Education for everyone. Constructing an oval along two axes How to draw an even ellipse

The two-dimensional circles in the previous pictures can be represented as coins, records, pancakes, lenses, etc. But circles are also components of three-dimensional objects such as cylinders and cones, and are also widely used in the visual arts. Cylinders are the basis for an infinite number of things such as cigarettes, tanks, thread spools, pipes, etc. Cones are the bases for ice cream cones, hourglasses, martini glasses, funnels, etc.

An ellipse is an oval with two unequal axes (major and minor), which always form a right angle with each other. The axes divide the ellipse into short and long arcs respectively, both arcs being perfectly symmetrical.

You need to learn how to draw ellipses freehand. Ellipses A and B are drawing attempts. Anyone familiar with ellipses can visually evaluate the major and minor axes and see that ellipse A is correct and ellipse B is not symmetrical enough. (If we draw two axes for B, we can see the errors more clearly. Notice how each sector is different.)

It may be useful for you to draw a rectangle according to the labels. This will create four more guides for evaluating and comparing the shape of the ellipse.

So, in order to learn how to draw (and represent) ellipses well, you first need to sketch out the axes. Mark with strokes equal segments on both sides of the center to define the edges.
Now let's try to draw four equal sectors. We always round the ends, do not make them sharp.

The center of a circle drawn in perspective does not coincide with the main axis of the ellipse - it is always farther (to the observer) than the main axis.

This amazing fact is often the cause of many difficulties. What is the relationship between the center of the circle and the axes of the ellipse?

A regular circle can always be described by a regular square. The center of the square (we will find it by drawing two diagonals) coincides with the center of the circle.

A circle in perspective can also be described by a perspective square. Drawing the diagonals will determine the center of both the square and the circle. We know from previous lessons that this point is not equidistant from the bottom and top lines. So, we draw the diameter of the circle through this central point - it is also not equidistant from the bottom and top.
We also know that the main axis of the ellipse must be equidistant from the top and bottom lines.
Now, by combining the two drawings, we see that the diameter of the circle is slightly higher than the main axis of the ellipse. Note also that the minor axis coincides in most cases with the perspective diameter of the circle.

The view from above explains this apparent paradox. The widest part of the circle (projected onto the plane of the drawing) is not a diameter, but a simple chord (shown with strokes). This chord will become the main axis of the ellipse, while the true diameter of the circle, lying further away, looks smaller.


So, don't make the mistake of drawing a square in perspective and using its center as the location of the major axis of the ellipse. The resulting figure will look like this

Also, if you want to draw half a circle (or cylinder), you cannot draw an ellipse and consider either side of the main axis to be half a circle in perspective. (The figure on the left is not half, although it seems equal)
But on the right are the correct halves, because the diameter of the circle was used as a dividing line.

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;
  • - circular.

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.

Video on the topic

Despite the fact that the ellipse and the oval are very similar in appearance, geometrically they are different 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. 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.

Video on the topic

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. Put a dot on the paper. 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 on 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. It is allowed to erect arc by origin, center, chord length or internal angle. An option is offered for two extreme points and a radius, for a central and final or starting point and an internal corner, etc. Choose the appropriate 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.

Video on the topic

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 About 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 arcs of conjugation 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 junction points will be determined.


In AutoCAD, an oval is constructed using two reference circles of the same radius, which are:

1. have a point of contact;

2. intersect;

3. do not intersect.

Let's consider the first case. A segment OO 1 =2R is built, parallel to the X axis, at its ends (points O and O 1) the centers of two reference 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, relative to the arcs CD and C 1 D 1, the inner parts of the support circles are cut off. In Figure ъъ the resulting oval is highlighted with a thick line.

Figure Constructing an oval with touching support circles of the same radius

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 we divide in half, and from the middle of this segment we restore the perpendicular to the intersection with the axes of the oval at the points O 1 And About 4(Fig. 2.17c);

4).Construct points symmetrical to the points O 1 And About 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 About 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) The results of the construction - fig. 2.17z.

