Engineering graphics oval. How to correctly draw a circle in perspective, draw an ellipse. Task: draw a cylinder in a horizontal non-facade position
For larger drawing scales, you need to move further away. When controlling the drawing, you need to squint your eyes, as with the clarification method. The paper should lie perpendicular to the direction of view. We invite you to consider. Drawing the perspective of a jug. Familiarity with the seven main ways in which we observe, identify, draw, depict and test objects is not an end in itself, but only serves as preparation for their use in solving specific problems. Drawers will be able to solve any promising problems without difficulty if the correct application of these seven methods becomes a skill for them. Without proper assimilation and practice of them, it is impossible to achieve satisfactory results. The examples of how draftsmen use the techniques and skills given below do not exhaust the possible options. The proposed drawing method is not the only one. Which of the described methods and skills and to what extent the teacher uses when working with children depends on the age of the students, on the degree of their development, on the teacher’s ability to use the material, on the learning objectives, on the intended degree of accuracy, etc. If the drawing should be precise, which means the action must be precise. The teacher himself decides which methods and in what sequence should be used according to the preparedness of the students. The use of all the above methods with students aged 12 years and older turned out to be positive. Even eleven-year-old students learned some methods. It is acceptable for self-taught people to drew from life, using seven methods when depicting vertical lines and divisions in perspective, but they must be able to depict facade planes and know the proportionality of surface dimensions. In school practice, this method has proven itself when teaching perspective drawing begins with the depiction of cylindrical bodies and only after that moves on to multifaceted ones.
Ways to depict small round bodies
This exercise is convenient to carry out on individual models that students have made in circles under the guidance of a teacher. A circle is cut out of a carbolite board and polished so that it can rotate freely around a horizontal axis (this can be pins stuck into the board). The board can be inserted into a wooden stand so that the rotation axis is vertical. A convenient square side size is 250 mm, the radius of the cut circle is 105 mm 7.1st drawing exercise
It is necessary to draw a moving circle in different positions. By the way, we have already discussed this in a separate article earlier. Place the model with a square on the front. Turn the inner circle in front of the students from the main position front to horizontal. Students see it as a circle tapering into an ellipse. They can compare the size CD with AB either visually or by comparing the sizes CD and AB (Fig. 1). Dimension CD can be obtained in the figure by rotating any circle with diameter AB. The CD to AB ratio should then be checked or measured on the model. After it has been established by measurement that CD is equal to half AB, in the figure AB is divided in half and the image size CD is obtained. When drawing at school, there is no need to pay attention to the theory of perspective reduction between DS and CS. We compare the entire radius CD.
Figure 1 - Movable circle in different planes. If you rotate the square out of the way, further variations will arise, but these should be explained to more advanced students. The square board will be in a vertical facade position, while the inner circle will be either in a vertical, horizontal, non-facade position, or in a façade position. These tasks should also not be considered an end in themselves. They should be considered as exercises to develop a system of independent work skills. How the size of the ellipse axis is formed and changes in accordance with the movement of a cylindrical body vertically to the plane of observation, above it or below it - all these phenomena must be observed with students individually and collectively before establishing dimensions by measurement. Drawing an ellipse
Drawing an ellipse should begin after this process becomes completely clear to students. Subsequently, students will better understand how to represent perspective circle in different positions, whether it will be horizontal, close to it, or distant from it. When it is determined that the minor axis of the ellipse can be plotted on the major axis dimension more than once or more than once, while often the minor axis of the ellipse is enlarged so that it is plotted on the major axis fewer times, students should not measure these quantities, and determine their ratio of sizes by division or multiplication - 1:4, 1:1, etc. In the same way, you should demonstrate to students movement of a circle along a horizontal plane in the direction from the eye into the distance and analyze these phenomena.
Figure 2 - Drawing an ellipse. Before drawing cylindrical bodies in a non-facade position, it is necessary to show and draw a vertical square with a circle inscribed in it in a non-facade position. The axis of the ellipse in the figure does not coincide with the axis of rotation, but will tilt towards the sharp corners of the perspective square. Correctly draw an ellipse- the task is not easy. In geometric drawing, to make it easier to construct an ellipse, the following methods are sometimes used: - Construct an ellipse using a paper strip. Segments equal to half the axis are applied to a strip of paper (Fig. 3, top left) so that MS = a (major semi-axis of the ellipse), PM = b (auxiliary semi-axis of the ellipse). If point S passes through the auxiliary axis, and at the same time point P passes through the main axis, point M forms the circle of an ellipse.
- If you need to inscribe an ellipse into a given quadrilateral (Fig. 3 at the top right) so that straight lines AB, CD are its axes, you can use the following methods:
- drawing an ellipse using thread. Using the distance AS = CF1 = CF2 = a (the length of the major semi-axis of the major radius), we determine the foci of the ellipse F1, F2. They contain the ends of a thread 2a long. By stretching the thread with a pencil lead, we simultaneously gradually inscribe a half-ellipse in half of the square AB, C and the second half of the ellipse in the square AB, D.
- method of inscribed circles at the vertices of the main and auxiliary axes.
