Views: 49 Author: Site Editor Publish Time: 2021-05-14 Origin: Site

If the surface of a form cannot be laid flat on the same plane without omission, overlap or crease, then it is a non-spreadable surface, which can be classified as a non-spreadable rotating surface or a straight non-spreadable surface according to their formation mechanism. A non-spreadable surface is a rotating surface made up of curved lines that rotate around a fixed axis, such as the (a) spherical surface and (b) parabolic surface shown below. It is customary to refer to the surface as a meridian, and the plane curve formed by the rotation of any point C on the bus line AB is called the latitude of the surface, and the circle formed by one week of rotation is called the latitude circle. This is the case for straight conical surfaces and (e) straight cylindrical surfaces, as shown in (d) below.

Although non-expandable surfaces cannot be unfolded with 100 per cent accuracy, they can be approximated. For example, the surface of a ping-pong ball can be approximated by tearing the surface into many small pieces, then considering each small piece as a small plane, and then laying these identified small planes onto the same plane. This is the principle behind the approximate unfolding of a non-spreadable surface: according to the size and shape of the surface to be unfolded, the surface is divided into several parts according to certain rules.

**Approximate unfolding of a non-expandable surface**

The methods used to divide a non-developable surface into smaller parts are warp, weft, and combined warp and weft, and are as follows.

**Warp splitting: **The principle of warp splitting is to divide the non-spreadable rotating surface into a number of sections in the direction of the warp, and then to treat the non-spreadable surface between each of the two adjacent warp lines as a one-way bend in the direction of the warp line. The diagram below shows a hemispherical surface unfolded by the warp division method.

The procedure for unfolding by meridional division is as follows.

⒈Divide the surface of the form using the meridian division method. By connecting the eight equal points A, B, C, ... on the outer circumference of the plan to the centre of the circle O, the rotating surface is divided into eight equal parts in the plan.

⒉Assume that the non-developable surfaces between two adjacent meridians are replaced by surfaces curved in one direction along the meridian, or, alternatively, that the non-developable surfaces between adjacent meridians are considered to be expandable surfaces curved along the meridian.

⒊To illustrate the use of the parallel line method for each of the subdivisions, the following is an example of the OAB section: First, add a set of parallel lines that cross the main view O "K° at any point 1, 2, 3 and K° and lead the plumb line to OB at 1', 2', 3', K' and to OA at 1", 2", 3", K", so that 1'1", 2'2", 3'3", K'K" are a set of mutually Then, in the direction of the vertical line of K'K', the K°O" in the main view is straightened and the points 1, 2 and 3 are photographed, and the parallel lines of K'K" are drawn through the photographed points and intersect with the vertical lines of K'K" drawn from the points O, 1', 1", 2', 2", ... K', K" in the same name. The points of intersection are connected in turn by a smooth curve, thus giving an approximate one-eighth of the non-expandable rotating surface.

**The latitudinal division method:** The principle of the latitudinal division method is to draw a number of latitudinal lines on the rotating surface; then assume that the non-spreadable rotating surface located between two adjacent latitudinal lines is approximated as the side surface of a positive conical table with the adjacent latitudinal lines as the upper and lower base, and then expand all the side surfaces of the positive conical table to obtain an approximate expansion of the non-spreadable rotating surface. The diagram below shows the unfolding of a hemispherical surface by the weft division method.

The procedure for unfolding with the latitudinal division method is as follows.

⒈Partition the surface of the form with the weft line division method. In the main view, make any three weft lines (that is, three horizontal lines), so that the rotating surface is divided into four parts.

⒉ Consider parts Ⅰ, Ⅱ and Ⅲ as the sides of three different sizes of a square conical table, and part Ⅳ as a flat circle.

⒊ Use the sector expansion method to make an expansion diagram of each part. Now take the diagram of the small part Ⅱ as an example, explain the following: first extend AB, EF, so that the intersection with the axis of rotation in O Ⅱ, O Ⅱ is the centre of the circle; then measure the size of AF, AF is the small cone table Ⅱ diameter of the bottom d; to O Ⅱ as the centre of the circle, O Ⅱ A, O Ⅱ B, respectively, as the radius of the arc, the outer arc intercept A 'A" long equal to πd, and then connect O Ⅱ A', O Ⅱ A" A' B' B" A" A ' is the expansion diagram of the second small part, and the other blocks are also expanded by the same method to obtain an approximate expansion diagram of the non-expandable rotating surface.

