# How do cross-section

At the intersection of the cutting plane geometric planes - cylindrical, conical, surfaces of revolution, etc. - Images of various kinds of cross-section. In particular, tapered.

### You will need:

Pencils, rulers, compasses, curves, triangle

## Instruction how to make the cross section

Step 1:

The lines of intersection of the cone with the cutting plane are called conic sections. Kind of depends on the position of the cutting plane relative to the projection planes.

Step 2:

In the particular case when the cutting plane Σ (Σ₂) parallel to the horizontal plane of projection P₁ - in the section will be a circle with a diameter 1₂2₂, 1₁, 2₁ (Figure 1.).

Step 3:

If the cutting plane Σ (Σ₁) passed through the vertex of the cone S (S₂ S₁) - in the section are two intersecting lines (Figure 2.).

Step 4:

If the cutting plane intersects all the generators of the cone angle to its axis, in cross-section is an ellipse (Fig. 3).

Step 5:

If a cutting plane parallel to the generatrix of the cone, the section is a parabola (fig. 4).

Step 6:

If the cutting plane is parallel to two generatrices of the cone, the cross section is a hyperbola (Fig. 5).

Step 7:

Example. Build cross-section of a circular cone frontal projecting plane (Fig. 6)

Step 8:

To solve this problem, apply the method of auxiliary section planes. Set the plane Σ (Σ₂) parallel to one of the cone, then, in the section will be a parabola. Front projection of the desired cross-section coincides with the projection plane and expressed Σ₂ straight.

Step 9:

First of all, define the so-called typical (reference) point of the line section: on the sketch projection cone - point 3₂ - the vertex of the parabola, on the projection of its base - point 1₂≡5₂.

Step 10:

Without additional constructions, using the link build horizontal projections of these points: 3₁, 1₁, 5₁.

Step 11:

Note 2₂≡4₂ points and through them draw an auxiliary plane F (G₂) parallel to the horizontal plane P₁ projections. It intersects the cone by a circle with a diameter equal to A₂V₂.

Step 12:

Build-it-section of a circle on a plane P₁, and its sketch line connection points define 2₁, 4₁.

Step 13:

Mark waypoints 6₂, 7₂, through them draw another reference plane, define a new section (a circle with a diameter C₂D₂), and find the horizontal projection points 6₁, 7₁.

Step 14:

For accuracy and smoothness of the curve defined by the draw additional support plane and determine the projection of new waypoints. By combining them, build a plan view of the desired cross-section of the cone projecting plane, in this case - a parabola.