If, instead of the molding terminating against a plane surface, as shown by F G in Fig. 238, it be required to develop a pattern to fit against an irregular surface, the method of procedure would be exactly the same, simply substituting for the straight line K G a representation of that surface. From this it will be seen that all that is required to develop the pattern of any miter is that a correct representation (elevation or plan) of the molding be made, showing the angle of the miter, and that a profile be so drawn that it shall be in line with the elevation of the molding - its face being so placed as to agree with the face of the molding - and that poiats from the subdivisions of the profile be carried parallel to the molding, their intersections with the miter line being marked by short lines.
Fig. 239. - Comparison Between a Butt Miter and a Miter Between Two Moldings at Any Angle.
In order to more clearly indicate the point desired by this summary of requirements, suppose that upon each of two pieces of molding made of wood, miters at the same angle be cut (right and left) by means of a saw, and that they be then placed together, as shown in Fig. 239. Now, if a piece of sheet iron, for example, be slipped into the joint, as shown by A, and then one arm of the miter be removed what is left will be exactly what is shown in Fig. 238. In other words, a miter between two straight pieces of molding of the same profile is exactly the same as a miter of the same mold against a plane, and, hence, the operation of cutting the pattern in such a case as shown in Fig. 239 is identical with that described in Figs. 237 and 238.
From this it is plain to be seen that the central idea in miter cutting is to bring the points from the profile against the miter line, no matter what may be its shape or position, and from the miter line into a stretch-out prepared to receive them. Inasmuch as all moldings, if they do not member or miter with duplicates of themselves, must either terminate square or against some dissimilar profile, it follows that the two illustrations given cover in principle the entire catalogue of miters.
The principles here explained are the fundamental principles in the art of pattern cutting, and their application is universal in sheet metal work. It would be difficult to compile a complete list of miter problems. New combinations of shapes and new conditions arc continually arising. The best that can be done, therefore, in a book of this character, is to present a selection of problems calculated to show the most common application of principles which, carefully studied, will so familiarize the student with them that he will have no difficulty afterward in working out the patterns for whatever shapes may come up in his practice, whether they be of those specifically illustrated or not.
From the foregoing the following summary of requirements, together with a general rule for cutting all miters whatsoever, are derived:
Requirements - There must be a plan, elevation or other view of the shape, showing the line of the joint or surface against which it miters, in line with which must be drawn a profile or sectional view of same, and this profile must be prepared for use by having all its curved portions divided into such a number of spaces as is consistent with accuracy and convenience.
Fig. 240. - Usual Method of Cutting a Square Return Miter.
It may be remarked here that the division of the profile into spaces is only an approximate method of obtaining a stretch-out. As theoretically the straight distance from one of the assumed points to another upon a curved line is less than the distance measured around the curve, and the shorter the radius of the
Principles of Pattern Cutting, curve the greater is this difference (a chord is less than the arc which it subtends.) hence the greater the number of points assumed the greater will be the accuracy, and a curve of short radius should be divided more closely than one of longer radius. The profile thus represents practically a succession of plane surfaces.
Rule. - 1. Place a stretch-out of the profile on a line at right angles to the direction of the molding, as shown in the plan, elevation or other view, through the points in which draw measuring lines parallel to the molding. 2. Drop lines from the points in the profile to the miter line or line of joint, carrying them in the direction of the molding till they intersect said line. 3. Drop lines from the intersections thus obtained with the miter or joint line on to the measuring lines of the stretch-out, at right angles to the direction of the molding.
In making the application of this rule the student must not forget that the word profile covers a vast range of outlines, varying from a simple straight line to an entire section of a roof or even more, where large curved surfaces are to be treated, and that a rule that applies to one can be applied to the others equally well.
Fig. 241. - Comparison Between the Short or Usual Method of Cutting a Square Miter and the Method Prescribed by the Rule.
The student who gives careful attention to these rules will at once remark that the operation of cutting a common square miter - that is, a miter between the moldings running across two adjacent sides of a square building, for example - does not employ a miter line, and, therefore, appears to be an exception. Yet it has been remarked that a thorough understanding of how a square miter is cut comprehends within itself the science of miter cutting. The square return miter - for such is the distinctive name applied to the kind of square miter in question - is an exception to the general rule only in the sense that it admits of an abbreviated method. The short rule for cutting it is usually the first thing a pattern cutter learns, and the operation is very generally explained to him without any reason being given for the several steps taken. In many cases it would bother him to cut the pattern by any other than the short method, even after he had obtained considerable proficiency in his art. Hence it is that, to all who have any previous knowledge of pattern cutting the rules above set forth seem inadequate, or, to put it otherwise, are a formula to which there are exceptions.