The construction of pattern curves is carried out as follows:

First, the points belonging to the curve are determined and then connected using a pattern. Pattern curves include the so-called conic sections of a parabola, hyperbola, ellipse obtained by cutting a circular cone with a plane, involute, sinusoid and others

1. Construction of an ellipse.

2. Ellipse focus

3. Construction of a parabola

6. Drawing pattern curves.

An ellipse is a conic section that belongs to the so-called pattern curves. An ellipse, hyperbola and parabola are obtained by cutting a circular cone with a plane, a sinusoid, an involute and other curves.

Figure 41. Intersection of a cone by a plane along an ellipse (a) and an ellipse (b).

In order to construct pattern curves (parabola, ellipse, hyperbola), the points that belong to the curve are determined and then all points are connected using a pattern. In the case when the surface of a circular cone is cut with an inclined plane -P, so that the inclined plane intersects all the generatrices of the circular cone, then an ellipse is formed in the section plane itself. (See Figure 41, a).

An ellipse is a flat closed curve in which the sum of the distances of each of its points - M to two given points F1 and F2 - is a constant value. This constant value is equal to the major axis of the ellipse MF1 + MF2 = AB. The minor axis of the ellipse CD and the major axis AB are mutually perpendicular and one axis divides the other in half.

Figure 42. Construction of an ellipse along the axes


Thus, the axes divide the ellipse curve into four pairwise symmetrical equal parts. If from the ends of the minor axis CD, as from the centers, we describe an arc of a circle with a radius equal to half the major axis of the ellipse R=OA=OB, then it will intersect it at points F1 and F2, which are called foci.

Figure 42 shows an example of constructing an ellipse along its axes. On the given axes AB and CD, as on diameters, we construct two concentric circles with the center at point O. We divide the large circle into an arbitrary number of parts and connect the resulting points with straight lines to the center O.

From intersection points 1; 2; 3; 4; with auxiliary circles we draw segments of horizontal and vertical lines until they intersect each other at points E, F, K, M, which belong to the ellipse. Next, using a pattern, the constructed points of a smooth curve are connected and the result is an ellipse.

Construction of pattern curves, parabola

Figure 43. Intersection of a cone by a plane along a parabola. Constructing a parabola using the focus and directrix.

If you cut a circular cone parallel to one of its generatrices with an inclined plane P, then a parabola is formed in the section plane (see Figure 43 a). A parabola is an open flat curved line. Each point of the parabola is located from the given straight line -MN, and from the focus -F at the same distance.

The straight line MN is a guide and is located perpendicular to the axis of the parabola. Between the guide -MN and the focus -F, the vertex of the parabola A is located right in the middle. In order to construct a parabola using the focus and a given guide, through the focus point -F, draw the axis of the parabola -X, perpendicular guide -MN.

Divide the segment-EF in half and get the vertex of the parabola-A. From the vertex of the parabola at an arbitrary distance, draw straight lines perpendicular to the axis of the parabola. From the point -F with a radius equal to the distance -L, from the corresponding straight line to the guide, for example CB, we make a straight line to this. In this case, points C and B.

Having thus constructed several pairs of symmetrical points, we draw a smooth curve through them using a pattern. Figure (43 c) shows an example of constructing a parabola tangent to two straight lines OA and OB at points A and B. The segments OA and OB are divided into the same number of equal parts (for example, divided into eight). After this, the resulting division points are numbered and connected by straight lines 1-1; 2-2; 3-3 (see Figure 43, c) and so on. These lines are tangent to the parabolic curve. A smooth tangent parabolic curve is then inscribed into the contour formed by the straight lines.

If you cut the direct and reverse cones with a plane parallel to its two generatrices or, in a particular case, parallel to the axis, then in the section plane you will get a hyperbola consisting of two symmetrical branches (see Figure 45, a).

Figure 45. Intersection of a cone by a plane along a hyperbola (a) and construction of a hyperbola (b).

A hyperbola (Figure 45,b) is a flat curve in which the difference in distances from each of its points to two given points F1 and F2, called foci, is a constant value and equal to the distance between its vertices a and b, for example SF1-SF2=ab. A hyperbola has two axes of symmetry - real AB and imaginary CD.

Two straight lines KL and K1 L1 passing through the center O of the hyperbola and touching its branches at infinity are called asymptotes. A hyperbola can be constructed from given vertices a and b and foci F1 and F2. We determine the vertices of the hyperbola by inscribing a rectangle in a circle constructed at the focal length (segment F1 and F2), as on the diameter.

