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88 Manual of Engineering Drawing
1 Draw a centre line parallel to line AB as part of an
auxiliary view.
2 Project point B to the circumference of the base in
the plan view, to give the points B
1
and B
2
. Mark
chord-width B
1
B
2
in the auxiliary view and in the
end elevation.
3 Project point A onto the other three views.
4 Take any horizontal section XX between A and B
and draw a circle in the plan view of diameter D.
5 Project the line of section plane XX onto the end
elevation.
6 Project the point of intersection of line AB and
plane XX to the plane view.
7 Mark the chord-width W on the plan, in the end
elevation and the auxiliary view. These points in
the auxiliary view form part of the parabola.
8 Repeat with further horizontal sections between A
and B, to complete the three views.
To draw a rectangular
hyperbola from part
of a cone
Figure 12.4 shows the method of drawing the hyperbola,
which is a true view on the line AB drawn parallel to
the vertical centre line of the cone.
1 Project point B to the circumference of the base in
the plan view, to give the points B
1
and B
2
.
2 Mark points B
1
and B
2
in the end elevation.
3 Project point A onto the end elevation. Point A
lies on the centre line in the plan view.
4 Take any horizontal section XX between A and B
and draw a circle of diameter D in the plan view.
5 Project the line of section XX onto the end elevation.
6 Mark the chord-width W in the plan, on the end
elevation. These points in the end elevation form
part of the hyperbola.
7 Repeat with further horizontal sections between A
and B, to complete the hyperbola.
The ellipse, parabola, and hyperbola are also the loci
of points which move in fixed ratios from a line (the
directrix) and a point (the focus). The ratio is known
as the eccentricity.
Eccentricity =
distance from focus
perpendicular distance from directrix
The eccentricity for the ellipse is less than one.
The eccentricity for the parabola is one.
The eccentricity for the hyperbola is greater than
one.
Figure 12.5 shows an ellipse of eccentricity 3/5, a
parabola of eccentricity 1, and a hyperbola of
eccentricity 5/3. The distances from the focus are all
radial, and the distances from the directrix are per-
pendicular, as shown by the illustration.
To assist in the construction of the ellipse in Fig.
12.5, the following method may be used to ensure that
the two dimensions from the focus and directrix are in
the same ratio. Draw triangle PA1 so that side A1 and
side P1 are in the ratio of 3 units to 5 units. Extend
both sides as shown. From any points B, C, D, etc.,
draw vertical lines to meet the horizontal at 2, 3, 4,
etc.; by similar triangles, vertical lines and their
corresponding horizontal lines will be in the same ratio.
A similar construction for the hyperbola is shown in
Fig. 12.6.
Commence the construction for the ellipse by drawing
a line parallel to the directrix at a perpendicular distance
of P3 (Fig. 12.6 (a)). Draw radius C3 from point F
1
to
X
B
B1
B2
W
A
φ
D
B1
W
B2
Parabola
A
B2
B1
W
A
X
A
φ
D
Fig. 12.3
X
φ
D
B
B1
X
A
A
W
Hyperbola
B2
B2
B1
φ
D
A
W
Fig. 12.4