Or
Where:
- radius of the circle,
- circle diameter,
- circumference,
- Pi (),
As a rule, the value () up to the second sign (3,14) is used for calculation, but in some cases, this may not be enough.
- Truncated cone with accessible vertex: A cone that can be used to determine the position of the vertex.
- A truncated cone with an inaccessible vertex: A cone, during the construction of which the position of the vertex is difficult to determine, in view of its remoteness.
- Triangulation: a method for constructing unfolded surfaces of non-developing, conical, general form and with a cusp.
- It should be remembered: Regardless of whether the surface under consideration is developable or non-developable, only an approximate development can be plotted graphically. This is due to the fact that in the process of taking and postponing dimensions and performing other graphic operations, errors are inevitable due to the design features of drawing tools, the physical capabilities of the eye, and errors from replacing arcs with chords and angles on the surface with flat corners. Approximate developments of curves of non-developing surfaces, in addition to graphical errors, contain errors obtained due to the mismatch of the elements of such surfaces with flat approximating elements. Therefore, in order to obtain a surface from such a development, in addition to bending, it is necessary to partially stretch and compress its individual sections. Approximate scans, when carefully performed, are accurate enough for practical purposes.
The material presented in the article implies that you have an idea about the basics of drawing, know how to divide a circle, find the center of a segment with a compass, take / transfer dimensions with a compass, use patterns, and relevant reference material. Therefore, the explanation of many points in the article is omitted.
Construction of a cylinder sweep
Cylinder
A body of revolution with the simplest unfolding, having the shape of a rectangle, where two parallel sides correspond to the height of the cylinder, and the other two parallel sides correspond to the circumference of the bases of the cylinder.
Truncated cylinder (fish)
truncated cylinder
Training:
- To create a sweep, draw a quadrilateral ACDE full size (see drawing).
- Let's draw a perpendicular BD, out of plane AC exactly D, cutting off from the construction the straight part of the cylinder ABDE which can be adjusted as needed.
- From the center of the plane CD(dot O) draw an arc with a radius of half the plane CD, and divide it into 6 parts. From the resulting points O, draw perpendicular lines to the plane CD. From points on a plane CD, draw straight lines perpendicular to the plane BD.
Building:
- Line segment BC transfer, and turn it into a vertical. From a point B, vertical BC, draw a ray perpendicular to the vertical BC.
- Take the size with a compass C-O 1 B, point 1 . We remove the size B1-C1 1 .
- Take the size with a compass O 1 -O 2, and set aside on the beam, from the point 1 , point 2 . We remove the size B2-C2, and set aside the perpendicular from the point 2 .
- Repeat until point is delayed D.
- The resulting verticals, from the point C, vertical BC, to the point D- connect with a curved curve.
- The second half of the sweep is mirrored.
Any cylindrical slices are constructed in a similar way.
Note: Why "Rybina"- if you continue building a sweep, while building half from the point D, and the second in the opposite direction from the vertical BC, then the resulting pattern will look like a fish, or a fish tail.
Construction of a development of a cone
Cone
The reaming of the cone can be done in two ways. (See drawing)
- If the size of the side of the cone is known, from the point O, an arc is drawn with a compass, with a radius equal to the side of the cone. Two points are plotted on the arc ( A 1 and B1 O.
- A life-size cone is built, from a point O, exactly A, a compass is placed, and an arc is drawn passing through the points A and B. Two points are plotted on the arc ( A 1 and B1), at a distance equal to the circumference and connected to a point O.
For convenience, half of the circumference can be set aside from, on both sides of the centerline of the cone.
A cone with a displaced apex is constructed in the same way as a truncated cone with displaced bases.
- Construct the circumference of the base of the cone in top view, full size. Divide the circle into 12 or more equal parts, and put them on a straight line one by one.
A cone with a rectangular (polyhedral) base.
Cones with polyhedral base
- If the cone has an even, radial base: ( When constructing a circle in a top view, by setting the compass to the center, and outlining the circle along an arbitrary vertex, all the vertices of the base fit on the arc of the circle.) Construct a cone, by analogy with the development of an ordinary cone (build the base in a circle, from a top view). Draw an arc from a point O. Put a point in an arbitrary part of the arc A 1, and alternately put all the faces of the base on the arc. The end point of the last face will be B1.
- In all other cases, the cone is built according to the principle of triangulation ( see below).
Truncated cone with accessible apex
Frustum
Construct a truncated cone ABCD full size (See drawing).
Parties AD and BC continue until the intersection point appears O. From the point of intersection O, draw arcs, with radius OB and OC.
On the arc OC, set aside the circumference DC. On the arc OB, set aside the circumference AB. Connect the resulting points with segments L1 and L2.
For convenience, half of the circumference can be set aside from, on both sides of the centerline of the cone.
How to plot the circumference of an arc:
- With the help of a thread, the length of which is equal to the circumference.
- With the help of a metal ruler, which should be bent “in an arc”, and put the appropriate risks.