On the real axis AB to the right of the focus F2 we mark arbitrary 1, 2, 3, 4, ... From the focuses F1 and F2 we draw arcs of circles, first with radius a-1, then b-1 until mutual intersection on both sides of the real axis of the hyperbola. Next, we will perform the mutual intersection of the next pair of arcs with radii a-2 and b-2 (point S) and so on.

The resulting intersection points of the arcs belong to the right branch of the hyperbola. The points of the left branch will be symmetrical to the constructed points relative to the imaginary axis CD.

A sinusoid is the projection of the trajectory of a point moving along a cylindrical helix onto a plane parallel to the cylinder axis. The motion of a point consists of a uniformly rotational movement (around the axis of the cylinder) and a uniformly translational movement (parallel to the cylinder).

Figure 46. Construction of a sinusoid

A sine wave is a flat curve that shows the change in the trigonometric sine function depending on the change in the magnitude of the angle. to construct a sinusoid (Figure 46), through the center O of a circle of diameter D, draw a straight line OX and on it plot a segment O1 A equal to the length of the circle π D. We divide this segment and circle into the same number of equal parts. From the obtained and numbered points we draw mutually perpendicular straight lines. We will connect the resulting intersection points of these lines using a smooth curve pattern.

Drawing pattern curves

Pattern curves are constructed by points. These points are connected using patterns, first drawing a curve by hand by hand. The principle of connecting individual points of a curve is as follows:

We select that part of the pattern arc that best coincides with the largest number of points of the outlined curve. Next, we will not draw the entire arc of the curve that coincides with the pattern, but only the middle part of it. After this, we will select another part of the pattern, but so that this part touches approximately one third of the drawn curve and at least two subsequent points of the curve, and so on. This ensures a smooth transition between the individual arcs of the curve.

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Construction of an ellipse

An ellipse is a closed flat convex curve, the sum of the distances of each point of which to two given points, called foci, lying on the major axis is constant and equal to the length of the major axis. The construction of an oval along two axes (Figure 23) is performed as follows:

• - draw axial lines on which segments AB and CD, equal to the major and minor axes of the ellipse, are laid symmetrically from the intersection point O;
• - construct two circles with radii equal to half the axes of the ellipse with the center at the point of intersection of the axes;
• - divide the circle into twelve equal parts. The division of the circle is performed as shown in paragraph 2.3;
• -diameter rays are drawn through the obtained points;
• - straight lines are drawn from the points of intersection of the rays with the corresponding circles parallel to the axes of the ellipse until they intersect each other at points lying on the ellipse;
• - the resulting points are connected by a smooth curved line using patterns. When constructing a pattern curve line, it is necessary to select and position the pattern so that at least four to five points are connected.

There are other ways to construct an ellipse.

Constructing a parabola

A parabola is a flat curved line, each point of which is equidistant from the directrix DD 1 - a straight line perpendicular to the axis of symmetry of the parabola, and from the focus F, a point located on the axis of symmetry. The distance KF between the directrix and the focus is called the parabola parameter p.

Figure 24 shows an example of drawing a parabola along vertex O, axis OK and chord CD. The construction is carried out as follows:

• - draw a horizontal straight line on which the vertex O is marked and the OK axis is plotted;
• - through point K, draw a perpendicular on which the length of the chord of the parabola is plotted symmetrically up and down;
• - construct a rectangle ABCD, in which one side is equal to the axis and the other is equal to the chord of the parabola;
• - side BC is divided into several equal parts, and segment KC into the same number of equal parts;
• - from the vertex of the parabola O, rays are drawn through points 1, 2, etc., and through points 1 1, 2 1, etc.;
• - draw straight lines parallel to the axes and determine the points of intersection of the rays with the corresponding parallel lines, for example, the point of intersection of the ray O1 with the straight line O1 1, which belongs to the parabola;
• - the resulting points are connected by a smooth curved line under the pattern. The second branch of the parabola is constructed in a similar way.

There are other ways to construct a parabola.

How to build a parabola? There are several ways to graph a quadratic function. Each of them has its pros and cons. Let's consider two ways.

Let's start by plotting a quadratic function of the form y=x²+bx+c and y= -x²+bx+c.

Example.

Graph the function y=x²+2x-3.

