How to draw an expanded cone. How to make a scan - a pattern for a cone or a truncated cone of a given size. Simple sweep calculation. Mutual intersection of surfaces

Instead of the word “pattern”, “sweep” is sometimes used, but this term is ambiguous: for example, a reamer is a tool for increasing the diameter of a hole, and in electronic technology there is the concept of a reamer. Therefore, although I am obliged to use the words “cone sweep” so that search engines can find this article using them, I will use the word “pattern”.

Building a pattern for a cone is a simple matter. Let us consider two cases: for a full cone and for a truncated one. On the picture (click to enlarge) sketches of such cones and their patterns are shown. (I note right away that we will only talk about straight cones with a round base. Cones with an oval base and inclined cones will be considered in the following articles).

1. Full taper

Designations:

Pattern parameters are calculated by the formulas:
;
;
where .

2. Truncated cone

Designations:

Formulas for calculating pattern parameters:
;
;
;
where .
Note that these formulas are also suitable for the full cone if we substitute .

Sometimes, when constructing a cone, the value of the angle at its vertex (or at the imaginary vertex, if the cone is truncated) is fundamental. The simplest example is when you need one cone to fit snugly into another. Let's denote this angle with a letter (see picture).
In this case, we can use it instead of one of the three input values: , or . Why "together about", not "together e"? Because three parameters are enough to construct a cone, and the value of the fourth is calculated through the values ​​of the other three. Why exactly three, and not two or four, is a question that is beyond the scope of this article. A mysterious voice tells me that this is somehow connected with the three-dimensionality of the “cone” object. (Compare with the two initial parameters of the two-dimensional circle segment object, from which we calculated all its other parameters in the article.)

Below are the formulas by which the fourth parameter of the cone is determined when three are given.

4. Methods for constructing a pattern

  • Calculate the values ​​on the calculator and build a pattern on paper (or immediately on metal) using a compass, ruler and protractor.
  • Enter formulas and source data into a spreadsheet (for example, Microsoft Excel). The result obtained is used to build a pattern using a graphic editor (for example, CorelDRAW).
  • use my program, which will draw on the screen and print out a pattern for a cone with the given parameters. This pattern can be saved as a vector file and imported into CorelDRAW.

5. Not parallel bases

As far as truncated cones are concerned, the Cones program still builds patterns for cones that have only parallel bases.
For those who are looking for a way to construct a truncated cone pattern with non-parallel bases, here is a link provided by one of the site visitors:
A truncated cone with non-parallel bases.

The development of the surface of the cone is a flat figure obtained by combining the side surface and the base of the cone with a certain plane.

Sweep construction options:

Development of a right circular cone

The development of the lateral surface of a right circular cone is a circular sector, the radius of which is equal to the length of the generatrix of the conical surface l, and the central angle φ is determined by the formula φ=360*R/l, where R is the radius of the circumference of the cone base.

In a number of problems of descriptive geometry, the preferred solution is the approximation (replacement) of a cone by a pyramid inscribed in it and the construction of an approximate sweep, on which it is convenient to draw lines lying on a conical surface.

Construction algorithm

  1. We inscribe a polygonal pyramid into the conical surface. The more side faces of the inscribed pyramid, the more accurate the correspondence between the actual and approximate scan.
  2. We build a development of the side surface of the pyramid using the triangle method. The points belonging to the base of the cone are connected by a smooth curve.

Example

In the figure below, a regular hexagonal pyramid SABCDEF is inscribed in a right circular cone, and an approximate development of its lateral surface consists of six isosceles triangles - the faces of the pyramid.

Consider a triangle S 0 A 0 B 0 . The lengths of its sides S 0 A 0 and S 0 B 0 are equal to the generatrix l of the conical surface. The value A 0 B 0 corresponds to the length A'B'. To build a triangle S 0 A 0 B 0 in an arbitrary place of the drawing, we set aside the segment S 0 A 0 =l, after which we draw circles with a radius S 0 B 0 =l and A 0 B 0 = A'B' from points S 0 and A 0 respectively. We connect the point of intersection of circles B 0 with points A 0 and S 0 .

