PROPOSITIONS OF GEOMETRY Table 2e. Geometric Constructions Machinery's Handbook, 31st Edition
70
To construct a parabola: Divide line segment AB into a number of equal parts and divide BC into the same number of parts. From the division points on AB , draw horizontal lines. From the division points on BC , draw lines to point A . The points of intersection of the lines drawn from points numbered alike are points on the parabola.
To construct a hyperbola: From focus F , lay off a distance FD to be the transverse axis, or the distance AB between the two branches of the curve. With F as center and any distance FE greater than FB as a radius, describe a circular arc. Then with F 1 as center and DE as radius, describe arcs intersecting at C and G the arc just described. C and G are points on the hyperbola. Any number of points can be found in a similar manner; when a sufficient number of points are found, draw a smooth curve through them.
C
A B
E F 1
F
D
G
To construct an involute: Divide the circumference of the base circle ABC into a number of equal parts. Through the division points 1, 2, 3, etc., draw tangents to the circle and make the lengths D -1, E -2, F -3, etc., of these tangents equal to the actual length of the arcs A -1, A -2, A -3, etc. Connect the ends of these tangents with a smooth curve.
F
3
2
E
D A 1
C
B
0 1 2 3 4 5 6
To construct a helix: Divide half the circumference of the cylinder on the surface of which the helix is to be described into a number of equal parts. Divide half the lead of the helix into the same number of equal parts. From the division points on the circle representing the cylinder, draw vertical lines using the straight edge, and from the division points on the lead, draw horizontal lines as shown. The intersections between lines numbered alike are points on the helix. Connect these points with a smooth curve.
2 3 4 5
1
6
0
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