Figure 3 - Freehand ellipse. The accuracy of a hand-drawn ellipse is usually checked by using a strip of paper or by drawing the ellipse using arcs at the end points of the major and minor axes. The basis for this second method is the inscribed circle method described above. Ellipses in drawing are not drawn, but drawn. Drawing a cylinder in a plane
1st method. In contour drawing for students of first-level schools, we depicted a cylinder in the form of a quadrangle, while its upper and lower circular areas were depicted as horizontal straight lines. Students most often determined the ratio of the dimensions of a quadrilateral by eye. When drawing a cylinder in perspective at the 2nd stage of training, students can again start from a profile image of the cylinder as a quadrilateral, the base of which they draw according to their own idea. On the upper base, the upper ellipse, its minor axis is compared with the major one. If the ellipse is five times smaller than the width of the top stem, students will divide it into five parts in their drawing. They will draw one fifth as a representation of the height of the upper ellipse. On the bottom base of the cylinder, which is drawn on paper placed under the cylinder, the minor axis is compared with the major one, if the cylinder was previously moved from the paper, which does not display the lower area of the cylinder. This ellipse will appear taller (Figure 4).
Figure 4 - Drawing a cylinder with a pencil. When compared, it turns out that the height fits into the width less times. Based on this mapping, you should divide the width display in the figure and draw an ellipse. 2nd method. An auxiliary quadrangle is installed into which the entire cylinder with both bases is drawn. This second possibility is easier to grasp in classes where students will later draw a rotating body in profile and this drawing will end in a side silhouette. Performance. Paper is placed under the base of the cylinder so that its front end is horizontally facing. The bottom of the cylinder is shaded on it. Students establish and indicate the highest point Y, the lowest X, the lateral left A, the lateral right B (Fig. 4). At the lowest point X, an auxiliary façade horizontal straight line АХВ is drawn on the underlying paper. The projections of the extreme points of the width of the cylinder are marked and indicated on it, while one eye is closed, the other is squinted. We move the pencil in a bent hand in a vertical position so that it coincides with the straight surface of cylinder A. Vertical pencil moves along the horizontal plane to position OAa. The horizontal straight line of the base of the cylinder, passing in the lower projection of the point of the cylinder, is already marked on the underlying paper at point A. If you need to show the projection of the surface of the cylinder b on the straight line, which is projected onto the lowest point X, then the horizontal-facade straight line b must first of all be looked at like this , so that the pencil held vertically coincides with the straight surface and its projection A. Then, without turning your head, you need to apply the pencil, held in a bent hand, to the straight surface b, just as it was done before on surface a. After this, we get projection B. We shade it on paper and check the correctness of the image by doing the opposite. Exercise Analysis . Students observe from one point. The pencil is held in a bent hand (if it is held in an outstretched hand, it is impossible to cover such a significant plane). Each student has a model of a cylinder in front of them. With action #1 he sets the highest point Y and the lowest point X. The distance between them is the height of the auxiliary quadrilateral. The side of the auxiliary quadrangle is established as follows: a vertically standing façade pencil with a straight surface a and an eye O formed an imaginary plane directed from the eye to a straight line a. The pencil, eye and straight surface a will intersect the horizontal plane at the line of intersection OA. If we apply a pencil to the center of the cylinder, we obtain the OXY plane; with further movement of the pencil so that it coincides with the straight surface b, an imaginary plane OBb is formed. The line of intersection of this plane with the horizontal plane is straight line OBb. The segment AB is the projection of the width of the cylinder and the basis of the auxiliary quadrilateral. This segment represents the auxiliary facade plane, in which the auxiliary quadrangle for this cylinder is located. In this case, the side of the auxiliary quadrilateral is a horizontal front line on the underlying paper, which passes through the projection of the lowest point of the body. Its extreme points form projections of the left and right extreme straight surface of the body. This is the situation with height. The height of the auxiliary quadrilateral is the distance of the perpendiculars lowered from the point that seems to us to be the highest to its projection on the auxiliary horizontal front line, which passes through the projection of the lowest point of the body. Students determine the dimensions of an auxiliary quadrilateral by comparing its smaller side with its larger one. If it turns out that in the long side of the quadrilateral its height fits 1.5 times, then it can be drawn like this: set aside a convenient arbitrary width of the quadrilateral and consider it the base, and taking its height 1.5 times less, build an auxiliary quadrilateral. You can do it another way: divide any height that is convenient in this case with a paper strip so that the height can be set aside 1.5 times on most of the part. This will be the desired width of the auxiliary quadrilateral. If during the measurement we met the height twice, we then need to divide the taken height into two parts in the figure, one of the parts will be the desired width. On the paper placed under the model, the straight line going to the eye is the direction of the vertical pencil. In the figure it is the vertical side or height of the auxiliary quadrilateral. In the figure, both vertical sides of the auxiliary quadrilateral are checked with a vertically placed pencil. At the third stage of education, students need to emphasize this rule. When built correctly auxiliary quadrilateral, students draw the vertical axis of the body. Then its bases are determined and depicted. On the vertical axis of the cylinder, the relationship between the axes of the upper, visible ellipse is established by comparison. If it is established that the smaller axis is laid on the larger one 6 times, under the figure they notice: 1: 6 (students usually forget this). Then the width of the projection of the cylinder is divided into 6 parts (approximately by eye, but checked with a piece of paper) and one sixth of the part is applied from the highest horizontal straight line to the vertical axis of the auxiliary quadrilateral. An ellipse is inscribed in the quadrilateral shown. If you need to determine the lower ellipse, measure its size as it is sketched on the underlying paper. The model is pushed back for this time. Sketching the base of the model is necessary when drawing groups of objects, especially polygonal ones. Exercise: determine and depict the projection of the lowest point of the body.
