is perpendicular to the bisector of the angle formed by the straight lines F 1 D And F 2 D, Where F 1 And F 2- focuses of the ellipse.

Figure 45

Figure 46 shows the construction tangent to the parabola at a given point M. A tangent connects a given point M with a dot TO, the position of which is determined by the relation AK=AN. Methods for constructing tangents to other given pattern curves can be studied in the recommended literature.

Figure 46

7.5 Self-test questions

Self-test questions for topic 1:

1. How many A4 sheets are contained in an A1 sheet?

2. How are additional drawing formats created?

3. What determines font size?

4. What is the height? lowercase letters compared with
in capitals?

5. Is it possible to use roman font in drawings?

6. What determines the choice of the thickness of the stroke line of the visible contour?

7. What type and thickness are the axial, center, extension, dimensional and invisible contour lines drawn?

8. How are the center lines of a circle of small diameter (less than 12 mm) drawn?

9. In what units are dimensions placed on drawings?

11. In what cases is the arrow of a dimension line replaced by a dot or a stroke?

12. How are the angle size numbers arranged?

13. In what cases is the diameter sign Æ put down?

14. What are the dimensions when making a drawing on a scale other than 1:1?

15. On what two positions of geometry is the construction of mates based?

16. List the elements of mates.

Introduction

Study of intensively developing and knowledge-intensive subject area, such as microelectronics and microprocessor technology, is an interesting and complex task that requires constant improvement, replenishment of acquired knowledge and familiarity with related scientific and technical fields. Due to the widespread use electronic systems management and for the purpose effective solution any applied problems modern specialist, professionally related and not related to computer technology, must have not only elementary representation about the basic concepts of building modern electronic systems, but also have an adequate understanding of the state and prospects for the development of the element base.

The development of computer technology - the highest achievement of electronics - over the last decade has progressed in such strides that today it is almost impossible to imagine any area of ​​life where microprocessors (MPs) are not used: from personal computers- to managing the most complex technological processes, from controlling household washing machines and cell phones- to design workstations and multiprocessor supercomputers.

In just over a quarter of a century of history, microprocessors have come a truly gigantic way.

The first MP microcircuit, released by INTEL in 1971, operated at a clock frequency of 108 kHz, contained 2300 transistors, was made using 10 micron technology and cost about $200. One of the latest modifications of the INTEL PENTIUM-4 chip is made using 0.09 micron technology and has 140 million transistors inside a semiconductor crystal measuring 87 sq. mm.

A comparison of the above data also confirms the figurative assessment of the success of the microprocessor industry given by the founder and chairman of the board of directors of INTEL, Gordon Moore: “If the automotive industry had evolved at the speed of the semiconductor industry, then today a Rolls-Royce would cost $3, could drive half a million miles on a gallon of gas, and it would be cheaper to throw it away than to pay for parking.”

It is not difficult to understand that today computerization is one of the main directions scientific and technological progress and its concentrated expression. MP embodies the most advanced achievements engineering thought, and on the extent to which they are saturated computer technology the most various industries production depends not only on the economic, but also on the military potential of the country.

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 common point for 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 in this example construction will be considered pairing right angle 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 from two arbitrary points There are two arcs 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 perpendiculars from the center of the mate to each of the sides and connect with an arc equal to the radius of the mate obtuse angle R, obtained points a and b.

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

First, let's find the center of conjugation. To do this, let's draw a straight line, parallel to the segment AB and spaced from it by a distance of 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.

When studying the discipline " descriptive geometry and engineering graphics" students must learn the rules and sequence of performing geometric constructions and connections. In this regard, the best way to acquire construction skills is through tasks to draw the contours of complex parts.

Before you begin the test task, you need to study the technique of performing geometric constructions and connections according to the methodological manual.

Line mates

Conjugation is a smooth transition from one line to another. To construct any mate with an arc of a given radius, you need to find:

1. Conjugation center – the center from which the arc is drawn;
2. Conjugation points (touching points) are points at which one line passes into another.

The center of the mate is located from the mate points at equal distances equal to the mate radius R. The transition from a straight line to a circle will be smooth if the straight line touches the circle. The conjugation point K lies on the perpendicular dropped from the center O of the circle to the straight line (Fig. 1)

The transition from one circle to another will be smooth if the circles touch.