The faces S 0 B 0 C 0 , S 0 C 0 D 0 , S 0 D 0 E 0 , S 0 E 0 F 0 , S 0 F 0 A 0 of the SABCDEF pyramid are built similarly to the triangle S 0 A 0 B 0 .

Points A, B, C, D, E and F, lying at the base of the cone, are connected by a smooth curve - an arc of a circle, the radius of which is equal to l.

Oblique cone development

Consider the procedure for constructing a sweep of the lateral surface of an inclined cone by the approximation method.

Algorithm

  1. We inscribe the hexagon 123456 in the circle of the base of the cone. We connect points 1, 2, 3, 4, 5 and 6 with the vertex S. The pyramid S123456, constructed in this way, with a certain degree of approximation, is a replacement for the conical surface and is used as such in further constructions.
  2. We determine the natural values ​​of the edges of the pyramid using the method of rotation around the projecting line: in the example, the i-axis is used, which is perpendicular to the horizontal projection plane and passes through the vertex S.
    So, as a result of the rotation of the edge S5, its new horizontal projection S'5' 1 takes a position in which it is parallel to the frontal plane π 2 . Accordingly, S''5'' 1 is the natural value of S5.
  3. We construct a development of the lateral surface of the pyramid S123456, consisting of six triangles: 0 1 0 . The construction of each triangle is performed on three sides. For example, △S 0 1 0 6 0 has the length S 0 1 0 =S''1'' 0 , S 0 6 0 =S''6'' 1 , 1 0 6 0 =1'6'.

The degree of correspondence of the approximate sweep to the actual one depends on the number of faces of the inscribed pyramid. The number of faces is chosen based on the ease of reading the drawing, the requirements for its accuracy, the presence of characteristic points and lines that need to be transferred to the scan.

Transferring a line from the surface of a cone to a development

The line n lying on the surface of the cone is formed as a result of its intersection with a certain plane (figure below). Consider the algorithm for constructing line n on the sweep.

Algorithm

  1. Find the projections of points A, B and C, in which the line n intersects the edges of the pyramid inscribed in the cone S123456.
  2. We determine the actual size of the segments SA, SB, SC by rotating around the projecting line. In this example, SA=S''A'', SB=S''B'' 1 , SC=S''C'' 1 .
  3. We find the position of points A 0 , B 0 , C 0 on the corresponding edges of the pyramid, setting aside segments S 0 A 0 =S''A'', S 0 B 0 =S''B'' 1 , S 0 C 0 =S''C'' 1 .
  4. We connect points A 0 , B 0 , C 0 with a smooth line.

Truncated cone development

The method for constructing a sweep of a right circular truncated cone, described below, is based on the principle of similarity.

There are 2 ways to build a cone sweep:

  • Divide the base of the cone into 12 parts (we enter a regular polyhedron - a pyramid). You can divide the base of the cone into more or less parts, because. the smaller the chord, the more accurate the construction of the sweep of the cone. Then transfer the chords to the arc of the circular sector.
  • Construction of a sweep of the cone, according to the formula that determines the angle of the circular sector.

Since we need to plot the lines of intersection of the cone and the cylinder on the development of the cone, we still have to divide the base of the cone into 12 parts and inscribe the pyramid, so we will immediately follow the 1st path for constructing the development of the cone.

Algorithm for constructing a sweep of a cone

  • We divide the base of the cone into 12 equal parts (we enter the correct pyramid).
  • We build the lateral surface of the cone, which is a circular sector. The radius of the circular sector of the cone is equal to the length of the generatrix of the cone, and the length of the arc of the sector is equal to the circumference of the base of the cone. We transfer 12 chords to the arc of the sector, which will determine its length, as well as the angle of the circular sector.
  • We attach the base of the cone to any point of the arc of the sector.
  • Through the characteristic points of intersection of the cone and the cylinder we draw generators.
  • Find the natural size of the generators.
  • We build data generators on the development of the cone.
  • We connect the characteristic points of intersection of the cone and the cylinder on the sweep.

More details in the video tutorial on descriptive geometry in AutoCAD.

During the construction of the sweep of the cone, we will use the Array in AutoCAD - a Circular array and an array along the path. I recommend watching these AutoCAD video tutorials. The AutoCAD 2D video course at the time of this writing contains the classic way to build a circular array and interactive when building an array along a path.


