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### Three ways to expand the developable surface

There are three main ways to expand the developable surface, namely the parallel line method, the radiation method and the triangle method. The method of its unfolding operation is as follows.

#### Parallel line method

According to the ridge line of the prism or the element line of the cylinder, the prismatic surface or cylindrical surface is divided into a number of quadrilaterals, and then they are flattened in sequence to make an expanded map. This method is called the parallel line method.

The principle of the parallel line method is: because the surface of the body is composed of a set of countless straight lines parallel to each other, we can divide the tiny area enclosed by the two adjacent lines and the upper and lower ends of the clamping line. Seen as an approximate flat trapezoid (or rectangle), when the divided small areas are infinite, the sum of the areas of the small planes is equal to the surface area of the body; when we put all the small plane areas in the original order and face up and down.

When the position is spread out without omission and without overlapping, the surface of the truncated body is unfolded. Of course, it is impossible for us to divide the surface of the frustum into an infinite number of small planes, but we can divide it into dozens or even several small planes.

All geometric objects with plain lines or shuttle lines parallel to each other, such as rectangular tubes, round tubes, etc., can be unfolded on the surface using the parallel line method.

**Figure 2-2 Unfolding of prismatic surface**

The steps to make an expanded graph are as follows:

- Make a front view and a top view;
- Make the reference line of the expanded drawing, that is, the extension line of 1′-4′ in the main view;
- From the top view, record the vertical distance of each ridge line 1-2, 2-3, 3-4, 4-1, and move it to the reference line to obtain points 10, 20, 30, 40, and 10. And draw vertical lines through these points;
- From the points 1′.2′.3′ and 4’in the main view, draw parallel lines to the right and intersect the corresponding vertical lines to obtain points 10, 20, 30, 40, and 10;
- Use straight lines Connect the points to get the expanded picture.

**Figure 2-3 shows the unfolding of the truncated cylinder**

The steps to make an expanded graph are as follows:

- Make the front view and top view of the truncated cylinder;
- Divide the horizontal projection into several equal parts, here is divided into 12 equal parts, the semicircle is divided into 6 equal parts, and the vertical line is drawn upward from each division point, and the corresponding prime line is obtained on the main view, and the oblique circle line is crossed at 1′, 2′, .., 7’each point;
- Expand the bottom circle of the cylinder into a straight line (the length can be calculated by πD), as a reference line, divide the straight line into 12 equal parts, and intercept the corresponding equal points (such as a”, b”, etc.);
- Draw a vertical line upwards from the equilateral point, that is, the plain line on the surface of the cylinder;
- Draw parallel lines from 1′, 2′, …, 7’on the main view respectively, and intersect the corresponding prime line at 1″, 2″, .,, 7″, that is, the endpoint of the prime line on the unfolded surface;
- Connect the endpoints of all the plain lines to form a smooth curve, and you can get an expanded view of the oblique truncated cylinder 1/2. Draw the other half of the expanded view in the same way, and get the desired expanded view.

From this, it can be clearly seen that the parallel line expansion method has the following characteristics.

- Only when the straight lines on the surface of the body are parallel to each other, and the real length is expressed on the projection map, the parallel line expansion method can be applied
- The specific steps of using the parallel line method to expand the entity are: a. Arbitrary equally divided (or arbitrary divided) top view, projected lines are drawn from each equally divided point to the main view, and a series of intersection points are obtained in the main view (this is actually (The surface of the body is divided into several small parts); b. Cut a line segment in the direction perpendicular to the straight line (front view) to make it equal to the section (circumference) length, and record the points on the top view, passing over this line segment The vertical line of the line segment drawn from the point of each photo in the main view intersects with the vertical line of the prime line drawn from the intersection point obtained in the first step in the main view, and then the intersection points are connected sequentially (this is actually the first The several small parts divided by the step are spread out one by one), and then the unfolded picture can be obtained.