To determine, where the point will be projected from space onto the paper placed under the model, you can use the methods shown in Fig. 5. It shows the projections of the points that appear lowest to the observer.
Figure 5 - Projections of points during perspective drawing. Define and draw: - a quadrangle for the entire object (without the ear) and its axis;
- ellipses (primarily their closest points F, G);
- contour lines and eyelet.
- On paper placed under the model, draw the bottom of the jug. Through the lowest point on the underlying paper we will draw an auxiliary facade line, and on it we will draw the projections of the extreme points. We measure the largest width of the jug up to EF on the model (one and a half times). If we choose a width that is convenient for us, that is, straight line BC, we plot this distance from E up one and a half times, designate point 1 and draw an auxiliary quadrilateral in which we draw 1E. This will complete point a) of the analysis.
- We measure on the model how many times a–b are deposited in E1 or AD. Here it will be most convenient to compare a – b with AD. In the figure we divide AD in half; a – b is equal to half AD. Using action No. 2 or measuring, we establish that the bottom of the jug has the same width as the neck. Let's draw the width of the bottom, determine the position of points G, F, and then point I. When we determine that GF is equal to one third of 1E, we divide in our drawing the segment 1E, which we have already depicted, into the corresponding number of parts. One division will be the desired image 1F. We will find the mapping of point G by measuring it on model 1G with a–b. Then we divide the width of the ellipse into the same number of parts. One whole part will be the desired height of the ellipse. Let's draw the visible topmost ellipse. Then we move the jug and compare it with the width of the lower ellipse (on the model). One part of the width will be the desired height of the lower ellipse. Since we cannot measure the average ellipse, we will determine its height by eye. We again depict the height using axes. We determine and plot their closest points, draw arcs on the major axis of the ellipse at the points of contact and connect them with arcs passing through the ends of the minor axis (Fig. 6).
- We draw the contour line so that the entire ellipse passing through point F is in the spherical part of the jug. If we want to draw a jug without observing proportions and measurements, then we must start with the largest parts, that is, with the spherical part, to which we already draw the neck and ear (Fig. 6).
Figure 6 - Drawing a jug with a pencil. The model can always be compared with what is shown in the figure. The most correct thing would be to always compare them with the same basic size. However, sometimes it is necessary to compare with the size on the model that is most convenient for this, that is, when dividing we get halves, thirds and even sixths, and when multiplying whole parts. Drawing the Perspective of a Glass
Exactly the same as with a jug. Same task performance. By measuring on the model we create a quadrilateral: AD is equal to half EJ. The width of the top hole and the depth of the glass are the same, AD = JF. Point F lies in the middle of JE. IG is equal to one third of 1E. When comparing ellipses, we establish that IJ is equal to one sixth of AD; KE is equal to one third of BC (Fig. 7).
Figure 7 - Drawing a glass with a pencil. - In the figure we have chosen an arbitrary image of a convenient height JE. Point F divides the height in half. AD is equal in length to JF. Let's draw an auxiliary quadrilateral and its axis.
- By measuring, we divided JF in half, identified and showed points F, I, K and G. We drew a quadrilateral for the ellipses, in them there are axes, and at the points of contact the arcs we need and then the ellipses of the upper hole and the base of the stand.
- The contour of the bowl precisely determines the width of the ellipse. The minor axis of this ellipse is equal to one fifth of the major axis, which is determined by eye.
Figure 8 - Drawing part of a glass. If the dimensions of the glass are different, you should draw according to the results obtained by measuring the bowl using an auxiliary quadrangle. When measuring through the lowest point of the glass, a horizontal front straight line is drawn on the underlying paper, and the extreme left and extreme right points are designed on it. This sets the desired width of the model or the width of the auxiliary quadrilateral AB on the underlying paper. In the model, AB is compared with EF. With this action we determine the height of the auxiliary quadrangle, and then by division we find the width of the depicted object. The displayed distance is measured at intermediate heights at points G, H, J. By comparison with the height, the width of the upper hole JK and leg AB is determined. The height in the figure is divided into such a number of parts that one whole part will be the desired size. FH is compared with JK. In the figure, JK is divided into the same number of parts, one segment is drawn down from F. An ellipse of the upper hole is drawn. JE is compared with AB. In the figure, AB is divided into such a number of parts that one part represents the desired size IE. The lower defined ellipse LM is drawn by measuring EF or comparing it with AB. In the figure, a curve is drawn from point G to J and to K. A part of the circle is drawn near point I, and then the leg. The details are completed (glass thickness, legs, intercircles, intermediate holes). Assignment: draw a cylinder in a horizontal non-facade position.