There are two cases of contact between arcs of circles: external (Fig. 2) and internal (Fig. 3).

When touching externally, the centers of the circles lie along different sides from their common tangent L (Fig. 2). The distance between their centers OO 1 is equal to the sum of the radii of the circles R+R 1 and the point of contact lies on the straight line OO 1 connecting their centers.

With an internal tangency, the centers of the circles lie on one side of their common tangent L. The distance between their centers OO 1 is equal to the difference between them radii R-R 1 and the tangency point K of the circles lies on the continuation of straight line OO 1 (Fig. 3).

Tangent arcs of circles:

rice. 2– conjugation of two circles (external tangency)

rice. 3– conjugation of two circles (internal tangency)

Conjugation of two intersecting lines

Given are straight lines intersecting at right, acute and obtuse angles.

It is required to construct mates of these straight lines with an arc of a given radius R.

1. To find the center of the conjugation, draw auxiliary straight lines parallel to the data at a distance equal to the radius R. The point of intersection of these straight lines will be the center of the conjugation arc (Fig. 4).
2. The perpendiculars dropped from the center of the conjugation arc t.O onto these straight lines determine the points of tangency K and N.
3. From point O, as the center, they describe an arc of a given radius R.

Note. For right angles, it is more convenient to find the center of mate using a compass (Fig. 5).

Conjugation of a circular arc and a straight line with an arc of a given radius.

External touch

Given a circle of radius R and a straight line AB. It is required to connect them with an arc of radius R1.

1. To find the center of mate, an arc is drawn from the center O of a given circle m radius R + R 1 and at a distance R 1 – straight n// AB. Point O 1 of the intersection of the line n and arcs m will be the center of conjugation.
2. To obtain the connecting points: K and K 1 draw a line of centers OO 1 and restore the perpendicular OK 1 to the straight line AB.
3. From the center of mate O 1 between points K and K 1 draw a mate arc of radius R 1

Inner Touch

In the case of internal contact, the same constructions are performed, but the arc m of the auxiliary circle is drawn with radius R - R 1.

Conjugation of two circles with an arc of a given radius

Two circles of radius R 1 and R 2 are given. It is required to construct a mate with an arc of a given radius R.

External touch

1. To determine the center of conjugation O, auxiliary arcs are drawn: from the center O 1 of a circle with radius R + R 1 and from the center O 2 of a circle of radius R + R 2. Point O of the intersection of these arcs is the center of conjugation.
2. By connecting the centers O and O 1, as well as O and O 2, the points of conjugation (touching) K 1 and K 2 are determined.
3. From center O with radius R, draw a conjugation arc between points K 1 and K 2

Inner Touch

With an internal touch, the same constructions are performed, but the arcs are drawn with radii

R - R 1 and R - R 2 .

Mixed touch

The center of mate O is located at the intersection of two arcs described from the center O 1 with radius R - R 1 and from the center O 2 with radius R + R 2

Note. In a mixed conjugation, the center O 1 of one of the mating arcs lies inside the conjugate arc of radius R, and the center O 2 of the other arc lies outside it.

Special cases

Finding the center of an arc of a given radius.

Given an arc of radius R connecting two parallel lines m And n and passing through point A ∈ m(Fig. 11). It is required to find the center O of a given arc.

The construction is based on finding point O, equidistant from the given lines (Fig. 11).

1. From point A ∈ m, as if from the center, draw an arc of an auxiliary circle with a given radius R.
2. Draw an auxiliary line l, parallel to the line n, at a distance equal to the given radius R.
3. Point O - the point of intersection of these auxiliary lines is the center of a given arc. (Fig. 12)

Literature

1. Bogolyubov S.K. Engineering graphics: Textbook for secondary specialized educational institutions. – 3rd ed., rev. And additional - M.: Mechanical Engineering, 2006. – p. 392: ill.
2. Kuprikov M.Yu. Engineering graphics: textbook for secondary educational institutions - M.: Bustard, 2010 - 495 pp.: ill.
3. Fedorenko V.A., Shoshin A.I. Handbook of mechanical engineering drawing L.: Mechanical engineering. 1976. 336 p.