#### Radiation method

On the surface of the cone, there are clusters of prime lines or ridges. These prime lines or ridges are concentrated at a point on the top of the cone. The method of drawing the expanded map by using the top of the cone and radioactive prime lines or ridges is called the radiation method.

The principle of the radiation method is to treat any two adjacent prime lines of the shape and the bottom line of the point between them as an approximate small plane triangle. When the bottom side of each small triangle is infinitely short and there are infinite small triangles. , Then the sum of the area of each small triangle is equal to the side area of the original cross section, and when all small triangles are not omitted, overlapped, and laid out in the original relative order and position of left, right, up and down, then The surface of the original body is also unfolded.

The radiation method is the surface expansion method of various cones, whether it is a right cone, an oblique cone or a pyramid, as long as there is a common cone tip, it can be expanded by the radiation method.

**Figure 2-4 shows the expansion of the oblique section of the top of the right conical tube.**

The steps to make an expanded graph are as follows:

- Draw the main view and fill in the upper truncated to form a complete right cone.
- Make the plain line on the cone surface. The method is to divide the bottom circle into several equal divisions. Here, make 12 equal divisions to get 1, 2, ., 7 points. From these points, draw the vertical line upward and project it with the bottom circle. When the lines intersect, connect the intersection point with the top of the cone 0, and intersect the inclined plane at points 1′, 2′, .., 7′. Among them, the 2’, 3’, ., 6’ lines are not really long.
- Draw a fan with O as the center and Oa as the radius. The arc length of the fan is equal to the circumference of the bottom circle. Divide the sector into 12 equal points, and intercept the equal points 1, 2…, 7. The arc length of the equal points is equal to the arc length of the bottom circumference. With O as the center of the circle, make a lead to each equal point (radiation)
- From the points 2, 3′, ., 7′, make a lead parallel to ab and intersect with Oa, which is the actual length of 02′, 03′, .., O7′.
- Take O as the center and the vertical distance between O and Oa as the radius to make an arc, and intersect the corresponding plain lines O1, O2, .., O, etc., and get the intersection points 1″, 2″, ., 7″ each point.
- Connect the points with a smooth curve to get an expanded view of the oblique section of the top of the right conical tube.

The radiation method is a very important expansion method, it is suitable for all cones and frustum components. Although the unfolded cones or truncated bodies have various shapes, the unfolding methods are similar, and the methods can be summarized as follows:

- In the second view (or only in a certain view), complete the lofting of the entire cone by extending the edge (edge), etc. Of course, this step is not necessary for the truncated body with vertices.
- Through the method of equally dividing (or unequally dividing and arbitrarily dividing) the perimeter of the top view, draw the prime line (including the side edge of the pyramid and the straight line passing the apex on the side surface) corresponding to each bisector point. The meaning of is to divide the cone or truncated surface into several small parts.
- Apply the method of finding the real long line (the rotation method is commonly used), and take all the prime lines, ridges, and straight lines related to the development of the drawing that do not reflect the real length-without missing the real length.
- Using the solid long line as the criterion, draw the expanded view of the entire cone side surface, and at the same time, draw the expanded view of the truncated body based on the expanded side view of the entire vertebral body.

#### Triangle method

According to the characteristics and complexity of the shape of the sheet metal product, the surface of the sheet metal product is divided into several groups of triangles, and then the real shape of each group of triangles is obtained, and they are arranged next to each other to draw the expanded view. This method of making the expanded view It is called the triangle method.

The principle of the triangle method is to divide the surface of the body (component) into many small triangles, and then spread these small triangles one by one according to the original position and sequence of the left and right sides, so that the surface of the body (component) is also Unfolded.

Although the radiation method also divides the surface of the sheet metal product into several triangles to expand, the main difference between it and the triangle method is the arrangement of the triangles. The radiation method uses a series of triangles around a common center (cone top) to form a fan shape to make the expansion diagram; while the triangle method divides the triangles according to the characteristics of the sheet metal products, and these triangles do not necessarily surround a common center. Arrangement, in many cases it is arranged in a W shape. In addition, the radiation method is only applicable to cones, while the triangle method can be applied to any shape.

