Intersection and Development 75

As the cylinder, strictly speaking, has no edges parallel to its length, lines or elements may be drawn on the cylinder by the aid of the half end view shown at the right. This half end view is sufficient because the intersection on the upper half of the cylinder will be per fectly symmetrical with that on the lower half. The semicircle is now divided into some num ber of equal parts, say six, by the points E, F, G, etc., to M; and these points, when projected over to the plan in T-square lines, become ele ments of the cylinder. These elements are at different distances above the center of the cylin der, as shown in the end view. Element F is at a height F-s above the horizontal center plane of the cylinder; element G, at a height G-t; and so on. The elements may then be drawn in elevation by making them at their respective heights above the center. These elements are all on the upper half of the cylinder, but a cor responding set should be drawn on the lower half, by spacing similarly below the center line.

It is now a simple matter to find in the plan the points where the elements of the cylinder qut the prism, then to project to the elevation and obtain the required curve. This construc tion is similar to that of Fig. 73. When pro jecting the points of intersection from the plan, as for example the point where F intersects the face AB of the prism, the point on the under side of the cylinder, on Fv,, should also be located at the same time.

It is always important, in such an inter section, to find the points in which the cylinder cuts the edges of the prism—in this case B and C; and if the elements L and G did not happen to pass through B and C, additional elements would have to be drawn through B and C, in order to determine their exact points of inter section with the cylinder.

Notice carefully the visible part of the inter section, which is in accordance with Article 93.

98. Intersection of Two Curved Surfaces. In Fig. 76, a cone standing on its base is shown intersecting a cylinder with its axis parallel to both V and H. At the left, A is the profile view of the cone and cylinder, showing the right hand end of the latter. This view A shows the relative position of the cone and cylinder, and would be drawn first if the figures were to be drawn from given data.

The distance X-y, for example, is the dis tance that the axis of the cylinder is in front of the axis of the cone as shown in plan.

From the position of the solids as shown in this view, it may be seen that the intersection will consist of a single closed curve. If, how

ever, the cylinder were smaller and shown in end view entirely within the cone, then there would be two separate curves.

As the cylinder is shown endwise in this profile view, the method of solution will be to draw lines or elements on the cone, and find in the profile view where they intersect the cylin der. These points of intersection may then be transferred to the plan and the elevation.

A number of elements of the cone are drawn in view A, as o-a, o-b, o-c, o-d, and o-f, o-a and o-f being the extreme left-hand and right-hand elements which have any contact with the cylin der. These elements are next drawn in plan by laying off the distances dP-a', dv-c', etc., from along the line X-Y. Through these points on X-Y, lines drawn perpendicular to X-Y locate points fh, fh„ ch, bh, and and the elements are then drawn through these points and oh.

The elements may now be drawn in eleva tion. The intersection in this case will be ex actly symmetrical in plan and elevation with respect to the line o-a. This is because each element in view A, except o-a, represents two elements equally spaced on either side of o-a- as o-c and o-c„ o-b and and so on.

The points of intersection on the right-hand half are numbered from 1 to 9 inclusive. Points 1 and 9 are projected with the T-square from view A to element o-a in elevation; points 4 and 6, to element o-d and also to element o-d, in elevation, thus locating points on both halves at the same time; and other points are found in the same way.

With the exception of points 1 and 9, the points are then projected directly to the plan onto the corresponding elements of the cone. Points 1 and 9 are transferred directly from the end view to the plan, their horizontal dis tances from the axis op-dv being laid off in plan from oh along X-Y.

Points m and n, where the curve in plan touches the outside element of the cylinder, might be found the same way as the other points. That, however, would require in view A an element drawn very close so another construction is used. In view A, the center plane S of the cylinder will contain both outside elements K and L, and will cut from the cone a circle S of radius r-t. This circle S is then drawn in the plan, and its intersection with K gives the points m and n required. This latter construction might be used, if desired, in find ing the entire intersection.