The same method is shown in [Fig. 2188], except that in this case, instead of dividing a b and k l, the divisions are made directly on the peripheries a 6 b and k vi. l by stepping round with the dividers. The cutter f c is shown in this case at an angle of 45°, in order that the change in form which the curve assumes with the cutter at different angles may be clearly seen by comparing the curve n p j of [Fig. 2187] with the same in [Fig. 2188]. The two figures are similar in other respects, and as the lettering is the same on each, the description of [Fig. 2187] will apply equally to [Fig. 2188].
Fig. 2189.
There remains one more case of cutters moving in right lines, and that is where, besides having an inclination backward, as at f c, [Fig. 2187], making a vertical angle to the line of motion, they are placed at an angle across the guiding piece also, or “skewing,” thus making an angle to the line of motion on a horizontal plane as well as on a vertical one. Thus, suppose an ordinary carpenter’s plane to have the cutter or “iron” turned partly round and placed so that the cutting edge, instead of lying at a right angle across the body, crosses it at some other angle. [Fig. 2189] represents an ordinary carpenter’s plane with the blade so placed. Here the edge, or rather side, d b, of the blade inclines back at an angle, as a b d, which is 45° in this case, to the perpendicular line a b on the side of the plane. For convenience call a b d the vertical angle. The lower or cutting edge e b of the blade also crosses the bottom of the plane at an angle e b c—30° in this instance—to a line b c, crossing the bottom at right angles. Now, it is evident that this latter angle e b c will influence the form of the cutter, if, instead of being a flat plane, as represented for clearness in [Fig. 2189], it had a cutting edge of curved outline for cutting mouldings or similar work. But in either case the angle that d b or one side of the blade makes to e b, or the cutting edge—that is, the angle d b e—must be found in order to cut off the blank for the cutter or knife at the right “slant.”
Fig. 2190.
The method given in [Figs. 2187] and [2188] of determining the form of cutter to produce a moulding of given profile now undergoes a modification where there are two angles to be taken into consideration instead of one. As an example, suppose a cutter is required that is to be fixed in such a position in its carrier or block that the handle a b d, or “vertical angle,” of [Fig. 2189] is, say, 45°, and the angle e b c, or “horizontal angle,” of [Fig. 2189] shall be 30°. Required the angle at which the bottom of the blank for the cutter must be cut off; or the angle that the side d b and lower edge b e of [Fig. 2189] would make to each other, measured on the face of the cutter, and required the outline of cutting edge to be traced on the face of cutter to cut the section of moulding a e b, [Fig. 2190]: draw a horizontal line, as a b c d, and erect a perpendicular, as c r. From c draw c f, making an angle to c r equal to the “vertical angle,” or angle a b d, [Fig. 2189], which is 45° in this case. Draw a profile of the required moulding, as a e b, with its back a b coincident to the horizontal line a b c d. Draw a horizontal line from the highest point of the profile, as e, to meet f c in g. Draw parallel lines c j and g h, from c and g respectively, of any convenient length and making right angles to f c. At right angles to g h and c j, and parallel to f c, draw k h j to represent one side or edge of the cutter, but the angle of the lower end or angle d b e of [Fig. 2189] must now be determined; to do this, draw an indefinite horizontal line, a b c, [Fig. 2191], and from any point, as b, drop a perpendicular b d; now, from b set off on a b c the distance c b of [Fig. 2190], obtaining point e, and from e extend a perpendicular above and below a b c, as f e h. From e on e f set off distance g b of [Fig. 2190], obtaining j on e f. From b draw a line, making the same angle to b d that the angle e b c is in [Fig. 2189], or 30° in this case, and cutting e h in k. Set off distance e k from e on a c, obtaining l; draw l j. Now, on [Fig. 2190], with centre at h, and radius l j of [Fig. 2191], describe arc w x, and from j as centre, on [Fig. 2190], and b k of [Fig. 2191] as radius, describe arc y z. Through the intersection v of arcs y z and w x, j l m must be drawn, making the proper angle to the side j h k of the cutter; this angle is 69° in this case, as found by construction. From h draw h n parallel to j l, and from h draw h o at the same angle to h n that b k is to b d, [Fig. 2191], or angle e b c, [Fig. 2189]. Place a duplicate of a e b, with its base coincident to h o and corner a at h, as h p r. From r draw r n at right angles to h r and cutting h n at n; through n draw s n l parallel to k h j. Then while k h j represents one edge of the cutter, s n l will be the other, and j l the cutting edge before the opening is cut out. Divide the curves e b and p r similarly, obtaining points 1, 2, 3, &c., and i., ii., iii., &c., respectively. From points 1, 2, 3, &c., lines are to be drawn parallel to e g, meeting g c, continued from g c parallel to g h, and meeting h j, and from h j parallel to h n, meeting n l. From points i., ii., iii., &c., lines are to be drawn perpendicular to h r, meeting h n and continued from h n, parallel to h j, to j l, thus intersecting the first series. Lines from points 1, 2, 3, &c., then determine the height of different points of the curve, and those from i., ii., iii., &c., their location laterally; hence, by tracing through the intersections of 1 with i., 2 with ii., &c., the curve h t l is obtained. The two outside lines k h j and s n l may now represent the edges of a piece of steel of which the cutter is to be made, and h t l will be the contour of cutting edge that must be given it in order that when, fixed for use at the angles named, it will form the required moulding a e b.