Using action No. 4 we plot the direction of straight lines p" and p" on the surface. Divide the distance between them in half and determine the o axis. The major axes of the ellipses a, b are perpendicular to the o axis. (For vertically standing cylinders, these axes are also perpendicular.) The ratio of the minor auxiliary axis to the main major axis of the front visible ellipse of the lying cylinder is determined by measurement. In the same way, we determine the ratio of the major axis of the front ellipse to the length of the cylinder. If the major axis of the ellipse is smaller, we compare a with AZ. Then in the figure we plot the segment a on the o axis in the same ratio as we obtained from the measurement.
Figure 8 - Size ratio. The ratio of sizes at the nearest base can be measured, while the ratio of the sizes of the distant base is determined by eye or measured along the axis of the body. 
If the vertical axis is closer to the ellipse a than to b, the same perspective rules apply to it as for a circle that is close to the horizontal in a horizontal position (Fig. 8). At a base close to the vertical, the major and minor axes can be determined by a ratio of several times, since the minor axis of the ellipse appears smaller (Fig. 9). At the base, distant from the vertical, the dimensions of the minor axis of the ellipse can be plotted fewer times. 
Figure 9 - Auxiliary axis of the ellipse. This situation seems very difficult for students, especially with a significant decrease in the value of AZ. This mainly happens in the position of the cylinder when we see the nearest base almost from the front. Students already know that when a circle is rotated into a non-facing position, one dimension is shortened, but they usually do not know which axis is shortened. The axis of the ellipse always seems shorter, the direction of which coincides with the axis of the body. First, we define and draw the axes of the body, then the perpendiculars to them, that is, the axes of the ellipse. Draw a ball and a hemisphere
We depict the ball as a circle. The plane cut passing through the center of the ball is not horizontal. We determine the direction of the major axis of the ellipse visually, using steps No. 3 and 4 known to us, and draw an image of a ball on the drawing. The minor axis of the ellipse is perpendicular to the major one. We compare the distance between the end points of the small ellipse with the major axis, and then in the same ratio we divide the major axis in the figure and depict the length of the minor axis of the ellipse.We depict a ring on a cylindrical vessel
To depict a ring, we must draw two ellipses on a low cylinder: an upper and a lower one. They seem to be connected by a round ring, covering the distance from the highest point of the picture to the lowest (Fig. 10 - left).
Figure 10 - Ring on the cylinder Drawing a Frustum
We draw the lower base of the truncated cone on a piece of paper, on which we then determine its dimensions. When the ellipse of the base is created, we put the model in its original place, draw the lateral directions, and by measuring we establish the ratio of the sizes of the lower and upper bases. We previously determined the lowest point of the upper ellipse R by measuring it on the model. We make sure that the lateral straight surfaces are tangent to the ellipse and that they do not come out either from point A or from point B, but from points C and D (Fig. 10 on the right).
Drawing a cone lying in a non-facade position
You should proceed as in task No. 6 when drawing a cylinder in a non-facade position. We plot directions a, b, YХВ. Let's draw CD - perpendicular to YB. Let's compare YB with CD and also compare AB with CD (AB will be shorter because it lies on the axis of the body, which seems to us to be shortening). Let's compare the measurement of YA with AB and YA with YB. Draw an ellipse and check.Draw a truncated cone lying on the convexity in a non-facade position
The beginning of work in this task is similar to the previous one: we draw the directions of the lateral lines to the axis. Then we lower the perpendiculars to the axis of the body, determine and draw a visible ellipse. (For the relationship between the axes of the invisible ellipse, see the analysis of task No. 6.) We find the extreme point of the visible ellipse Y by comparing AB with XY. We compare by eye and measure. The body axis is again the axis of symmetry of the vertex angle.Drawing household items
Drawing geometric solids is a preparation for depicting everyday objects, the shapes of which are usually combinations of the shapes of various geometric bodies. You can draw small objects, kitchen utensils, glass, household items in various positions, machine parts, etc. Sequence of work. We depict the shapes of geometric bodies in a well-known way. In the same way, we draw the largest part of a given object, then we complete the details. We go from the whole to the parts. We divide the basis of the subject into parts. When drawing kitchen utensils, we pay attention to the fact that the ear of the object is located symmetrically with respect to the middle of the ellipse (Fig. 11). It should be noted that the upper plane of the ear with its axis is directed towards the center of the auxiliary ellipse. We mark and draw a straight line that determines the dimensions of the entire object including the ear.
Figure 11 - Drawing household items with a pencil. In this work, auxiliary directions for lying objects are especially necessary. In Fig. 11 shows two such positions. The upper edges of the mug are ring-shaped. When depicting machine parts, you need to show the perspective between the circles of the upper and lower bases. In Fig. Figure 11 shows part of a gear wheel. When all the ellipses are accurately drawn, we determine the position of the top of the teeth by eye and measurement. From their vertices to the center of the ellipse we draw connecting straight lines. These will be the axes of the teeth. Their width is precisely determined on both ellipses between the circles. When we have depicted the heights of the teeth, we draw the shape of the teeth on a distant base, drawing perspective straight lines from the nearest points of the teeth to distant points. If we need to depict the surface of spherical bodies, we draw auxiliary axes through the center of the ball, as when drawing a bowl. If the planes of these circles are perpendicular to each other, it is necessary that the axes of the ellipses are also perpendicular. In Fig. 12 fruits are also approximately spherical in shape. Action No. 1 determines how much higher the pear seems to us than the apple. Students should be warned against possible error– incorrect schematization of forms. 