Although the triangle method is suitable for any body, it is more cumbersome, so it is only used when necessary. When there are no parallel lines or ridges on the surface of the part, the parallel line method cannot be used to expand, and there is no concentration of all the vertices of the prime lines or ridges, and the radial method can not be used to expand, the triangle method is used to make the surface expansion map.

** Figure 2-5 shows the expansion of the convex five-pointed star**

The steps to use the triangle method to make an expanded graph are as follows:

- Use the method of making a regular pentagon in a circle to draw a top view of a convex five-pointed star;
- Draw the front view of the convex five-pointed star. In the figure, O’A’, O’B’ are the actual lengths of the OA and OB lines, and CE is the actual length of the base of the convex five-pointed star;
- With O”A” as the large radius R, O”B” as the small radius r, make the concentric circles of the expanded diagram;
- Measure 10 times on the large and small arcs with the length of m, and get 10 intersections of A” and B” on the large and small circles respectively;
- Connect these 10 intersections to get 10 small triangles (△A”O”C” in the picture), this is the expanded view of the convex five-pointed star.

The “round sky” component shown in Figure 2-6 can be seen as a combination of 4 partial surfaces of cones and 4 flat triangles. The expansion of this type of component is possible if the parallel line method or the radial method is used, but it is more troublesome to do. For simplicity and ease, you can use the triangulation method to expand.

**Figure 2-6 Expansion of the “”TianYuan Place”” component**

The steps to use the triangle method to make an expanded graph are as follows:

- Divide the circumference into 12 equal parts in the plan, connect the equal points 1, 2, 2 and 1 with the adjacent corner point A or B, and then make the vertical line from the equal point to the front view at 1′, 2 ‘, 2′, 1’each point, and then connect with A’or B’. The significance of this step is to divide the side surface of the sky circle into several small triangles. In this example, it is divided into 16 small triangles.
- From the perspective of the symmetrical relationship between the front and rear of the two views, the 1/4 of the lower right corner of the plan is the same as the other three parts. The upper and lower mouths reflect the real shape and the real length in the plan, and GH is in the main view because it is a horizontal line The corresponding line segment projection 1’H’ reflects the real length; however, B1 and B2 do not reflect the real length in any of the projections. This requires the application of the method of seeking real long lines to find the real length. Here, a right-angled triangle is used. : A1 is equal to B1, A2 is equal to B2), beside the main view, make two right-angled triangles, make the right-angled side CQ equal to h, and the other right-angled sides are A2 and A1, then the hypotenuse QM and QN are real long lines. The significance of this step is to find out the length of all small triangles, and then analyze whether the projection of each side line reflects the real length. If it does not reflect the real length, the real length must be obtained without missing one by one.
- Make an expanded view. Make the line segment Ax B x equal to a, A x and B x are respectively the center of the circle, and the solid long line QN (i.e. l1) is the radius and the arcs are drawn and intersected by 1x. The center of the circle, the S arc length in the plan view is the radius, and the arc is drawn with the Ax as the center and the real length QM (that is, l2) radius. Draw an arc with 2x as the center and S length as the radius, and intersect the arc drawn with Ax as the center and real length QM as the radius at 2x. Until the expanded view of the small triangle. Ex is intersected by the arc drawn with Ax as the center a/2 as the radius, 1× as the center and 1’B’ (that is, l3) as the radius. Only half of the entire expanded view is drawn in the expanded view.

The significance of choosing FE as the seam in this example is: to divide all the small triangles on the surface of the shape (truncated body), according to their actual size, according to the original left and right adjacent positions, uninterrupted, without omission, without overlapping, Spread it on the same plane without wrinkles, so that all the surface of the shape (truncated body) is unfolded.

