When crossing a secant the plane of the geometrical planes – cylindrical, conic, the surfaces of rotation, etc. – forms sections of a various type. In particular, conic.
It is required to you
- Pencil, ruler, compasses, curves, triangle
Instruction
1. Lines of crossing of a cone from a secant the plane are called conic sections. Their look depends on the provision of a secant of the plane concerning the planes of projections.
2. In that specific case, if the secant the plane Σ (Σ ₂) is parallel to the horizontal plane of projections P ₁ – in section there will be a circle with a diameter of 1₂2 ₂, 1 ₁, 2 ₁ (fig. 1).
3. If the secant the plane Σ (Σ ₁) passed through S cone top (S ₂ S ₁) – in section will be two crossed direct (fig. 2).
4. If the secant the plane crosses everything forming a cone at an angle to its axis, then in section there will be an ellipse (fig. 3).
5. If the secant the plane is parallel one forming a cone, then in section there will be a parabola (fig. 4).
6. If the secant the plane is parallel two forming a cone, then in section there will be a hyperbole (fig. 5).
7. Example. Construct the section of a circular cone the frontal projecting plane (fig. 6)
8. Apply a way of auxiliary secants of the planes to the solution of this task. The set plane Σ (Σ ₂) is parallel one forming a cone, so in section there will be a parabola. The frontal projection of required section coincides with a plane projection Σ ₂ and is expressed by a straight line.
9. First of all define so-called characteristic (basic) points of the line of section: on a cone projection essay – a point 3 ₂ – parabola top, on a projection of its basis – a point 1 ₂≡ 5 ₂.
10. Without additional constructions, by means of the communication line construct horizontal projections of these points: 3 ₁, 1 ₁, 5 ₁.
11. Note points 2 ₂≡ 4 ₂ and through them carry out auxiliary plane G (₂), parallel to the horizontal plane of projections P ₁. It crosses a cone on a circle diameter equal to AV ₂.
12. Construct this section circle on plane P ₁, and on its essay determine by the communication line points 2 ₁, 4 ₁.
13. Note intermediate points 6 ₂, 7 ₂, through them carry out other auxiliary plane, determine the new section (it is a circle diameter of C₂D ₂), and find horizontal projections of points 6 ₁, 7 ₁.
14. For the accuracy and smoothness of the defined curve carry out the additional auxiliary planes and define projections of new intermediate points. Having connected them, construct a horizontal projection of required section of a cone the projecting plane, in this case – a parabola.