**Warp-weft joint partitioning method: **warp-weft joint partitioning method is used in the expansion of a member of the warp partitioning method and weft partitioning method at the same time, the warp-weft joint partitioning method is applicable to the approximate expansion of large rotating surfaces, such as the diameter of more than ten metres or even dozens of metres of housing cover, large oil tanks and so on. The diagram below shows a large semi-circular spherical sphere with a joint warp-weft division method.

The steps of the joint division method with warp and weft lines are as follows.

⒈with the warp, weft lines jointly divided into a number of parts of the rotating surface, the outer circumference of the plan eight equal parts (the more the number of equal parts will be more accurate), and then the equal points and the centre O 'connected (this is the warp division), over the main view O "K ° on any point 1, 2, 3, 4, make a plumb line cross the plan O 'E in 1', 2', 3', 4 'points, cross O 'E' in 1", 2", 3", 4 Connect 1234 with a dash and make a horizontal line through 1, 2, 3 and 4. Then, with O' as the centre of the circle, draw circles with O'1' (O'1"), O'2' (O'2"), O'3' (O'3") and O'4' (O'4") as radii, thus dividing the rotating surface by the weft method; in the plan, connect the points of intersection of the warp and weft lines in turn with a dash; if the central octagon is treated as a piece of underlay, then each of the above connecting lines divides the rotating The surface is divided into twenty-five small pieces, e.g. 1'2'2"1"1', 2'3'3"2"2', 3'4'4"3"3' are three of these pieces.

⒉Treat the twenty-five non-expandable surfaces as planar, i.e. twenty-four of them are planar trapezoids and the other (top) is a planar octagon.

⒊Expand each of the small planes separately. Obviously, the top of the piece of material is the centre of the planar surface of the orthoctagon, the other small pieces of planar trapezoid expansion can be derived from the parallel line method, this to expand 1'2'2"1"1' as an example of the following: 1'1" in the direction of the vertical line intercepted 1 ° 2 °, so that 1 ° 2 ° is equal to the corresponding arc length 12 in the main view, over 1 °, 2 ° for 1'1" parallel line, and by 1' 2', 2', 2", 1" made by the 1'1" vertical line with the same name corresponding to intersect 1X, 2X, 2XX and 1xx, connecting 1x2x2xx1xx1x, and so get 1'2'2"'1"1' part of the unfolding diagram. From the main view, the eight small trapezoids in each layer are all equal from bottom to top, so by drawing one piece of unfolded material in each layer separately, the other pieces of unfolded material become known as well.

**Approximate unfolding of a straight non-developable surface**

The triangulation method can be used to approximate the unfolding of a straight, non-developable surface. The rules of surface division are exactly the same as those used in the triangulation method, i.e. the non-developable straight surface is divided using the triangulation method. The diagram below shows the triangular method of unfolding a non-expandable straight-grained conical surface.

The steps for unfolding with the triangle method are as follows.

⒈Divide the surface of the form into a number of small triangles. A "B" in the plan is divided into six equal parts, over each equal point lead plumb line intersection A "B" in 1', 2', 3', ... The line is drawn through the points of each equal division to intersect AB and A'B' in 1°° to 5°°, 1° to 5°, and then, as shown in the diagram, to form twelve small triangles.

⒉Find the real length. The upper edge of this component reflects the real length, the lower edge in the plan reflects the real length, the left and right edges in the main view reflect the real length; only eleven lines can not reflect the real length, which can be used to find the real length of the straight triangle method, in seeking the real length of the diagram, only marked the right angle edge length 11' and 1A", the other is not marked, where the real length are indicated in brackets, such as 1A" of the real length with (1A").

⒊According to the triangle method shown in the previous section to expand, you can get a non-expandable straight conical surface of the approximate expansion of the diagram.