Shortcut http://bibt.ru

Development of a truncated cylinder and a cone.

To build a scan of a truncated cylinder, a truncated cylinder is drawn in two projections (front view and top view), then the circle is divided into an equal number of parts, for example, into 12 (Fig. 243). On the right side of the first projection, a straight line AB is drawn, equal to the straightened circumference, and divided into the same number of equal parts, i.e. 12. From the division points 1, 2, 3, etc., on the AB line, restore perpendiculars, and from points 1, 2, 3, etc., lying on a circle, draw straight lines parallel to the axial line until they intersect with an inclined section line.

Rice. 243. Construction of a flat pattern of a truncated cylinder

Now, on each perpendicular, segments are laid with a compass upward from the line AB, equal in height to the segments indicated on the projection of the front view by the numbers of the corresponding points. For clarity, two such segments are marked with curly brackets. The obtained points on the perpendiculars are connected by a smooth curve.

The construction of a development of the lateral surface of the cone is shown in fig. 244, a. A life-size lateral projection of the cone is drawn according to the given dimensions of diameter and height. The length of the generatrix of the cone, indicated by the letter R, is measured with a compass. An arc is drawn with a compass with a fixed radius around the center O, which is the extreme point of an arbitrarily drawn straight line OA.

From point A along the arc lay off (with a compass in small segments) the length of the unfolded circle, equal to πD. The resulting extreme point B is connected to the center O of the arc. The figure AOB will be a development of the lateral surface of the cone.

The development of the lateral surface of the truncated cone is built, as shown in Fig. 244b. According to the height and diameters of the upper and lower bases of the truncated cone, a life-size profile of the truncated cone is drawn. The generators of the cone continue until they intersect at point O. This point is the center, arcs are drawn from it, equal to the circumferences of the base and apex of the truncated cone. To do this, divide the base of the cone into seven parts. Each such part, i.e. 1/7 of the diameter D, is laid along a large arc 22 times and a straight line is drawn from the resulting point B to the center of the arc O. After connecting the point O with points A and B, a scan of the lateral surface of the truncated cone is obtained.

You will need

  • Pencil Ruler square compasses protractor Formulas for calculating the angle from the length of the arc and radius Formulas for calculating the sides of geometric shapes

Instruction

On a sheet of paper, build the base of the desired geometric body. If you are given a box or , measure the length and width of the base and draw a rectangle on a piece of paper with the appropriate parameters. To build a sweep of a or a cylinder, you need the radius of the base circle. If it is not specified in the condition, measure and calculate the radius.

Consider a parallelepiped. You will see that all its faces are at an angle to the base, but the parameters of these faces are different. Measure the height of the geometric body and, using a square, draw two perpendiculars to the length of the base. Set aside the height of the parallelepiped on them. Connect the ends of the resulting segments with a straight line. Do the same on the opposite side of the original.

From the points of intersection of the sides of the original rectangle draw perpendiculars and to its width. Set aside the height of the parallelepiped on these straight lines and connect the obtained points with a straight line. Do the same on the other side.

From the outer edge of any of the new rectangles, the length of which is the same as the length of the base, build the upper face of the box. To do this, draw perpendiculars from the intersection points of the length and width lines located on the outside. Set aside the width of the base on them and connect the points with a straight line.

To build a sweep of a cone through the center of the base circle, draw a radius through any point on the circle and continue it. Measure the distance from the base to the top of the cone. Set aside this distance from the point of intersection of the radius and the circle. Mark the top point of the side surface. Based on the radius of the side surface and the length of the arc, which is equal to the circumference of the base, calculate the angle of development and set aside it from the straight line already drawn through the top of the base. Using a compass, connect the intersection point of the radius and circle found earlier with this new point. The reaming of the cone is ready.

To build a pyramid sweep, measure the heights of its sides. To do this, find the middle of each side of the base and measure the length of the perpendicular dropped from the top of the pyramid to this point. Having drawn the base of the pyramid on the sheet, find the midpoints of the sides and draw perpendiculars to these points. Connect the obtained points with the points of intersection of the sides of the pyramid.

The development of a cylinder consists of two circles and a rectangle located between them, the length of which is equal to the length of the circle, and the height is equal to the height of the cylinder.

Loading...Loading...