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Fig. 31.

Fig. 32.

Fig. 33.

the nature of the surface, meet it in one of its right-line
elements; but it will meet the plane of the base in a
straight line, as PT (fig. 31,) which will be a tangent
to the periphery of the base.
If therefore we draw through the proposed point M,
the straight line MP parallel to the generatrix AD, and
construct the tangent PT to the point P of the curvilin-
ear directrix, the tangent plane will be determined by the
two lines PM and PT.
Remark 1st. If the proposed point through which the
tangent plane is to be drawn, is without the cylinder, it is
evident that there will be two planes which will answer
to the conditions of the problem. To construct this so-
lution find the point where a straight line passing through
the given point and parallel to a rectilinear element of
the cylinder, meets the plane of projection. A line
drawn through this point, tangent to the intersection of
the cylindrical surface with this plane, will be the trace
of the tangent plane upon this plane of projection.
Remark 2d. If we would find the rectilinear tangent
to a curve made by the intersection of two curve surfa-
ces, we should construct tangent planes to these surfaces
at the proposed point ; and the intersection of the two
planes would be the right-line tangent required.

86. PROBLEM. To construct a tangent plane to a C07062,

The only difference between this construction and that of the last problem, is, that the line MP must be drawn through the summit S instead of being parallel to the generatrix (fig. 32).

87. PROBLEM. To draw a tangent plane to a surface of revolution.

The most convenient method of determining a tangent plane to surfaces of this class, is by means of the rectilinear tangents to the meridian curve and circular section, at the proposed point of contact.

The plane tangent to the point M (fig. 33) will therefore be determined by the straight lines MT and Mt ; the first being tangent to the generating curve MX, and the other to the circle MZ. If therefore we can draw a tangent to the meridian curve at the proposed point of contact, it is easy to construct the plane tangent to the surface at that point.

The tangent plane to a surface of revolution, is always perpendicular to the meridian plane passing through the point of contact. It therefore follows that, a meridian plane to a surface of revolution is always normal to that surface.

Plane Sections of Curve Surfaces.

88. Plane sections of a cylindrical surface parallel to its axis and of a conical surface through its summit, are all straight lines (61, 63).

Plane sections of these surfaces parallel to the bases of these bodies are similar to these bases (El. 268, 275).

In the right cylinder and right cone these sections are perpendicular to the axis; if the bases are circles these sections will be circles. But an oblique section of a right come by a plane which cuts all its rectilinear elements on the same side of the apex, is an ellipse (58). And it is easy to show to those acquainted with the properties of this curve, that a plane section of a right cylinder, oblique to its axis, is also an ellipse.

These sections are easily determined ; for having these surfaces given, their rectilinear elements are known ; and knowing these right lines and the cutting plane, the curves themselves are determined by finding o: points where these lines meet this cutting plane, 33).

89. A plane section of a sphere is always a circle. The determination of these sections belongs to the solution of the general problem—To find the position and magnitude of the circle which is the intersection of a given sphere and a given plane.

By drawing a perpendicular from the centre of the sphere to the cutting plane, and determining the meeting of this line and the proposed plane, we have the centre of the circle required.

This solution will be very simple, if we take the plane of the vertical projections, DAB (fig. 34) perpendicular Fig. 34. to the horizontal trace of the proposed plane, which may always be done. Then, O' and O'' being the projections of the centre of the sphere, if we suppose it cut by a vertical plane drawn through the line H'O' perpendicular to AC; this plane will pass through the centre of the

Fig. 34.

sphere, and will contain the perpendicular drawn from this point to the proposed plane DAC ; but it is parallel to the vertical plane DAB : We may therefore suppose it to be applied to this vertical plane without any change of the lines which it contains, either in magnitude or in position relative to its horizontal trace H' O' which will then coincide with AB. This being premised, M''E'N' will be the great circle which results from the section of the sphere by the vertical plane just mentioned; the perpendicular O'G'" will determine the vertical projection G" of the centre of the section sought. We thence obtain its horizontal projection G' ; and, revolving the plane DAC about it horizontal trace, as this centre will fall upon the point G in the horizontal plane, we may describe from this point, with the radius G"M", the circle MN which will be the required intersection of the sphere and plane proposed.

Of the Intersections of Curve Surfaces.

90. The most simple method of constructing the intersections of curve surfaces, is to suppose a series of plane sections of the two surfaces, and to determine the section made by each plane with each of these surfaces; the points common to the two curves, in any one of these plane sections, are of course points in the intersection required.

