To make machine casings, machine enclosures, ventilation devices, and pipelines, it is necessary to cut out their developments from sheet material.
Surface development polyhedron is a flat figure obtained by combining with the drawing plane all the faces of the polyhedron in the sequence of their location on the polyhedron.
To construct a development of the surface of a polyhedron, you need to determine the natural size of the faces and draw all the faces sequentially on the plane. The true dimensions of the edges of the faces, if they are not projected in full size, are found by the methods of rotation or changing projection planes (by projecting onto an additional plane) given in the previous paragraph.
Let's consider the construction of surface developments of some simple bodies.
Development of the surface of a straight prism is a flat figure made up of side faces - rectangles and two equal base polygons. For example, a regular right hexagonal prism is taken (Fig. 176, a). All side faces of the prism are rectangles, equal in width a and height H; The bases of the prism are regular hexagons with a side equal to a. Since we know the true dimensions of the faces, it is not difficult to construct a development. To do this, six segments are sequentially laid on a horizontal line equal to the side of the base of the hexagon, i.e. 6a. From the obtained points, perpendiculars equal to the height of the prism H are constructed, and a second horizontal line is drawn through the end points of the perpendiculars. The resulting rectangle (H x 6a) is a development of the lateral surface of the prism. Then the base figures are placed on one axis - two hexagons with sides equal to a. The outline is outlined with a solid main line, and the fold lines are outlined with a dash-dotted line with two dots.
In a similar way, you can construct developments of straight prisms with any figure at the base.
Development of the surface of a regular pyramid is a flat figure composed of lateral faces - isosceles or equilateral triangles and a regular base polygon. For example, a regular quadrangular pyramid is taken (Fig. 176, b). Solving the problem is complicated by the fact that the size of the side faces of the pyramid is unknown, since the edges of the faces are not parallel to any of the projection planes. Therefore, the construction begins with determining the true value of the inclined edge SA. Having determined by the method of rotation (see Fig. 173, c) the true length of the inclined edge SA, equal to s"a` 1 (Fig. 176, b), an arc of radius s"a` 1 is drawn from an arbitrary point O, as from the center. Four segments are laid on the arc, equal to the side of the base of the pyramid, which is projected in the drawing to its true size. The found points are connected by straight lines to point O. Having obtained a development of the lateral surface, a square equal to the base of the pyramid is attached to the base of one of the triangles.
Development of the surface of a right circular cone is a flat figure consisting of a circular sector and a circle (Fig. 176, c). The construction is carried out as follows. Draw an axial line and from a point taken on it, as from the center, with a radius Rh equal to the generatrix of the cone sfd, outline an arc of a circle. In this example, the generator, calculated using the Pythagorean theorem, is approximately equal to
We often encounter surface developments in everyday life, in production and in construction. To make a case for a book (Fig. 169), sew a cover for a suitcase, a tire for a volleyball, etc., you must be able to construct developments of the surfaces of a prism, ball and other geometric bodies. A development is a figure obtained by combining the surface of a given body with a plane. For some bodies, scans can be accurate, for others they can be approximate. All polyhedra (prisms, pyramids, etc.), cylindrical and conical surfaces, and some others have precise developments. Approximate developments have a ball, a torus and other surfaces of revolution with a curved generatrix. We will call the first group of surfaces developable, the second - non-developable.
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When constructing developments of polyhedra, you will have to find the actual size of the edges and faces of these polyhedra using rotation or changing projection planes. When constructing approximate developments for non-developable surfaces, it will be necessary to replace sections of the latter with developable surfaces close in shape to them.
To construct a scan of the lateral surface of the prism (Fig. 170), it is assumed that the scan plane coincides with the face AADD of the prism; other faces of the prism are aligned with the same plane, as shown in the figure. The face ССВВ is preliminarily combined with the face ААВВ. Fold lines in accordance with GOST 2.303-68 are drawn with thin solid lines with a thickness of s/3-s/4. Points on the scan are usually denoted by the same letters as on the complex drawing, but with index 0 (zero). When constructing a development of a straight prism according to a complex drawing (Fig. 171, a), the height of the faces is taken from the frontal projection, and the width from the horizontal one. It is customary to build a scan so that the front side of the surface is facing the observer (Fig. 171, b). This condition is important to observe because some materials (leather, fabrics) have two sides: front and back. The bases of the ABCD prism are attached to one of the faces of the side surface.
If point 1 is specified on the surface of the prism, then it is transferred to the development using two segments marked on the complex drawing with one and two strokes, the first segment C1l1 is laid to the right of point C0, and the second segment is laid vertically (to point l0).