Figure 12 - Draw fruits with a pencil. In the same way, Figure 13 shows how using action No. 1 it is convenient to clarify your observation and begin a perspective depiction of the forms that are closest to the eye of the front façade of the object. As when drawing geometric bodies, so when depicting various objects, we should not complete the final lines until we draw the entire object in a simplified manner, at least in rough form or its basic shape. And when drawing turned or tilted heads, it is convenient and advisable to start with the location of spherical objects in the plane of the general drawing. 
Figure 13. In the last one we see that sometimes, in order to maintain the placement of an object in a drawing, auxiliary ellipses or polygons should be used, with the help of which you can better determine the general appearance and dimensions of the depicted objects. We can also use the so-called “blocking” if it is not understood formalistically. Sequence of constructions (Fig. 2.17)
1). Asked big AB and small CD oval axis (Fig. 2.17a);
2).Let's connect the dots A And WITH. On this line we plot a point M: SM=AO-OS=SK(Fig.2.17b);
3).Segment AM divide in half, and from the middle of this segment we restore the perpendicular until it intersects with the axes of the oval at the points O 1 And O 4(Fig. 2.17c);
4).Construct points symmetrical to the points O 1 And O 4, we get O 2 And O 3(Fig. 2.17d);
5).Draw the lines of centers O 1 O 3, O 1 O 4, O 2 O 3, O 2 O 4(Fig. 2.17d);
6).From the center O 4 draw an arc with radius R 1 =O 4 C until it intersects with the center lines О 4 О 1 And O 4 O 2 at points 1 and 2. Similarly, we find points 3 and 4 (Fig. 2.17e);
7). We draw the closing arcs of the oval from the centers O 1 And O 2 radius R 2 =O 1 A(Fig. 2.17g).
8) Construction results - Fig. 2.17z.
Execution of drawings of parts with mates
The construction of a drawing of such a part (Fig. 2.18) should begin with an analysis of the geometric elements that make up the image of the part and determination of its overall dimensions. Then you should think about what geometric constructions need to be made in the drawing. The image scale is selected according to the overall dimensions of the part. It is recommended to carry out the construction in the following sequence (Fig. 2.19):
1).Draw axial and center lines (Fig. 2.19a);
2).Draw circles whose centers are located at the intersection of the center lines (Fig. 2.19b);
3).Perform conjugations indicating the auxiliary constructions necessary to determine the centers and conjugation points:
a) between circles Ø32, construct an external joint with a radius R24 similar to the constructions in Fig. 2.13;
b) between circles Ø32 and Ø44, construct an internal joint with a radius of R76 similar to the constructions in Fig. 2.13;
c) carry out constructions to draw a tangent to circles Ø32 and Ø44, construct a tangent similar to the constructions in Fig. 2.16. The constructions are shown in Fig. 2.19 in, city
4).Draw dimension lines and enter size numbers.
ATTENTION!
Auxiliary constructions must be left on the drawing.
Slope
Slope is the tangent of the angle of inclination of one straight line to another (Fig. 2.20).
Let's take an arbitrary scale segment ( A). Let's construct a right triangle
i = tg α = =15:75=20%
In the drawing, the slope is specified either as a percentage (Fig. 2.21) or as a ratio of numbers (Fig. 2.22). A slope of 1:5 means that for every five units of length we have one unit of height. Those. straight line AC has a slope to BC of 20% or 1:5.
In the drawings, slopes are indicated with a special sign, see GOST 2.304-81. The acute angle of the slope sign should be directed towards the decrease in height, one side of the angle is parallel to the shelf of the leader line.
Fig.2.21 Fig.2.22
The slope is used, for example, in the manufacture shaped steel: channels, I-beams, T-profiles, etc.
Let's consider an example of constructing the slope of the inner face of the lower flange of a channel (Fig. 2.23).
1. Using these dimensions, we find point A through which the given slope will pass (Fig. 2.24).
3. On the free field of the drawing, we build a slope of 10% (1:10 = 10:100) and through point A we draw a straight line parallel to the slope line.
Select a scale segment of any size.
3. An arc of radius 3 is the junction between the slope line and the vertical straight line. We build according to the rules for constructing connections between straight lines (Fig. 2.26).
Fig.2.26 Fig.2.27
4. An arc with a radius of 8 is the junction between the slope line and the vertical line of the rack (Fig. 2.27).
5. Similarly, we build the upper flange of the channel.
6. Since the height of the channel post is very large compared to the length of the shelf, and the post has a constant cross-section, a gap can be made, as shown in Figure 2.28.
7. We put down the dimensions. We save all constructions in the drawing.
2.9. Taper
Taper is the ratio of the difference in diameters of two cross sections of a truncated cone to the length between them (Fig. 2.29).
In the drawing, taper is most often expressed as percentages or ratios. The taper sign with an acute angle is directed towards the smaller diameter. The taper is placed either on the shelf of the leader line (Fig. 2.30) or above the center line (Fig. 2.31).