From this, it can be clearly seen that the triangle method has omitted the original relationship between the two element lines (parallel, intersecting, and different planes) of the shape, and replaced it with a new triangle relationship, so it is an approximate expansion method. The specific steps of the method are as follows

- Divide the surface of the sheet metal component into a number of small triangles correctly. The correct division of the surface of the body is the key to the triangle method. Generally speaking, the division that should meet the following 4 conditions is the correct division, otherwise it is the wrong division :

- All vertices of all small triangles must be located on the upper and lower mouth edges of the component;
- The edges of all small triangles must not pass through the internal space of the component, but can only be attached to the surface of the component
- All adjacent two small have and can only have one common edge
- Two small triangles separated by a small triangle in the middle can only have one common vertex; two small triangles separated by two or more triangles in the middle, or have a common vertex or no common vertex.

- Consider the sides of all small triangles, and see which ones reflect the actual length and which ones do not reflect the actual length. Those who cannot reflect the actual length must find the actual length one by one according to the method of seeking the actual length.
- Based on the adjacent positions of the small triangles in the figure, use the known or calculated real length as the radius, and draw all the small triangles in turn, and finally all the intersection points, depending on the specific shape of the component, use a curve Or connect them with a broken line to get an expanded view.

#### Comparison of three expansion methods

According to the above analysis, it can be seen that the triangle expansion method can expand the surface of all developable bodies, while the radiation method is limited to the expansion of the components where the element lines meet at one point, and the parallel line method is also limited to the expansion of the elements parallel to each other. Radiation method and parallel line method can be regarded as special cases of triangle method. From the point of view of the simplicity of drawing, the expansion steps of triangle method are more cumbersome. Generally speaking, the three expansion methods are selected according to the following conditions.

- If all the element lines on a certain plane or curved surface of the component (regardless of whether the section is closed or not) are projected on a projection surface, they will appear as real long lines parallel to each other, and the projection on the other projection surface is only The performance is a straight line or curve, then the parallel line method can be used to expand at this time.
- If a cone (or part of a cone) is projected on a certain projection surface, its axis reflects the real length, and the bottom surface of the cone is perpendicular to the projection surface, then the most favorable conditions for applying the radiation expansion method are available ( “The most favorable condition” does not refer to the necessary conditions, because there is a step to find the real length in the radiation expansion method, so no matter what projection position the cone is in, the real length of all necessary elements can always be obtained, and then the side of the cone can be expanded).
- When a certain plane or a certain surface of the component is represented as a polygon in the three views, that is, when a certain plane or a certain surface is neither parallel nor perpendicular to any projection surface, the triangle method is used to expand. Especially when drawing irregular shapes, the effect of triangle method is more significant.

### Approximate expansion of non-developable surfaces

From the above analysis, it can be seen that if the surface of a body cannot be flattened on the same plane without missing, overlapping, and without wrinkles, then it is a non-expandable surface, which can be divided into non-developable and rotating surfaces according to their different formation mechanisms. There are two kinds of non-developed and ruled surfaces. The non-developable rotating surface is the surface of the rotating body formed by the generatrix (primary line) formed by the curve rotating around the fixed axis, the spherical surface shown in Figure 2-7 (a) and the parabolic surface shown in Figure 2-7 (b) It is a non-expandable rotating surface.

The generatrix forming the rotating surface is also called the warp. The plane curve formed by any point C on the generatrix AB with the rotation of the generatrix is called the latitude line of the rotating surface, and the circle formed by one revolution is called the latitude circle, see Figure 2-7 ( c); Straight-grained non-developable surface refers to a surface where at least one straight line can be made at any point on the surface. These straight lines are neither parallel nor intersect (even if they are extended, they never intersect), but are different in space. State, the ruled tapered surface shown in Figure 2-7(d) and the ruled cylindrical surface shown in Figure 2-7(e) belong to the ruled non-developable surface.

