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A regular pyramid is one whose base is a regular polygon and whose vertex lies in a perpendicular erected at the center of the base, Fig. 75.

A truncated pyramid is the portion of a pyramid included between the base and a plane not parallel to the base.

A frustum of a pyramid is the solid included between the base and a plane parallel to the base, Fig. 76; its altitude is the perpendicular distance between the bases.

CYLINDERS

A cylinder is a solid having as bases two equal parallel surfaces bounded by curved lines, and as its lateral face the continuous

Fig. 77. Cylinder

Fig. 78. Right Cylinder

Fig. 79. Inscribed Cylinder

surface generated by a straight line connecting the bases and moving along their circumferences. The bases are usually circles and such a cylinder is called a circular cylinder, Fig. 77.

A right cylinder, Fig. 78, is one whose side is perpendicular to the bases.

The altitude of a cylinder is the perpendicular distance between the bases.

A prism whose base is a regular polygon may be inscribed in or circumscribed about a circular cylinder, Fig. 79.

CONES

A cone is a solid bounded by a conical surface and a plane which cuts the conical surface. It may be considered as a pyramid with an infinite number of sides, Fig. 80.

The conical surface is called the lateral area and it tapers to a point called the vertex; the plane is called the base.

The altitude of a cone is the perpendicular distance from the vertex to the base.

An element of a cone is any straight line from the vertex to the circumference of the base.

A circular cone is a cone whose base is a circle.

A right circular cone, or cone of revolution, Fig. 81, is a cone

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Fig. 80. Cone

Fig. 81. Right Circular
Cone

Fig. 82.

Frustum of Cone

whose axis is perpendicular to the base. It may be generated by the revolution of a right triangle about one of the legs as an axis.

A frustum of a cone, Fig. 82, is the portion of the cone included between the base and a plane parallel to the base; its altitude is the perpendicular distance between the bases.

SPHERES

A sphere is a solid bounded by a curved surface, every point of which is equally distant from a point within called the center.

The diameter is a straight line drawn through the center and having its extremities in the curved surface. The radius- diameter -is the straight line from the center to a point on the surface.

A plane is tangent to a sphere when it touches the sphere in only

SMALL

GREAT CIRCLE

Fig. 83. Plane Tangent to Sphere

Fig. 84. Great and Small Circle

one point. A plane perpendicular to a radius at its outer extremity is tangent to the sphere, Fig. 83.

An inscribed polyhedron is a polyhedron whose vertices lie in the surface of the sphere.

A circumscribed polyhedron is a polyhedron whose faces are tangent to a sphere.

A great circle is the intersection of the spherical surface and a plane passing through the center of the sphere, Fig. 84.

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Fig. 85. Intersections of Plane with Cone and Cylinder Giving Ellipses as Shown in (b) and (d)

A small circle is the intersection of the spherical surface and a plane which does not pass through the center, Fig. 84.

CONIC SECTIONS

If a plane intersects a cone at various angles with the base the geometrical figures thus formed are called conic sections. A plane perpendicular to the base passing through the vertex of a right circular cone forms an isosceles triangle. If the plane is parallel to the base, the intersection of the plane and the conical surfaces will be the circumference of a circle.

Ellipse. If a plane AB, Fig. 85a, cuts a cone oblique to the axis of the cone, but not cutting the base, the curve formed is called an ellipse, as shown in Fig. 85b, this view being taken perpendicular to the plane AB. If the

FOCUS

MAJOR

MINOR AXIS

FOCUS

AXIS

plane cuts a cylinder as shown in Fig. Fig. 86. Diagram Showing Constants 85c, the ellipse shown in Fig. 85d is

of Ellipse

the result, this view being also taken perpendicular to the plane AB. An ellipse may be defined as a curve generated by a point

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Fig. 87. Intersection of Plane with Cone,
Parallel to Element of Cone and Para-

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moving in a plane in such a manner that the sum of the distances from the point to two fixed points shall always be constant.

The two fixed points are a) called foci, Fig. 86, and shall lie on the longest line that can be drawn in the ellipse which is called the major axis; the shortest line is called the minor axis; and is perpendicular to the major axis at its middle point, called the center. An ellipse may be constructed if the major and minor axes are given or if the foci and one axis are known.

Parabola. If a plane AB, Fig. 87a, cuts a cone parallel to an element of the cone, the curve resulting from this intersection is called a parabola, as shown in Fig. 87b, the view being taken perpendicular to the plane AB. This curve is not a closed curve for the branches approach parallelism.

Intersection of Plane with Cone, Parallel to Axis and Hyperbolic Section Produced

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A parabola may be defined as a curve every point of which is equally distant from a line and

The point is called the focus, Fig. 88, and the given line, the directrix. The line perpendicular to the directrix and passing through the focus is the axis. The intersection of the axis and the curve is the vertex.

Hyperbola. If a plane AB, Fig. 89a, cuts a cone parallel to its axis, the resulting curve is called a hyperbola,

Fig. 89b, the view being taken perpendicular to the plane AB. Like the parabola, the curve is not closed, the branches constantly diverging.

A hyperbola is defined as a plane curve such that the difference between the distances from any point in the curve to two fixed points is equal to a given distance.

The two fixed points are the foci and the line passing through them is the transverse axis, Fig. 90.

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Rectangular Hyperbola. The form of hyperbola most used in Mechanical Engineering is called the rectangular hyperbola because it is drawn with reference to rectangular coördinates. This curve is constructed as follows: In Fig. 91, OX and OY are the two coördinate axes drawn at right angles to each other. These lines are also called asymptotes. Assume A to be a known point on the curve. Draw AC parallel to OX and AD' perpendicular to OX. Mark off any convenient

Fig. 91.

D' E' F' G' H' X Construction of Rectangular Hyperbola

points on AC such as E, F, G, and II, and through these points draw EE', FF', GG', 'and HH', perpendicular to OX. Connect E, F, G, H, and C with O. Through the points of intersection of the oblique lines and the vertical line AD' draw the horizontal lines LL', MM', N N', PP', and QQ'. The first point on the curve is the assumed point A, the second point is R, the intersection of LL' and EE', the third the intersection S, and so on.

In this curve the products of the coördinates of all points are equal. Thus LRXRE' = MSXSF' NTX TG'.

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ODONTOIDAL CURVES

Cycloidal Curves. Cycloid. The cycloid is a curve generated by a point on the circumference of a circle which rolls on a straight line tangent to the circle, as shown at the left, Fig. 92.

The rolling circle is called the describing or generating circle, the point on the circle, the describing or generating point, and the

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