Solution:

y=x²+2x-3 is a quadratic function. The graph is a parabola with branches up. Parabola vertex coordinates

From the vertex (-1;-4) we build a graph of the parabola y=x² (as from the origin of coordinates. Instead of (0;0) - vertex (-1;-4). From (-1;-4) we go to the right by 1 unit and up by 1 unit, then left by 1 and up by 1; then: 2 - right, 4 - up, 2 - left, 3 - up; 9 - up, 3 - left, 9 - up If. these 7 points are not enough, then 4 to the right, 16 to the top, etc.).

The graph of the quadratic function y= -x²+bx+c is a parabola, the branches of which are directed downward. To construct a graph, we look for the coordinates of the vertex and from it we construct a parabola y= -x².

Example.

Graph the function y= -x²+2x+8.

Solution:

y= -x²+2x+8 is a quadratic function. The graph is a parabola with branches down. Parabola vertex coordinates

From the top we build a parabola y= -x² (1 - to the right, 1- down; 1 - left, 1 - down; 2 - right, 4 - down; 2 - left, 4 - down, etc.):

This method allows you to build a parabola quickly and does not cause difficulties if you know how to graph the functions y=x² and y= -x². Disadvantage: if the coordinates of the vertex are fractional numbers, it is not very convenient to build a graph. If you need to know the exact values ​​of the points of intersection of the graph with the Ox axis, you will have to additionally solve the equation x²+bx+c=0 (or -x²+bx+c=0), even if these points can be directly determined from the drawing.

Another way to construct a parabola is by points, that is, you can find several points on the graph and draw a parabola through them (taking into account that the line x=xₒ is its axis of symmetry). Usually for this they take the vertex of the parabola, the points of intersection of the graph with the coordinate axes and 1-2 additional points.

Draw a graph of the function y=x²+5x+4.

Solution:

y=x²+5x+4 is a quadratic function. The graph is a parabola with branches up. Parabola vertex coordinates

that is, the top of the parabola is the point (-2.5; -2.25).

Are looking for . At the point of intersection with the Ox axis y=0: x²+5x+4=0. The roots of the quadratic equation x1=-1, x2=-4, that is, we got two points on the graph (-1; 0) and (-4; 0).

At the point of intersection of the graph with the Oy axis x=0: y=0²+5∙0+4=4. We got the point (0; 4).

To clarify the graph, you can find an additional point. Let's take x=1, then y=1²+5∙1+4=10, that is, another point on the graph is (1; 10). We mark these points on the coordinate plane. Taking into account the symmetry of the parabola relative to the straight line passing through its vertex, we mark two more points: (-5; 6) and (-6; 10) and draw a parabola through them:

Graph the function y= -x²-3x.

Solution:

y= -x²-3x is a quadratic function. The graph is a parabola with branches down. Parabola vertex coordinates

The vertex (-1.5; 2.25) is the first point of the parabola.

At the points of intersection of the graph with the x-axis y=0, that is, we solve the equation -x²-3x=0. Its roots are x=0 and x=-3, that is (0;0) and (-3;0) - two more points on the graph. The point (o; 0) is also the point of intersection of the parabola with the ordinate axis.

At x=1 y=-1²-3∙1=-4, that is (1; -4) is an additional point for plotting.

Constructing a parabola from points is a more labor-intensive method compared to the first one. If the parabola does not intersect the Ox axis, more additional points will be required.

Before continuing to construct graphs of quadratic functions of the form y=ax²+bx+c, let us consider the construction of graphs of functions using geometric transformations. It is also most convenient to construct graphs of functions of the form y=x²+c using one of these transformations—parallel translation.

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Constructing a parabola is one of the well-known mathematical operations. Quite often it is used not only for scientific purposes, but also for purely practical ones. Let's find out how to perform this procedure using the Excel application tools.

A parabola is the graph of a quadratic function of the following type f(x)=ax^2+bx+c. One of its remarkable properties is the fact that a parabola has the form of a symmetrical figure consisting of a set of points equidistant from the directrix. By and large, constructing a parabola in Excel is not much different from constructing any other graph in this program.

Creating a table

First of all, before you start building a parabola, you should build a table on the basis of which it will be created. For example, let's take the construction of a graph of a function f(x)=2x^2+7.

Plotting a graph

As mentioned above, now we have to build the graph itself.

Editing a chart

Now you can slightly edit the resulting graph.

In addition, you can perform any other types of editing of the resulting parabola, including changing its name and the names of the axes. These editing techniques do not go beyond the scope of working in Excel with other types of diagrams.