If the drawing indicates taper, then the dimensions on the rod and in the hole are set differently, based on the cone manufacturing technology, since normal taper is established on computer-controlled machines. Therefore, the normal taper must be indicated, and the “extra” size removed.
On a conical rod, the larger of the two diameters is indicated, since to manufacture the part you need to take a workpiece of a larger diameter. The small diameter is not indicated (Fig. 2.31).
In a hole of two diameters, the smaller one is indicated, since to obtain a taper you must first drill a hole with a diameter equal to the small diameter, and then bore the tapered hole (Fig. 2.32).
General purpose tapers are standardized. Their meaning can be found in GOST 8593-81.
In the task you need to construct a taper according to the dimensions and instead of a letter n enter the numerical value obtained by calculating using the formula in Fig. 2.29. Enter dimensions (Fig. 2.33)
Control questions
1. Formulate the concept of “conjugation”.
2. What pairing is called external, internal and mixed?
3. How are junction points determined?
4. What is called a slope and how to determine the magnitude of the slope?
5. What is called taper?
Applying dimensions
(GOST 2.307-68)
The basis for determining the size of the depicted product and its elements are the dimensional numbers printed on the drawing.
The rules for drawing dimensions on drawings and other technical documents for products from all branches of industry and construction are established by GOST 2.307 - 68. Dimensions are a very important part of the drawing. An omission or error in at least one of the dimensions makes the drawing unusable.
Therefore, dimensioning is one of the most critical stages in the preparation of a drawing.
When completing the first training drawings, the student needs to know the basic rules for drawing dimensions on the drawings.
Oval is a closed box curve that has two axes of symmetry and consists of two support circles of the same diameter, internally conjugate by arcs (Fig. 13.45). An oval is characterized by three parameters: length, width and radius of the oval. Sometimes only the length and width of the oval are specified, without defining its radii, then the problem of constructing an oval has a large variety of solutions (see Fig. 13.45, a... d).
Methods for constructing ovals based on two identical reference circles that touch (Fig. 13.46, a), intersect (Fig. 13.46, b) or do not intersect (Fig. 13.46, c) are also used. In this case, two parameters are actually specified: the length of the oval and one of its radii. This problem has many solutions. It's obvious that R > OA has no upper bound. In particular R = O 1 O 2(see Fig. 13.46.a, and Fig. 13.46.c), and the centers O 3 And O 4 are determined as the points of intersection of the base circles (see Fig. 13.46, b). According to the general point theory, mates are determined on a straight line connecting the centers of arcs of osculating circles.
Constructing an oval with touching support circles(the problem has many solutions) ( rice. 3.44). From the centers of the reference circles ABOUT And 0 1 with a radius equal, for example, to the distance between their centers, draw arcs of circles until they intersect at points ABOUT 2 and O 3.
Figure 3.44
If from points ABOUT 2 and O 3 draw straight lines through the centers ABOUT And O 1, then at the intersection with the support circles we obtain the connecting points WITH, C 1, D And D 1. From points ABOUT 2 and O 3 as from centers of radius R 2 draw arcs of conjugation.
Constructing an oval with intersecting reference circles(the problem also has many solutions) (Fig. 3.45). From the intersection points of the reference circles C 2 And O 3 draw straight lines, for example, through centers ABOUT And O 1 until they intersect with the reference circles at the junction points C, C 1 D And D 1, and radii R2, equal to the diameter of the reference circle - the conjugation arc.
Figure 3.45 Figure 3.46
Constructing an oval along two specified axes AB and CD(Fig. 3.46). Below is one of many possible solutions. A segment is plotted on the vertical axis OE, equal to half the major axis AB. From point WITH how to draw an arc with a radius from the center SE to the intersection with the line segment AC at the point E 1. Towards the middle of the segment AE 1 restore the perpendicular and mark the points of its intersection with the axes of the oval O 1 And 0 2 . Build points O 3 And 0 4 , symmetrical to the points O 1 And 0 2 relative to the axes CD And AB. Points O 1 And 0 3 will be the centers of reference circles of radius R1, equal to the segment About 1 A, and the points O2 And 0 4 - centers of conjugation arcs of radius R2, equal to the segment O 2 C. Straight lines connecting centers O 1 And 0 3 With O2 And 0 4 At the intersection with the oval, the connecting points will be determined.
In AutoCAD, an oval is constructed using two reference circles of the same radius, which:
1. have a point of contact;
2. intersect;
3. do not intersect.
Let's consider the first case. A segment OO 1 =2R is constructed, parallel to the X axis; at its ends (points O and O 1) the centers of two supporting circles of radius R and the centers of two auxiliary circles of radius R 1 =2R are placed. From the intersection points of the auxiliary circles O 2 and O 3, arcs CD and C 1 D 1 are built, respectively. The auxiliary circles are removed, then the inner parts of the support circles are cut off relative to the arcs CD and C 1 D 1. In Figure ъъ the resulting oval is highlighted with a thick line.
Figure Constructing an oval with touching support circles of the same radius
Oval is a closed convex plane curve. The simplest example of an oval is a circle. Drawing a circle is not difficult, but you can construct an oval using a compass and ruler.
