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7.5.1 AUXILIARY SURFACE METHOD

System of auxiliary surfaces can be chosen as system of elementary surfaces. Auxiliary
surfaces can be planes or spheres.
There are cutting-plane method and cutting-sphere method (concentric cutting- sphere
method and eccentric cutting- sphere method).
Method of construction of the intersection curve depends on the type of intersecting surfaces.
System of auxiliary surfaces can be chosen in such way, that the intersection curves of
surfaces and both given surfaces are elementary curves (lines, or circles).
Arbitrary point 1 (and 2) on the intersection curve is the point located on both surfaces Σ and
Δ, it is therefore located on some curve k on the surface Σ and on some curve l on the surface Δ,
while k ∩ l = 1 (Fig. 7.21).

Figure 7.21

Curves k, l are located on one surface α, which intersects surface Σ in the curve k and surface
Δ in the curve l.
In the construction of points on the intersection curve of surfaces Σ and Δ we choose the
system of auxiliary surfaces, for which the following is valid:
Common points of curves k and l (if they exist) are points on the intersection curve.
Auxiliary surfaces α from the system are usually aptly chosen planes or spheres, with regard
to the intersecting surfaces Σ and Δ, in order to receive points on the intersection in the most precise
and easy way.

Cutting-plane method: The two solids are assumed to be cut by a series of cutting planes. The
cutting planes may be vertical (i.e. perpendicular to the  1), edgewise (i.e. perpendicular to the  2)
or oblique. The cutting planes are so selected as to cut the surface of one of the solids in straight
lines and that of the other in straight lines or circles.
In general the method of finding the line of intersection of any two surfaces is to pass a series
of planes through them in such a way as to cut from each the simplest lines. The intersection of the
lines cut from each surface by a plane will give one or more points on the line of intersection.
The intersection of cone and sphere is shown in Fig. 7.22.
Auxiliary cutting plane passes at right angle to the axis of the cone and parallel to its base.
Points 1 and 2 are at the intersection of outlines of cone and sphere. Points 1 and 2 are
uppermost and lowermost points of the curves of intersection. Points 5 and 6 are constructed by
plane Σ. Points 3 and 4 are constructed by plane α. Points 3 and 4 are points at which curve changes
visibility on horizontal projection.
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