As you can see, constructing a parabola in Excel is no fundamentally different from constructing another type of graph or diagram in the same program. All actions are performed on the basis of a pre-generated table. In addition, you need to take into account that the scatter diagram is most suitable for constructing a parabola.

Ellipse. If you cut the surface of a circular cone with an inclined plane R so that it intersects all its generators, then an ellipse will be obtained in the section plane (Figure 65).

Figure 65

Ellipse(Figure 66) – a flat closed curve in which the sum of distances from any of its points (for example, from a point M ) up to two given points F 1 And F 2 – the foci of the ellipse – there is a constant value equal to the length of its major axis AB (For example, F 1 M + F 2 M = AB ).Line segment AB is called the major axis of the ellipse, and the segment CD - its minor axis. The axes of the ellipse intersect at the point O- the center of the ellipse, and its size determines the lengths of the major and minor axes. Points F 1 And F 2 located on the major axis AB symmetrical about the point O and are removed from the ends of the minor axis (points WITH And D ) to a distance equal to half the major axis of the ellipse .

Figure 66

There are several ways to construct an ellipse. The easiest way is to construct an ellipse along its two axes using auxiliary circles (Figure 67). In this case, the center of the ellipse is specified - the point O and two mutually perpendicular straight lines are drawn through it (Figure 67, a). From the point ABOUT describe two circles with radii equal to half the major and minor axes. The large circle is divided into 12 equal parts and the division points are connected to the point ABOUT . The drawn lines will also divide the smaller circle into 12 equal parts. Then, horizontal lines (or straight lines parallel to the major axis of the ellipse) are drawn through the division points of the smaller circle, and vertical lines (or straight lines parallel to the minor axis of the ellipse) are drawn through the division points of the larger circle. The points of their intersection (for example, the point M ) belong to the ellipse. By connecting the resulting points with a smooth curve, an ellipse is obtained (Figure 67, b).

Figure 67

Parabola. If a circular cone is cut by a plane R , parallel to one of its generatrices, then a parabola will be obtained in the section plane (Figure 68).

Figure 68

Parabola(Figure 69) – a flat curve, each point of which is the same distance from a given straight line DD 1 , called headmistress, and points F – focus of a parabola. For example, for a point M segments MN (distance to the headmistress) and M.F. (distance to focus) are equal, i.e. MN = M.F. .

A parabola has the shape of an open curve with one axis of symmetry, which passes through the focus of the parabola - the point F and is located perpendicular to the director DD 1 .Accurate A , lying in the middle of the segment OF , called the vertex of the parabola. Distance from focus to directrix - segment OF = 2´OA – denoted by a letter R and call parabola parameter. The larger the parameter R , the more sharply the branches of the parabola move away from its axis. A segment enclosed between two points of a parabola located symmetrically relative to the axis of the parabola is called chord(for example, chord MK ).

Figure 69

Constructing a parabola from its directrix DD 1 and focus F(Figure 70, a) . Through the point F draw the axis of the parabola perpendicular to the directrix until it intersects the directrix at the point ABOUT. Line segment OF = p divide in half and get a point A – the top of the parabola. On the axis of the point parabola A lay down several gradually increasing sections. Through division points 1, 2, 3 it. D. draw straight lines parallel to the directrix. Taking the focus of the parabola as the center, they describe arcs with a radius R 1 =L 1 1 ,radius R2 = L2 until it intersects a line through a point 2 , etc. The resulting points belong to the parabola. First, they are connected by a thin smooth line by hand, then traced along the pattern.

Construction of a parabola along its axis, vertex A and intermediate point M(Figure 70, b).Through the top A draw a straight line perpendicular to the axis of the parabola, and through the point M – straight line parallel to the axis. Both lines intersect at a point B . Segments AB And B.M. are divided into the same number of equal parts, and the division points are numbered in the directions indicated by the arrows. Through the top A and dots 1 , 2 , 3 , 4 conduct rays, and from points I , II , III ,IV – straight lines parallel to the axis of the parabola. At the intersection of lines marked with the same number, there are points belonging to the parabola. Both branches of the parabola are the same, so the other branch is built symmetrically to the first using chords.