You will need
- – compass;
- - ruler;
- - pencil.
Instructions
1. Let us know the width of the oval, i.e. its horizontal axis. Let us construct a segment AB different from the horizontal axis. Let's divide this segment into three equal parts by points C and D.
2. From points C and D as centers we construct circles with a radius equal to the distance between points C and D. We denote the intersection points of the circles with the letters E and F.
3. Let's unite points C and F, D and F, C and E, D and E. These lines intersect the circles at four points. Let's call these points G, H, I, J respectively.
4. Note that the distances EI, EJ, FG, FH are equal. Let's denote this distance as R. From point E as the center, draw an arc of radius R, connecting points I and J. Let's connect points G and H with an arc of radius R centered at point F. Thus, the oval can be considered constructed.
5. Let the length and width of the oval be known now, i.e. both axes of symmetry. Let's draw two perpendicular lines. Let these lines intersect at point O. On the horizontal line, plot a segment AB with the center at point O, equal to the length of the oval. On a vertical line we plot a segment CD with a center at point O, equal to the width of the oval.
6. Let's unite the straight points C and B. From point O as the center, draw an arc of radius OB connecting straight lines AB and CD. Let's call the point of intersection with the straight line CD point E.
7. From point C we draw an arc of radius CE so that it intersects the segment CB. Let's denote the intersection point by point F. Let's denote the distance FB by Z. From points F and B as from centers we'll draw two intersecting arcs of radius Z.
8. We connect the points of intersection of 2 arcs of a straight line and call the points of intersection of this line with the axes of symmetry points G and H. Let's put the point G* symmetrically to the point G tangent to the point O. And put the point H* symmetrically to the point H tangent to the point O.
9. We connect points H and G*, H* and G*, H* and G with straight lines. Let us denote the distance HC as R, and the distance GB as R*.
10. From point H as the center we draw an arc of radius R intersecting lines HG and HG*. From the point H* as the center we draw an arc of radius R intersecting the lines H*G* and H*G. From points G and G* as centers we draw arcs of radius R*, closing the resulting figure. The construction of the oval is completed.
Not everyone knows that an ellipse and an oval are different geometric shapes, even though they are similar in appearance. Unlike an oval, an ellipse has a regular shape, and it will not be possible to draw it using a compass alone.
You will need
- - paper;
- - pencil;
- - ruler;
- - compass.
Instructions
1. Take paper and pencil, draw two straight lines perpendicular to each other. Place a compass at the point where they intersect and draw two circles of different diameters. In this case, the smaller circle will have a diameter equal to the width, that is, the minor axis of the ellipse, and the huge circle will correspond to the length, that is, the major axis.
2. Divide the huge circle into twelve equal parts. Using straight lines that will pass through the center, connect the division points that are located in reverse. As a result, you will also divide the smaller circle into twelve equal segments.
3. Number it. Do this so that the highest point in the circle is called point 1. Then draw vertical lines down from the points on the large circle. In this case, skip points 1, 4, 7 and 10. From the points on the small circle corresponding to the points on the large circle, draw horizontal lines that will intersect with the verticals.
4. Connect the points with a smooth oblique where the verticals and horizontals intersect and points 1, 4, 7, 10 on the small circle. The result was a correctly constructed ellipse.
5. Try another method of constructing an ellipse. On paper, draw a rectangle with a height and width equal to the height and width of the ellipse. Draw two intersecting lines that will divide the rectangle into four parts.
6. Using a compass, draw a circle that intersects the long line in the middle. Place the rod of the compass in the center of the side of the rectangle. The radius of the circle should be equal to half the length of the side of the figure.
7. Mark the points where the circle intersects the vertical center line, stick two pins into them. Place a third pin at the end of the middle line and tie all three with linen thread.
8. Take out the third pin and put a pencil in its place. Draw a curve using thread tension. An ellipse will be obtained if all actions were performed correctly.
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Despite the fact that the ellipse and the oval are very similar in appearance, geometrically it is various figures. And if an oval can be drawn only with the help of a compass, then it is impossible to draw a true ellipse with the help of a compass. It turns out that we will consider two methods for constructing an ellipse on a plane.
Instructions
1. The first and most primitive method of drawing an ellipse: Draw two straight lines perpendicular to each other. From the point of their intersection with a compass, draw two circles of different sizes: the diameter of the smaller circle is equal to the given width of the ellipse or the minor axis, the diameter of the larger circle is equal to the length of the ellipse, the major axis.
2. Divide the huge circle into twelve equal parts. Connect the division points located opposite each other with straight lines passing through the center. The smaller circle will also be divided into 12 equal parts.
3. Number the points clockwise so that point 1 is the highest point on the circle.
4. From the division points on the larger circle, in addition to points 1, 4, 7 and 10, draw vertical lines downwards. From the corresponding points lying on the small circle, draw horizontal lines intersecting the vertical ones, i.e. the vertical line from point 2 of the larger circle must intersect with the horizontal line from point 2 of the small circle.
5. Combine with a smooth oblique the intersection points of the vertical and horizontal lines, as well as points 1, 4, 7 and 10 of the small circle. The ellipse is built.
