Construct two concentric circles equal in diameter to the major and minor axes of the required ellipse. Let these diameters be AB and CD in Fig. 10.1.
Divide the circles into any number of parts; the parts do not necessarily have to be equal. The radial lines now cross the inner and outer circles.
Where the radial lines cross the outer circle, draw short lines parallel to the minor axis CD. Where the radial lines cross the inner circle, draw lines parallel to AB to intersect with those drawn from the outer circle. The points of intersection lie on the ellipse. Draw a smooth connecting curve.
2 Trammel method
Draw major and minor axes at right angles, as shown in Fig. 10.2.
Take a strip of paper for a trammel and mark on it half the major and minor axes, both measured from the same end. Let the points on the trammel be E, F, and G.
Position the trammel on the drawing so that point F always lies on the major axis AB and point G always lies on the minor axis CD. Mark the point E with each position of the trammel, and connect these points to give the required ellipse.
Note that this method relies on the difference between half the lengths of the major and minor axes, and where these axes are nearly the same in length, it is difficult to position the trammel with a high degree of accuracy. The following alternative method can be used.
Draw major and minor axes as before, but extend them in each direction as shown in Fig. 10.3.
Take a strip of paper and mark half of the major and minor axes in line, and let these points on the trammel be E, F, and G.
Position the trammel on the drawing so that point G always moves along the line containing CD; also, position point E along the line containing AB. For each position of the trammel, mark point F and join these points with a smooth curve to give the required ellipse.
major
Fig. 10.2 Trammel method for ellipse construction
ÎD | ||
2 minor |
2 major | |
axis |
axis | |
E |
F |
Fig. 10.3 Alternative trammel method
Draw major and minor axes intersecting at point O, as shown in Fig. 10.4. Let these axes be AB and CD. With a radius equal to half the major axis AB, draw an arc from centre C to intersect AB at points F1 and F2. These two points are the foci. For any ellipse, the sum of the distances PF1 and PF2 is a constant, where P is any point on the ellipse. The sum of the distances is equal to the length of the major axis.
Fig. 10.4 Ellipse by foci method
line QPR. Erect a perpendicular to line QPR at point P, and this will be a tangent to the ellipse at point P.
The methods of drawing ellipses illustrated above are all accurate. Approximate ellipses can be constructed as follows.
Approximate method 1 Draw a rectangle with sides equal in length to the major and minor axes of the required ellipse, as shown in Fig. 10.6.
Fig. 10.4 Ellipse by foci method
Divide distance OF1 into equal parts. Three are shown here, and the points are marked G and H.
With centre F1 and radius AG, describe an arc above and beneath line AB.
With centre F2 and radius BG, describe an arc to intersect the above arcs.
Repeat these two steps by firstly taking radius AG from point F2 and radius BG from F1
The above procedure should now be repeated using radii AH and BH. Draw a smooth curve through these points to give the ellipse.
It is often necessary to draw a tangent to a point on an ellipse. In Fig. 10.5 P is any point on the ellipse, and F1 and F2 are the two foci. Bisect angle F1PF2 with
Divide the major axis into an equal number of parts; eight parts are shown here. Divide the side of the rectangle into the same equal number of parts. Draw a line from A through point 1, and let this line intersect the line joining B to point 1 at the side of the rectangle as shown. Repeat for all other points in the same manner, and the resulting points of intersection will lie on the ellipse.
Approximate method 2 Draw a rectangle with sides equal to the lengths of the major and minor axes, as shown in Fig. 10.7.
Bisect EC to give point F. Join AF and BE to intersect at point G. Join CG. Draw the perpendicular bisectors
lines at points H and J.
Using radii CH and JA, the ellipse can be constructed by using four arcs of circles.
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