Note: It is not at all necessary that the segments L1 and L2, if they continue, will converge at a point O. To be completely honest, they should converge, but taking into account the corrections for the errors of the tool, material and eye, the intersection point may be slightly lower or higher than the top, which is not a mistake.
Truncated cone with a transition from a circle to a square
Cone with a transition from a circle to a square
Training:
Construct a truncated cone ABCD full size (see drawing), build a top view ABB 1 A 1. Divide the circle into equal parts (in the above example, the division of one quarter is shown). points AA 1-AA 4 connect segments with a dot A. Hold Axis O, from the center of which draw a perpendicular O-O 1, with a height equal to the height of the cone.
Below, the primary dimensions are taken from the top view.
Building:
- Remove size AD and build an arbitrary vertical AA0-AA1. Remove size AA0-A, and put an "approximate point" by making a go-ahead with a compass. Remove size A-AA 1, and on the axis O, from the point O O 1 AA 1, to the expected point A. Connect dots with line segments AA 0 -A-AA 1.
- Remove size AA 1-AA 2, from the point AA 1 put a "approximate point", making a go-ahead with a compass. Remove size A-AA 2, and on the axis O, from the point O, postpone the segment, remove the size from the received point to the point O 1. Make a go-ahead with a compass from a point A, to the expected point AA 2. Draw a segment A-AA 2. Repeat until the segment is delayed A-AA 4.
- Remove size A-AA 5, from the point A set a point AA5. Remove size AA 4-AA 5, and on the axis O, from the point O, postpone the segment, remove the size from the received point to the point O 1. Make a go-ahead with a compass from a point AA 4, to the expected point AA5. Draw a segment AA 4-AA 5.
Build the rest of the segments in the same way.
Note: If the cone has an accessible vertex, and SQUARE foundation - then the construction can be carried out according to the principle truncated cone with an accessible vertex, and the base is cones with a rectangular (polyhedral) base. The accuracy will be lower, but the construction is much simpler.
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. We will consider cones with an oval base and inclined cones 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.
16.1. Drawings of unfolded surfaces of prisms and cylinders.
For the manufacture of fences for machine tools, ventilation pipes and some other products, their reamers are cut out of sheet material.
The development of the surfaces of any straight prism is a flat figure composed of side faces - rectangles and two bases - polygons.
For example, in the development of the surfaces of a hexagonal prism (Fig. 139, b), all faces are equal rectangles with a width a and height h, and the bases are regular hexagons with a side equal to a.
Rice. 139. Construction of a drawing of a sweep of the surfaces of a prism: a - two types; b - development of surfaces
Thus, it is possible to build a drawing of a sweep of the surfaces of any prism.
The development of the surfaces of the cylinder consists of a rectangle and two circles (Fig. 140, b). One side of the rectangle is equal to the height of the cylinder, the other is the circumference of the base. In the drawing of the sweep, two circles are attached to the rectangle, the diameter of which is equal to the diameter of the bases of the cylinder.
Rice. 140. Construction of a drawing of a development of the surfaces of a cylinder: a - two types; b - development of surfaces
16.2. Drawings of developments of the surfaces of the cone and pyramid.
The development of the surfaces of the cone is a flat figure consisting of a sector - the development of the lateral surface and a circle - the base of the cone (Fig. 141, 6).
Rice. 141. Construction of a drawing of a development of the surfaces of a cone: a - two types; b - development of surfaces
The builds are done like this:
- An axial line is drawn and from the point s "on it they describe with a radius equal to the length s" a "generator of the cone, an arc of a circle. The circumference of the base of the cone is plotted on it.
Point s" is connected to the end points of the arc.
- A circle is attached to the resulting figure - the sector. The diameter of this circle is equal to the diameter of the base of the cone.
The circumference when constructing a sector can be determined by the formula C = 3.14xD.
Angle a is calculated by the formula a = 360°xD/2L, where D is the diameter of the base circle, L is the length of the generatrix of the cone, it can be calculated using the Pythagorean theorem.
Rice. 142. Construction of a drawing of the development of the surfaces of the pyramid: a - two types; b - development of surfaces
The drawing of the development of the surfaces of the pyramid is built as follows (Fig. 142, b):
From an arbitrary point O, an arc of radius L is described, equal to the length of the side edge of the pyramid. On this arc lay four segments equal to the side of the base. The extreme points are connected by straight lines to point O. Then a square is attached equal to the base of the pyramid.
Pay attention to how the sweep drawings are drawn up. A special sign is placed above the image. From the fold lines, which are drawn with dash-dotted lines with two points, they draw leader lines and write on the “Fold Lines” shelf.
- How to build a drawing of a development of the surfaces of a cylinder?
- What inscriptions are applied to the drawings of the surface scans of objects?
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 use a square to 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.
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
- 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.
- 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
- We inscribe 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. 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.
- 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. - 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
- 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.
- 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 .
- 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 .
- 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.