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Similarly, a development of the surface of the cylinder of rotation is constructed (Fig. 172). Divide the surface of the cylinder into a certain number of equal parts, for example 12, and unfold the inscribed surface of a regular dodecagonal prism. The sweep length with this construction turns out to be slightly less than the actual sweep length. If significant accuracy is required, then a graphic-analytical method is used. The diameter d of the circumference of the base of the cylinder (Fig. 173, a) is multiplied by the number π = 3.14; the resulting size is used as the development length (Fig. 173, b), and the height (width) is taken directly from the drawing. The bases of the cylinder are attached to the development of the side surface.
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If point A is given on the surface of the cylinder, for example, between the 1st and 2nd generatrices, then its place on the development is found using two segments: a chord marked with a thick line (to the right of point l1), and a segment equal to the distance of point A from the upper base of the cylinder , marked in the drawing with two strokes.
It is much more difficult to construct the development of a pyramid (Fig. 174, a). Its edges SA and SC are straight lines in general position and are projected onto both projection planes by distortion. Before constructing the development, it is necessary to find the actual value of each edge. The size of the edge SB is found by constructing its third projection, since this edge is parallel to the plane P3. The ribs SA and SC are rotated around a horizontally projecting axis passing through the vertex S so that they become parallel to the frontal plane of projections P (the actual value of the rib SB can be found in the same way).
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After such a rotation, their frontal projections S 2 A 2 and S 2 C 2 will be equal to the actual size of the ribs SA and SC. The sides of the base of the pyramid, like horizontal straight lines, are projected onto the projection plane P 1 without distortion. Having three sides of each face and using the serif method, it is easy to construct a development (Fig. 174, b). Construction begins from the front face; a segment A 0 C 0 = A 1 C 1 is laid out on a horizontal straight line, the first notch is made with a radius A 0 S 0 - A 2 S 2 the second - with a radius C 0 S 0 = = G 2 S 2 ; at the intersection of the serifs, point S„ is obtained. Accept the order side A 0 S 0 ; from point A 0 make a notch with radius A 0 B 0 =A 1 B 1 from point S 0 make a notch with radius S 0 B 0 =S 3 B 3 ; at the intersection of the serifs, point B 0 is obtained. Similarly, the face S 0 B 0 C 0 is attached to the side S 0 G 0 . Finally, the base triangle A 0 G 0 S 0 is attached to side A 0 C 0 . The lengths of the sides of this triangle can be taken directly from the development, as shown in the drawing.
The development of a cone of rotation is constructed in the same way as the development of a pyramid. Divide the circumference of the base into equal parts, for example into 12 parts (Fig. 175, a), and imagine that a regular dodecagonal pyramid is inscribed in the cone. The first three faces are shown in the drawing. The surface of the cone is cut along the generatrix S6. As is known from geometry, the development of a cone is represented by a sector of a circle whose radius is equal to the length of the cone generatrix l. All generatrices of a circular cone are equal, therefore the actual length of the generatrix l is equal to the frontal projection of the left (or right) generatrix. From the point S 0 (Fig. 175, b) a segment of 5000 = l is laid vertically. An arc of a circle is drawn with this radius. From the point O 0, the segments Ol 0 = O 1 l 1, 1 0 2 0 = 1 1 2 1, etc. are laid off. By setting aside six segments, we get point 60, which is connected to the vertex S0. The left part of the scan is constructed in the same way; The base of the cone is attached below.
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If you need to put point B on the scan, then draw the generatrix SB through it (in our case S 2), apply this generatrix to the scan (S 0 2 0); rotating the generatrix with point B to the right until it aligns with the generatrix S 3 (S 2 5 2), find the actual distance S 2 B 2 and set it aside from the point S 0. The found segments are marked on the drawings with three strokes.
If it is not necessary to plot points on the cone scan, then it can be constructed faster and more accurately, since it is known that the scan sector angle is a=360°R/l, the radius of the base circle, and l is the length of the cone generatrix.
Cylinder (straight circular cylinder) is a body consisting of two circles (the bases of a cylinder), combined by parallel translation, and all the segments connecting the corresponding points of these circles during parallel translation. The segments connecting the corresponding points of the base circles are called generators of the cylinder.
Here's another definition:
Cylinder- a body that is limited by a cylindrical surface with a closed guide and two parallel planes intersecting the generatrices of this surface.
Cylindrical surface- a surface that is formed by the movement of a straight line along a certain curve. The straight line is called the generatrix of the cylindrical surface, and the curved line is called the guide of the cylindrical surface.
Lateral surface of the cylinder- part of a cylindrical surface that is limited by parallel planes.
Cylinder bases- parts of parallel planes cut off by the side surface of the cylinder.
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The cylinder is called direct(Cm. Fig.1), if its generators are perpendicular to the planes of the bases. Otherwise the cylinder is called inclined.
Circular cylinder- a cylinder whose bases are circles.
Right circular cylinder (just a cylinder) is a body obtained by rotating a rectangle around one of its sides. Cm. Fig.1.
Cylinder radius is the radius of its base.
Generator of the cylinder- generatrix of a cylindrical surface.