**Figure 2-7** **Types of non-developable surfaces**

Although the non-developable surface cannot be spread out 100% accurately, it can be spread out approximately. For example, for a ping-pong ball, you can tear its surface into many small pieces, and then treat each small piece as a small plane, and then spread these identified small planes on the same plane. In this way, the ping-pong The surface of the sphere is approximately expanded. According to this assumption, the principle of the approximate expansion of the non-developable surface can be obtained: according to the size and shape of the curved surface, the surface is divided into several parts according to a certain rule, and then it is assumed that each small part is divided into several parts. Finally, apply an appropriate expansion method to unfold each of the identified small developable surfaces one by one, so as to obtain an approximate unfolded view of the non-developable surface.

#### Approximate expansion of non-developable rotating surface

According to the different rules used to divide the non-developable rotating surface into several small parts, the methods used for the non-developable rotating surface are divided into the warp division method, the weft division method, and the line and weft joint division method.

- The expansion principle of the warp division method is to divide the non-developable rotating surface into several parts along the direction of the warp, and then treat the non-developable surface between every two adjacent warps as one-way curved along the warp direction. Expand the surface so that each small surface can be unfolded by the parallel line method. Figure 2-8 shows the expansion of the hemispherical meridian division method.

**Figure 2-8** **Expansion of hemispherical surface by longitude division method**

The steps to expand with the warp division method are as follows:

- Use the warp division method to divide the surface of the body. Connect the eight equal points A, B, C… of the outer circumference of the plan view to the center O, and then divide the rotating surface into eight equal parts in the plan view.
- Assume that the non-developable curved surface between two adjacent warps is replaced by a unidirectional curved surface along the direction of the warp. In other words, the non-developable curved surface between adjacent warps is regarded as a developable curved surface that is curved along the direction of the warp.
- Use the parallel line method to make an expanded view of each small block. Now take the OAB part as an example to illustrate the following: first add a set of parallel lines, pass any point 1, 2, 3 and K° on the main view O”K° In the plan view, the plumb line crosses OB at 1′, 2′, 3′, K’, and crosses OA at 1″, 2″, 3″, K”, then 1’1″, 22″, 3’3″, K’K” is a group of parallel, and in the plan view the real length can be developed. The surface of the surface, and then in the direction of the vertical line of K’K”, straighten the K°O” in the main view and record the points 1, 2, and 3 on it, and then quote the K’K” Parallel lines intersect with the vertical lines of the same name of K’K” cited by points O, 1′, 1″, 2′, 2″, …, K’, K” in the plan, and connect the intersection points in sequence with smooth curves. Thus, an approximately one-eighth unfolded image of the non-developable rotating surface is obtained.
- The expansion principle of the latitude division method is to draw a number of latitude lines on the rotating surface; then it is assumed that the non-developable rotating surface located between two adjacent latitude lines is approximately the side of a right truncated cone with the adjacent latitude lines as the upper and lower bases. Then, all the side surfaces of each right truncated cone are unfolded to obtain an approximate unfolded view of the non-developing rotating surface.

**Figure 2-9** **Expansion of hemispherical surface by dividing latitude**

The steps to expand with the latitude division method are as follows:

- Use the weft division method to divide the surface of the body. In the main view, make 3 latitude lines (that is, 3 horizontal lines), and then divide the rotating surface into 4 parts.
- The parts I, II, and III are regarded as the side faces of three right truncated cones of different sizes, and the part IV is regarded as a flat circle.
- Use the fan-shaped expansion method to make an expanded view of each part. Now take the expansion of the second small part in the figure as an example, the explanation is as follows: first extend AB and EF so that they intersect the axis of rotation at OⅠ, OⅡ is the center of the expanded drawing; then measure the size of AF, AF is the small truncated cone II Diameter d of the bottom base of OⅡ; draw an arc with OⅡA and OⅡB as the radius respectively, and intercept A’A” with length equal to πd on the outer arc, and then connect OⅡA’, OⅡA”, so A’B’B”A” A’is the expanded view of the second small part, even after the other pieces are expanded in the same way, an approximate expanded view of the non-expandable rotating surface is obtained.
- The warp and weft joint division method The warp and weft joint division method is to use both the warp division method and the weft division method in the expansion of a component. The warp and weft joint division method is suitable for the approximate expansion of a large rotating surface, like a diameter of more than ten meters or even dozens of meters. Rice house covers, large oil tanks, etc. Figure 2-10 shows the expansion of the joint division method of warp and weft of a large-size semicircular arc sphere.

