Fig 208. - To Draw on Ellipse by Means of a Trammel.

Fig. 209.

Fig. 210.

To Draw an Ellipse by Means of a Square and a Strip of Wood.

One definition of an ellipse is " a figure bounded by a regular curve, which corresponds to an oblique section of a cylinder."

This can be practically illustrated by assuming a piece of stove pipe as the representative of the cylinder. If the piece of pipe is cut square across, the end placed upon a board, and a line drawn around it, the resulting figure will be a circle. If now the pipe be cut obliquely, as in making an elbow at any angle, and the end thus cut be placed upon a board and a line drawn around it, as mentioned in the first case, the figure drawn will be an ellipse. "What has thus been roughly done by mechanical means may be also accomplished upon the drawing board in a very simple and expeditious manner. The demonstration which follows is of especial interest to the pattern cutter, because the principles involved in it lie at the root of many practical operations which he is called upon to perform. For example, the shape to cut a piece to stop up the end of a pipe or tube which is not cut square across, the shape to cut a flange to fit a pipe passing through the slope of a roof, and other similar requirements of almost daily occurrence, depend entirely upon the principles here explained.

68. To Describe the Form or Shape of an Oblique Section of a Cylinder, or to Draw an Ellipse as the Oblique Projection of a Circle. - The two propositions which are stated above are virtually one and the same so far as concers the pattern cutter, and they may be made quite the same so far as a demonstration is concerned. The explanation of the engraving is confined to the idea of the cylinder, believing it in that shape to be of more practical service to the readers of this book than in any other. In Fig. 211, let G E F H represent any cylinder, and ABCD the plan of the same. Let I K represent the plane of any oblique cut to be made through the cylinder. It is required to draw the shape of the section as it would appear if the cylinder were cut in two by the plane I K, and either piece placed with the end I K flat upon paper and a line scribed around it. Divide one-half of the plan ABC into any convenient number of equal parts, as shown by the figures 1, 2, 3, 4, etc. Through these points and at right angles to the diameter A C draw lines as shown, cutting the opposite side of the circle. Also continue these lines upward until they cut the oblique line I K, as shown by 1', 2', 3l, etc. Draw I' K', making it parallel to I K for convenience in transferring spaces. With the T-square set at right angles to I K, and brought successively against the points in it, draw lines through I' K', as shown by 12, 22, 32, etc. With the dividers take the distance across the plan A B C D on each of the several lines drawn through it, and set the same distance off on corresponding lines drawn through I1 K'. In other words, taking A C as the base for measurement in the one case and I1 K1 as the base of measurement in the other, set off from the latter, on each side, the same length as the several lines measure on each side of A C. Make 2' equal to 2, and 3' equal to 3, and so on. Through the points thus obtained trace a line, as shown by I' M K1 and the opposite side, thus completing the figure.

Fig. 211. - The Ellipse as an Oblique Section of a Cylinder.

To make this problem of practical use it is necessary that the diameter of the cylinder shall be equal to the short diameter of the required ellipse, and that the line 1 K he drawn at such an angle that the distance 11 91 shall be equal to its long diameter.

Another definition of the ellipse is that "it is a figure bounded by a regular curve, corresponding to an oblique section of a cone through its opposite sides." It is this definition of the ellipse that classes it among what are known as conic sections. It is generally a matter of surprise to students to find that an oblique section of a cylinder, and an oblique section of a cone through its opposite sides, produce the same figure, but such is the case. The method of drawing an ellipse upon this definition of it is given in the following demonstration. The principles upon which this rule is based, no less than those referred to in the last demonstration, are of especial interest to the pattern cutter, because so many of the shapes with which he has to deal owe their origin to the cone.

Fig. 212. - The Ellipse as an Oblique Section of a Cone.

69. To Describe the Shape of an Oblique Section of a Cone through its Opposite Sides, or to Draw an Ellipse as a Section of a Cone. - In Fig. 212, let B A C represent a cone, of which E D G F is the plan at the base Let H I represent any oblique cut through its opposite sides. Then it is required to draw the shape of the section represented by II I, which will be an ellipse. At any convenient place outside of the figure draw a duplicate of II I parallel to it, upon which to construct the figure sought, as II1 I1. Divide one-half of the plan, as E D G, into any convenient number of equal parte, as shown by 1, 2, 3, 4, etc. From the center of the plan M draw radial lines to these points. From each of the points also erect a perpendicular line, which produce until it cuts the base line B C of the cone. From the base line of the cone continue each of these lines toward the apex A, cutting the oblique line H I. Through the points thus obtained in H I, and at right angles to the axis A.D of the cone, draw lines, as shown by 11, 22, 31, 41, etc., cutting the opposite sides of the cone. From the same points in H I, at right angles to it, draw lines cutting H1 I1, as shown by 12, 22, 32, 42, etc., thus transferring to it the same divisions as have been given to other parts of the figure. After having obtained these several sets of lines, the first step is to obtain a plan view of the oblique cut, for which proceed as follows: With the dividers take the distance from the axial line A D to one side of the cone, on each of the lines l1, 21, 31, 41, etc., and set off like distance from the center of the plan M on the corresponding radial lines 1, 2, 3, 4, etc. A line traced through the points thus obtained will give the plan view of the oblique cut, as shown by the inner line in the plan.