Making 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. According to the overall dimensions of the part, the scale of the image is selected. The construction is recommended to be performed in the following sequence (Fig. 2.19):

1). Apply axial and center lines (Fig. 2.19a);

2). Draw circles whose centers are located at the intersection of center lines (Fig. 2.19b);

3).Perform conjugations indicating the auxiliary constructions necessary to determine the centers and conjugation points:

a) between the circles Ø32 construct an external conjugation with a radius R24 similar to the constructions in Fig. 2.13;

b) between the circles Ø32 and Ø44 construct an internal conjugation with radius R76 similar to the constructions in Fig. 2.13;

c) perform constructions for drawing a tangent to the 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 flange of the leader line.

Fig.2.21 Fig.2.22

The slope is used, for example, in the manufacture of shaped steel: channels, I-beams, T-sections, etc.

Consider an example of constructing a slope of the inner face of the lower flange of the channel (Fig. 2.23).

1. Based on 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 a conjugation 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 shelf 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. All constructions on the drawing are saved.

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, the taper is most often expressed as a percentage or ratio. The sign of the taper 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. 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 build a taper in size 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 conjugation 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.

"No fish or sausages! You have to draw the right ovals!"

This is exactly what my teacher, Sergei Ivanovich Poluychik, said when he looked at our first still lifes. Thanks to this phrase, I immediately remembered what regular ovals should look like when constructing cylindrical shapes.

So, let's get acquainted with fish, sausages, and regular ovals.

FISH- irregular oval with sharp corners.

An oval is a circle that lies on a plane, so no matter from which side we look, it cannot have sharp corners.

SAUSAGE- an incorrectly drawn oval with parallel sides.

Once again, just to remember: an oval is a circle on a plane; a circle has no parallel sides.

CORRECT OVAL, without sharp corners and parallel sides.

Following the rules of perspective, the far part of the oval is drawn smaller (red line), the one closest to the viewer is drawn larger (blue line in the figure).

Almost all cylindrical and cone-shaped shapes (jugs, jars, vases, bottles, mugs, etc.) are drawn according to the same pattern. Here, using this jug as an example, we will analyze step by step this diagram of drawing cylindrical bodies.

The entire construction is done with light, barely noticeable lines so that you don’t have to erase with an eraser, since erasing deteriorates the top layer of paper. Both the paint in a painting and the strokes in a drawing fall unevenly onto the paper after erasing.

Determine the place of the object on the sheet. We draw a central center line to construct the jug.

Determine the location of the center lines for constructing ovals. That is, using the sighting method, we clarify the proportions and sizes between the centers of the ovals of the jug. Let's draw these lines.

With the help of sighting, we determine the size of the ovals. We set this size aside with a pencil, mark equal segments from the point of intersection of the center lines.

Set aside the points of the width of the ovals.

When noting these dimensions, we do not forget about the rules of perspective: the side of the oval that is further from us will be a little smaller, which means that the one that is closer to us will be larger.

We also remember that the lower the oval is at eye level, the more it wants to become a circle.

Finally, we draw the ovals of our cylindrical object.

We connect the extreme points of the ovals and our jug ​​is almost ready.

It remains to finish the handle and nose. When drawing a handle and a spout, we try to remember that usually they are opposite each other, that is, on the same line.

HOW TO DRAW OVALS DEPENDING ON THE ARTIST'S EYE LEVEL

This is how the construction of the jug will look like if we put it higher than the one whose construction we analyzed.

This is how the construction of the jug will look like if the upper edge of the jug is at eye level, so we depict it as a line. But the bottom of the jug is below eye level, therefore, in order to see the bottom line, we build an oval for the bottom. draw a jug above eye level

This is how the construction of the jug will look if its middle coincides with the line of the eyes. The top of the jug will be above the eye line - draw an oval, which will have the top line closer to us. The bottom of the jug turns out to be slightly below eye level, so we build a regular oval. But! If the jug (vase) is located far from the viewer (artist), then both the top edge and the bottom line will be drawn with a simple straight line, as if they were at eye level. Beginning artists very often make mistakes when constructing ovals, which spoils the impression of the whole picture as a whole.


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