Cylinder height is called the distance between the planes of the bases. Cylinder axis called a straight line passing through the centers of the bases. The section of a cylinder by a plane passing through the axis of the cylinder is called axial section.
The axis of the cylinder is parallel to its generatrix and is the axis of symmetry of the cylinder.
A plane passing through the generatrix of a straight cylinder and perpendicular to the axial section drawn through this generatrix is called tangent plane of the cylinder. Cm. Fig.2.
Development of the lateral surface of the cylinder- a rectangle with sides equal to the height of the cylinder and the circumference of the base.
Cylinder side surface area- development area of the lateral surface. $$S_(side)=2\pi\cdot rh$$ , where h is the height of the cylinder, and r– radius of the base.
Total surface area of a cylinder- area, which is equal to the sum of the areas of the two bases of the cylinder and its side surface, i.e. is expressed by the formula: $$S_(full)=2\pi\cdot r^2 + 2\pi\cdot rh = 2\pi\cdot r(r+h)$$ , where h is the height of the cylinder, and r– radius of the base.
Volume of any cylinder equal to the product of the area of the base and the height: $$V = S\cdot h$$ Volume of a round cylinder: $$V=\pi r^2 \cdot h$$ , where ( r- base radius).
A prism is a special type of cylinder (the generators are parallel to the side ribs; the guide is a polygon lying at the base). On the other hand, an arbitrary cylinder can be considered as a degenerate (“smoothed”) prism with a very large number of very narrow faces. In practice, a cylinder is indistinguishable from such a prism. All properties of the prism are preserved in the cylinder.
You will need
- Pencil Ruler square compass protractor Formulas for calculating angles using arc length and radius Formulas for calculating sides of geometric figures
Instructions
On a sheet of paper, build the base of the desired geometric body. If you are given a parallelepiped or, measure the length and width of the base and draw a rectangle on a piece of paper with the appropriate parameters. To construct a development 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 located 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. Plot 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 one.
From the intersection points of the sides of the original rectangle, draw perpendiculars to its width. Plot the height of the parallelepiped on these straight lines and connect the resulting 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 coincides with the length of the base, construct the top face of the parallelepiped. 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 construct a development 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 intersection point of the radius and the circle. Mark the vertex point of the side surface. Using the radius of the side surface and the length of the arc, which is equal to the circumference of the base, calculate the sweep angle and set it aside from the straight line already drawn through the top of the base. Using a compass, connect the previously found intersection point of the radius and the circle with this new point. The cone scan is ready.
To construct a pyramid scan, 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 drawn from the top of the pyramid to this point. Having drawn the base of the pyramid on a sheet of paper, find the midpoints of the sides and draw perpendiculars to these points. Connect the resulting points with the intersection points 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 the height of the cylinder.
Curved surfaces that can be completely aligned with a plane, without stretching or compression, without tears or folds, are called developable. These surfaces include only ruled surfaces and only those in which adjacent generatrices intersect each other or are parallel. This property is possessed by torsi (surfaces formed by straight lines tangent to a directing spatial curve), conical and cylindrical surfaces. The remaining ruled surfaces, as well as all non-ruled surfaces, are not expandable.
Construction of a complete development of a right circular truncated cylinder of revolution
(Fig. 10.41).
To construct a development of a cylinder, it is enough to imagine it as a prism with a large number of faces (in fact, 12-16 such faces are enough), evenly dividing the circumference of the base of the cylinder into an equal number of parts.
If there is any line on the surface of the cylinder, then this line can be transferred to the development of the cylinder along the points belonging to the corresponding generators of this surface.
Constructing a scan of the full surface of a right circular cone (Fig. 10.42).
To construct a development of a right circular cone, it is enough to imagine its surface as a regular pyramid with a large number of faces and then construct its development by finding the actual size of one of the faces, which is an isosceles triangle, along its side and base. The construction of the development of the cone can be seen from the drawing, where the base of the “face” S01 is equal to the chord 0 ` 1 `. The development of the lateral surface of the cone, in this case, contains 12 such “faces”.
The development of the lateral surface will be found more accurately if we determine the angle j 0 at point S on the development using the formula:
j 0 =R/l 360 0, where R is the radius of the base of the cone, and l is the length of the generatrix of the cone.
The points of a certain ABCDE curve belonging to the lateral surface of the cone can be found by the belonging of these points to the corresponding generators of the conical surface. To do this, it is enough to use a rotation method, as shown in the example of point C belonging to the generatrix S2, to find the segments S``B`` 0 =SB, S``D`` 0 =SD and S``E`` 0 =SE .. Place the found segments along the corresponding generators on the development of the cone and draw a line ABCDE through them. To obtain a complete development of the cone surface, it must be supplemented with the base of the cone, tangent at the corresponding point of development of the lateral surface.
Development of the lateral surface of an inclined cone be like the development of an inclined pyramid with a large number of faces, each of which is found on three sides - two lateral “edges” and a “base” (Fig. 10.43).