Development is the surface of a solid developed on the plane. For creation of development of any surface it is necessary to combine consistently all its flat elements with one plane.
It is required to you
- Pencil, compasses, curves, triangle, ruler
Instruction
1. Example. To construct development of the truncated cone. The side surface of the truncated cone has no flat elements since is a curve surface. For receiving approximate development execute the following constructions (figure 1).
2. Enter a polyhedron in a cone. For this purpose on horizontal projections circle of the lower basis of a cone divide into arches 12(1₁2 ₁), 23(2₁3 ₁), etc. And divide a circle of the top basis into arches 67(6₁7 ₁), 78(7₁8 ₁), etc. Connect these arches chords. As a result receive the octahedral truncated pyramid entered in this truncated cone. Sides represent it trapezes which have parties of the basis – a chord 1₁2 ₁, 6₁7 ₁, etc., and two other opposite sides – side edges 1₁6 ₁, 2₁7 ₁, etc. These sides trapezes are also flat elements which are combined with the drawing plane at development.
3. In each side carry out diagonals 1₁7 ₁, 2₁8 ₁, etc., dividing them into two triangles. Determine the full size (N of century) of diagonal by the 17th way of a rectangular triangle. For this purpose note height of a frontal projection of the truncated h cone. At right angle to h postpone a horizontal projection of diagonal 1₁7 ₁. The received hypotenuse 1₀7 ₁ is equal to the full size (N of century) of diagonal 17.
4. At creation of development all sizes have to have full size. 1672 entered pyramids all elements are presented to sides without distortion: the full size of an edge 16 is equal to its frontal projection 1₂6 ₂, chords 67 (6₁7 ₁), 12 (1₁2 ₁) were designed full-scale on plane P ₁. The full size of diagonal 1₀7 ₁ is found way of a rectangular triangle.
5. Creation of development. On a vertical straight line (or a straight line of any situation) postpone a piece 1₀6 ₀=1₂6 ₂. Of a point 6 ₀ with a radius of 6₁7 ₁ make a notch, and of 1 ₀ with a radius of 1₀7 ₁ (N of century) – the second. 7 ₀ connect the received point straight lines to 1 ₀ and 6 ₀. Of a point 1 ₀ make a notch with a radius of 1₀2 ₀=1₁2 ₁, and of a point 7 ₀ with a radius of 7₀2 ₀=1₀6 ₀. Receive a point 2 ₀, connect it to points 1 ₀ and 7 ₀. The constructed trapeze 1₀6₀7₀2 ₀ is the side of the pyramid entered in this truncated cone combined with the drawing plane.
6. All sides of the entered pyramid are equal among themselves therefore, using the same sizes, construct all adjacent sides and connect straight lines of a point 1 ₀, 2 ₀, 3 ₀, etc. The received flat figure will be development of a side surface of the pyramid entered in the truncated cone.
7. Connect the constructed points 1 ₀, 2 ₀, 3 ₀, etc. the lower basis and a point 6 ₀, 7 ₀, 8 ₀, etc. the top basis of the truncated cone a lekalny curve. The received figure is development of the truncated cone.