Figure 70

Construction of a parabola tangent to two straight lines OA and OB at points A and B given on them(Figure 71, b). Segments O.A. And OB divided into the same number of equal parts (for example, into 8 parts). The resulting division points are numbered and points of the same name are connected by straight lines. 1–1 , 2 –2 , 3 –3 etc . d . These lines are tangent to the parabolic curve. Next, a smooth tangent curve – a parabola – is inscribed into the contour formed by the straight lines. .

Figure 71

Hyperbola. If you cut the direct and reverse cones with a plane parallel to its two generatrices or, in a particular case, parallel to the axis, then in the section plane you will get a hyperbola consisting of two symmetrical branches (Figure 72, a).

Hyperbole(Figure 72, b) is called an open plane curve, which is a set of points, the difference in distances from two given points is a constant value.

Figure 72

Constant points F 1 And F 2 are called tricks , and the distance between them is focal length . Line segments ( F 1 M And F 2 M ), connecting any point ( M ) curve with foci are called radius vectors hyperboles . Difference between point and focus distances F 1 And F 2 is a constant value and equal to the distance between the vertices A And b hyperbole; for example, for a point M will have: F 1 M -F 2 M = ab. A hyperbola consists of two open branches and has two mutually perpendicular axes - valid AB And imaginary CD. Direct pq And rs, passing through the center O ,are called asymptotes .

Constructing a hyperbola using these asymptotes pq And rs, tricks F 1 And F 2 shown in Figure 72, b.

Real axis AB a hyperbola is the bisector of the angle formed by the asymptotes. Imaginary axis CD perpendicular AB and passes through the point ABOUT. Having tricks F 1 And F2, define the vertices A And b hyperbolas, why on a segment F 1 F 2 construct a semicircle that intersects the asymptotes at points m And P. From these points perpendiculars are lowered onto the axis AB and at the intersection with it we get vertices A And b hyperbole.

To construct the right branch of a hyperbola on a line AB to the right of focus F 1 mark arbitrary points 1 , 2 , 3 , ..., 5. Points V And V1 hyperbolas are obtained if we take the segment a5 beyond the radius and from the point F2 draw an arc of a circle, which is marked from the point F 1, radius equal to b5. The remaining points of the hyperbola are constructed by analogy with those described.

Sometimes you have to construct a hyperbola whose asymptotes OH And OY mutually perpendicular (Figure 73). In this case, the real and imaginary axes will be bis With ectrices of right angles. To construct, one of the points of the hyperbola is specified, for example, the point A.

Figure 73

Through the point A carry out direct AK And A.M. , parallel to the axes Oh And ou .From point O re With concepts about With they give her direct With straight lines A.M. And AK at points 1 , 2 , 3 , 4 And 1" , 2" , 3" , 4" . Next, vertical and horizontal segments are drawn from the points of intersection with these lines until they intersect each other at the points I, II, III, IV etc. The resulting points of the hyperbola are connected using a pattern . Points 1, 2, 3, 4 located on a vertical line are taken arbitrarily .

Involute of a circle or development of a circle. Involute of a circle is called a flat curve that is described by each point of a straight line if this straight line is rolled without sliding along a stationary circle (the trajectory of points of a circle formed by its deployment and straightening) (Figure 74).

To construct an involute, it is enough to specify the diameter of the circle D and the initial position of the point A (point A 0 ). Through the point A 0 draw a tangent to the circle and plot the length of the given circle on it D . The resulting segment and the circle are divided into the same number of parts and tangents to it are drawn in one direction through the dividing points of the circle. On each tangent, segments taken from the horizontal line and correspondingly equal are laid 1A 1 = A 0 1 , 2A 2 = V A 0 2 , 3A 3 = A 0 3 etc.; The resulting points are connected according to the pattern.

Figure 74

Archimedes spiral- a flat curve described by a point A , uniformly rotating around a fixed point – poles ABOUT and at the same time evenly moving away from it (Figure 75). The distance traveled by a point when turning a straight line by 360° is called the spiral pitch. The points belonging to the Archimedes spiral are constructed based on the definition of the curve, specifying the step and direction of rotation.

Construction of an Archimedes spiral using a given pitch (segment OA) and direction of rotation clockwise(Figure 75).Through a point ABOUT draw a straight line and mark the spiral pitch on it O.A. and, taking it as a radius, describe a circle. Circle and segment O.A. divided into 12 equal parts. Radii are drawn through the dividing points of the circle O1 , O2 , O3 etc. and on them from the point ABOUT are laid using arcs, respectively, 1/12, 2/12, 3/12, etc., of the radius of the circle. The resulting points are connected along a pattern with a smooth curve.