This method will be the most simple if we draw the cutting planes parallel to one of the co-ordinate planes. Suppose them parallel to the vertical plane ; the sections which they make with the proposed surfaces will have their horizontal projections respectively in the horizontal traces of these cutting planes; and their vertical projections will not only show the height of any required point above the horizontal plane, but will be exactly equal to the auxiliary curves whose respective projections they 8. I’62.

A few examples will illustrate these principles, and enable the reader to apply them generally to the construction of analogous problems.

91. PROBLEM. To construct the intersection of a cylinder and a sphere.

We shall take for the plane of the horizontal projec

tions a plane perpendicular to the rectilinear elements of
the cylinder ; the vertical plane will therefore be parallel
to these elements. We suppose cutting planes parallel
to the vertical plane; their intersections with the cylin-
drical surface will be right-line elements of this surface;
and their intersections with the sphere will be circles
whose centres will have their vertical projection in the
same point with that of the centre of the sphere. The
radii of these circles may be easily found from the dis-
cussion in article 89.
The details of these constructions will be mostly left
to the reader, as they will be found easy when the prin-
ciples already discussed are well understood.
In the present case we draw straight lines G’ I’, g' i', (fig.
35) in the horizontal plane, parallel to the ground line,
These lines are the horizontal traces of the auxiliary planes,
which are supposed parallel to the vertical plane of pro-
jection; go i is the radius of the circle which is the sec-
tion made by one of these planes in the sphere whose
centre has its horizontal projection in E' and its vertical
projection in E". The points P'i, Pa, where the line
g' i' meets the base of the cylinder, are the horizontal
projections of two rectilinear elements of this surface
which are met each in two points by the circumference
of the circle which is the section of the sphere by this
auxiliary plane whose horizontal trace is g' i'. If, there-
fore, from the point E" as a centre and with a radius
equal to go i, we describe a circumference, the points
p", p"a, P’, , P’a, where it meets the vertical projec-
tions of the rectilinear elements just mentioned, will be-
long to the intersection of the cylinder and sphere.
By a similar process we may determine as many other
points as we wish of this intersection. There is, how-
ever, one circumstance in the general problem under dis-
cussion, which demands particular attention; this is the
case in which the cylinder entirely penetrates the sphere,
which it will do if the radius of a circular section of the
cylinder is less than the radius of the sphere minus the
distance of the axis of the cylinder from the centre of the
sphere. In this case there will be two intersections of
the surfaces in question, the one where the cylinder en-
ters the sphere, and the other where it emerges from it.
The horizontal projections of these two curves will of
course be the same ; and it will be readily seen that the

Fig. 35. Fig. 35.

Fig. 36.

Fig. 37.

parts P'i, P's, P'', , P'a, belong to the vertical projection
of the curve, where the cylinder enters the sphere ; and
that p", p"a, p", , p", belong to the vertical projection
of the curve where the cylinder emerges from the spheri-
cal surface.
The following figure represents the case in which the
distance of the axis of the cylinder from the centre of the
sphere is greater than the difference of the radii of the
sphere and cylinder. In this case the cylinder does not
entirely penetrate the sphere ; there is therefore but one
intersection of the two surfaces.

92. PROBLEM. To find the projection of the curve made by the intersection of the sphere and the cone.

Suppose the cone to be cut by planes passing through its apex, and perpendicular to the horizontal plane. The sections thus made in the conical surface, will be straight lines which are easily determined ; the sections which these auxiliary planes make in the sphere, will be circles whose centres and radii are also easily found (89).

Let S K (fig. 37) be the horizontal trace of one of these cutting planes. Instead of revolving about this trace to coincide with the horizontal plane, we suppose it applied to the vertical plane DAB by bringing the line STK' to AB, so that the point S will fall upon S ; then by taking Sk equal to S K we can draw the straight lines S*k, which will be the sections of the conical surface by the cutting plane. Making mg equal to SG", the point g will be the position of the centre of the circle in which the sphere is met by the cutting plane, and whose radius is g h equal to G'H', (89).

The points p where the circumference of this circle meets the straight lines S." k, belong to the intersection of the proposed cone and sphere. There are four of these points; two where the cone enters the sphere, and two where it emerges from it.

The points p are situated in the cutting plane. To have their projections, we must take SP equal to p i, and the point P will be the horizontal projection ; by drawing P'P" perpendicular to AB, the vertical projection P" will be found at the meeting of this straight line with p i ; for the point p being taken in a vertical plane, is at the same height with its projection upon every other vertical plane.

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