**Figure 2-10** **Joint segmentation of longitude and latitude for large hemispherical surface**

The steps of using the joint division method of warp and weft are as follows:

- Use warp and latitude to split the rotating surface into several parts, divide the outer circumference of the plane into eight equal parts (the more the number of equal parts, the more accurate), and then connect the equal points to the center O'(this is the warp division); Go past any points 1, 2, 3, 4 on the front view O”K°, make a plumb line to intersect O’E at 3’ and 4’ points in the plan view, and intersect O’E’ at 1”, 2”, 3 “, 4″ points, connect 1234 with a broken line, pass 1, 2, 3, 4 Make a horizontal line. Then, with O’as the center of the circle, with O’1′ (O’1″), O’2′ (O’2”), O’3(O’3″), O’4′ (O’4”) Draw circles for the radii respectively, so the rotating surface is divided by the latitude method; in the plan view, the intersection of the warp and the latitude is connected in turn with a polyline; If the octagon in the center is regarded as a piece of blanking, then the above connections will divide the rotating surface into 25 small pieces, such as 1’2″1″1′, 2’3’3″2″2′, 3 ‘4’4″3″3′ is 3 of them.
- The 25 non-developable surfaces divided into are regarded as planes, that is, 24 of them are flat trapezoids, and the other (top) is a plane regular octagon.
- Expand each small plane separately. Obviously, the expanded view of the top piece of material is the regular octagon in the center of the plan view. The expanded views of the other small plane trapezoids can be obtained by the parallel line method. Today, it is expanded 1’2’2″1″1. ‘As an example, the description is as follows: intercept 1°2° in the vertical direction of 1’1″, make 1°2° equal to the corresponding arc length 12 in the front view, and make 1’1″ parallel lines through 1° and 2° ,And the 1’1″ vertical line made by 1′, 2′, 2″, 1″ with the same name, intersect at 1x, 2x, 2xx and 1xx, connect them, then get 1’2’2″1″1’ part From the main view, from bottom to top, the eight small trapezoids on each layer are all equal. Therefore, as long as you draw one piece of unfolded material in each layer separately, the other pieces of unfolded material are also Becomes known.

#### Approximate development of non-developable ruled surface

The approximate expansion of the ruled non-developable surface can use the triangle line expansion method. Its surface division rules are exactly the same as those in the triangle expansion method, that is, the non-developable ruled surface segmentation method is the triangle method. As shown in Figure 2-11, it is the unfolding of the straight-grained cone-shaped surface by the triangle method.

**Figure 2-11** **Triangle expansion of undevelopable ruled conical surface**

The steps to expand with the triangle method are as follows:

- Divide the surface of the body into several small triangles. Divide A”B” into six equal parts in the plan view, and cross the vertical line A”B” at 1′, 2′, 3’…, 5′, and cross AB and A’B’ in the front view through each of the equal points. At each point of 1°°~5°°, 1°~5°, and then as shown in the figure, connect them into 12 small triangles.
- Be realistic and long. The upper edge of this component reflects the actual length, the lower edge reflects the actual length in the plan view, and the left and right side lines reflect the actual length in the main view; only 11 connections cannot reflect the actual length. This can be obtained by the straight triangle method. On the real-length graph, we only marked the right-angle side length 11′ and 1A”. Others are not marked. All real lengths are indicated by brackets. For example, the real length of 1A” is indicated by (1A”).
- Expand according to the triangle method shown in the previous section, and you can get an approximate expanded view of the undeveloped straight-grained cone-shaped surface.

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