The Archimedes spiral is an open curve, and if necessary, you can construct any number of its turns. To construct the second turn, describe a circle with a radius R = 2 OA and repeat all previous constructions.

Figure 75

Sine wave.Sine wave is called the projection of the trajectory of the point moving With I'm cylindrical With which helix, on a plane parallel to the cylinder axis . The motion of a point consists of uniform rotational motion (around the cylinder axis) and uniform translational motion (parallel to the cylinder axis) . A sine wave is a flat curve that shows the change in the trigonometric sine function depending on the change in angle .

To build a sinusoid (Figure 76) through the center ABOUT circle diameter D carry out direct OH and a segment is laid on it O 1 A , equal to the circumference D. This segment and the circle are divided into the same number of equal parts. Mutually perpendicular straight lines are drawn from the obtained and numbered points. The resulting intersection points of these lines are connected using a smooth curve pattern.

Figure 76

Cardioid. Cardioid(Figure 77) calls With I am a closed trajectory of a point in a circle With that rolls without slipping along a stationary circle of the same radius .

Figure 77

From the center ABOUT draw a circle of a given radius and take an arbitrary point on it M. A series of secants are drawn through this point. On each secant, on both sides of the point of intersection of it with the circle, segments equal to the diameter of the circle are laid M1. Yes, secant III3МIII 1 intersects the circle at a point 3 ;segments are laid off from this point 3III And 3III 1, equal to diameter M1. Points III And III 1 , belong to the cardioid . Similarly, With current IV4MIV 1 re With circle at a point 4; segments are laid from this point IV4 And 4IV 1, equal to diameter M1, get points IV And IV 1 etc.

The found points are connected by a curve, as shown in Figure 77.

Cycloidal curves. Cycloids – plane curved lines described by a point belonging to a circle rolling without slipping along a straight line or circle . If the circle rolls in a straight line, then the point describes a curve called cycloid.

If a circle rolls along another circle, being outside it (along the convex part), then the point describes a curve called epicycloid .

If a circle rolls along another circle, being inside it (along the concave part), then the point describes a curve called hypocycloid . The circle on which the point is located is called producing . The line along which the circle rolls is called guide .

To construct a cycloid(Figure 78) draw a circle of a given radius R ; take the starting point on it A and draw a guide line AB, along which the circle rolls .

Figure 78

Divide the given circle into 12 equal parts (points 1" , 2" , 3" , ..., 12"). If the point A change With tit With I'm in a position A 12 , then the segment AA 12 will be equal to the given circumferential length With ty, i.e. . Draw a line of centers O – O 12 producing circumferentially With ti, equal , and divide it into 12 equal parts. Get points O 1 ,O2 ,O 3 ,..., O 12 , which are the centers of the generating circle With you . From these points draw in a circle With ty (or arcs around With tey) of a given radius R , which touch the line AB at points 1,2, 3, ..., 12. If from each point of contact we plot on the corresponding circle an arc length equal to the amount by which the point has moved A , then we obtain points belonging to the cycloid. For example, to get a point A 5 cycloids follows from the center O 5 draw a circle from the point of contact 5 lay an arc around the circumference A5, equal to A5", or from point 5" draw a straight line parallel AB, to the intersection at the point A 5 with a drawn circle . All other points of the cycloid are constructed similarly. .

The epicycloid is constructed as follows. Figure 79 shows the generating circle radius With A R with center O 0 , starting point A on it and the arc of the guide around With you radio With A R 1 along which it rolls With I am a circle. The construction of an epicycloid is similar to the construction of a cycloid, namely: divide a given circle into 12 equal parts (points 1" , 2" , 3" , ...,12"), each part of this circle is laid off from a point A along an arc AB 12 times (dots 1 , 2 , 3 , ..., 12) and get the arc length AA 12 . This length can be determined using the angle .

Further from the center ABOUT radius equal to OOO 0 , draw a line of centers of the generating circle and, drawing radii 01 , 02 , 03 , ...,012 , continued until they intersect with the line of centers, get centers O 1, O 2, ..., O 12 generating circle . From these centers with a radius equal to R , draw circles or arcs of circles on which they build and With which points of the curve; So, to get the point A 4 s should be checked With arc around With tee radius O4" until it intersects with a circle drawn from the center O4. Other points are constructed similarly, which are then connected by a smooth curve .

Figure 79

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