6. For another method of drawing an ellipse, you will need a compass, 3 pins and strong linen thread. Draw a rectangle whose height and width are equal to the height and width of the ellipse. Using two intersecting lines, divide the rectangle into 4 equal parts.
7. Using a compass, draw a circle intersecting the long center line. To do this, the support rod of the compass must be installed in the center of one of the sides of the rectangle. The radius of the circle is determined by the length of the side of the rectangle, divided in half.
8. Mark the points where the circle intersects the vertical center line.
9. Insert two pins into these points. Insert the third pin into the end of the midline. Tie linen thread around all three pins.
10. Remove the third pin and use a pencil instead. Using even thread tension, outline the curve. If everything is done correctly, you should end up with an ellipse.
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The designer is repeatedly faced with the need to build arc given curvature. Parts of buildings, spans of bridges, and fragments of parts in mechanical engineering can have this shape. The thesis of building an arch in any type of design is no different from what a schoolchild has to do in a drawing or geometry lesson.
You will need
- - paper;
- - ruler;
- – protractor
- – compass;
- – computer with AutoCAD program.
Instructions
1. In order to build arc with the help of ordinary drawing tools, you need to know 2 parameters: the radius of the circle and the angle of the sector. They are either specified in the conditions of the problem, or they need to be calculated based on other data.
2. Place a dot on the sheet. Designate it as O. Draw a straight line from this point and plot the length of the radius on it.
3. Align the zero division of the protractor with point O and set aside this angle. Draw a straight line through this new point with the beginning at point O and plot the length of the radius on it.
4. Spread the legs of the compass to the size of the radius. Place the needle at point O. Connect the ends of the radii with an arc using a compass pencil.
5. The AutoCAD program allows you to build arc according to several parameters. Open the program. In the top menu you will find the main tab, and in it the “Drawing” panel. The program will offer several types of lines. Select the Arc option. You can also do it through the command line. Enter the command _arc there and press enter.
6. You will see a list of parameters according to which you can build arc. There are quite a lot of options: three points, the center, the beginning and the end. Allowed to build arc by origin, center, chord length or inner corner. There is an option for two extreme points and radius, by central and final or starting points and internal corner, etc. Select suitable option depending on what you are famous for.
7. Whatever you prefer, the program will prompt you to enter the necessary parameters. If you are building arc using any three points, you can indicate their location with cursor support. It is also possible to indicate the coordinates of any point.
8. If among the parameters by which you build arc, you have a corner, you will have to call the context menu a 2nd time. First, mark the points specified in the conditions with a cursor or with coordinate support, then call up the menu and enter the angle size.
9. The algorithm for constructing an arc using two points and a chord length is exactly the same as using two points and an angle. True, in this case it should be borne in mind that the chord subtends 2 arcs of one circle. If you are building a smaller arc, enter the correct value, large - negative.
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An oval is a geometric figure that is used to display individual parts of interior items, draw animals and much more. Many people are interested in how to draw an oval correctly by hand.
How to draw an oval yourself correctly
In order for the drawing to turn out beautiful and harmonious, it is necessary to correctly and accurately draw all its elements. However, not everyone knows how to make an ellipse by hand correctly and beautifully.
To make an ellipse, you need to take:
- album sheet;
- ruler;
- pencil;
- eraser.
Initially, you need to draw a rhombus in the middle of the sheet, all sides of which will be equal, and the opposite sides will be parallel. The rhombus should be such that an oval of the required size fits well into it. Then you need to fit an oval into the resulting diamond. After this, the diamond must be erased with a pencil.
Draw an oval evenly and beautifully
To look great, you need to know how to draw an oval with a compass in a few minutes. To make an ellipse using a compass, you need to take:
- album sheet;
- pencil;
- compass;
- thread;
- pins.
To draw an ellipse beautifully, you must initially draw two straight lines that will be perpendicular. Place the point of the compass at the intersection of the two lines and then draw a circle.
The diameter of such a circle will correspond to the width of the ellipse. Then, leaving the compass in the same place, you need to draw a slightly larger circle to get the length of the ellipse. Then you need to connect the two circles, erasing the extra lines. This way it will turn out beautiful and smooth oval, from which you can then draw various animals and birds. Knowing how to draw an oval by hand, you can make very beautiful and original drawings without much difficulty.
How to Draw a Guinea Pig Based on an Oval
Drawings of animals and birds drawn from simple geometric shapes look very interesting. Many are interested in coming out of the oval with their children.
Using an oval you can quickly and beautifully draw a guinea pig. To make a drawing, you need to draw two ovals in a horizontal position, one of which will be slightly smaller than the other.
One oval should intersect with another, and then the outer corners formed when the two shapes intersect should be closed with lines. This will create the neck of the animal. In the center of a small oval you need to draw a point, from which you will then get an eye.
After this, you need to draw small ears at the top of the small oval. When everything is ready, you should draw the guinea pig's paws. It is worth considering that the front legs should be slightly shorter than the hind legs and almost invisible.
When everything is completely drawn, the extra lines can be erased, and then you need to color the guinea pig, making the animal spotted.
Knowing how to draw an oval, you can make a very beautiful drawings animals and birds.
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