Should any part of a plan be a circle, it will when projected on to an
oblique plane be an ellipse.
Thus, take any one round piece of wood, cut one end off square to its
side, this end will be a true circle. Cut the other end to any angle,
which will then be an ellipse, and when the piece is stood on end will
be vertical over the circle, its plan. Fig. 1 shows this. And it will
be seen, no matter what angle the oblique plane may be, the minor axis
never changes, it is always the same length as on plan, as are all
lines parallel to it; but not so with the major axis, which always
lengthens as the angle increases.
Fig. 2 shows a quarter of a circle projected on to an oblique plane,
inclined 45° to the horizontal plane.
Fig. 3 shows a plank inclined 45° to the horizontal plane, with a
quarter of an ellipse traced on its oblique surface. At any point on
the curve draw a section line and a normal tangent. Cut the plank
square through to the section line. Draw a level line on the square
cut, and produce a bevel that will, when the stock is held to the
section line on the oblique surface, produce with its blade a line
across the cut that will be perpendicular to the level line on the
square cut.
At any point on the curve, say at S, draw a line square to, and to
cut the major axis in P; draw the level line on the edge to cut the
vertical line from the centre O in N; draw R N square to the major
axis. Join R S, which is the section line; draw the tangent square to
it through S. Cut the plank through to the lines S R and R N; join N S
after the cut is made. To get the bevel, draw A A parallel to, and at
a distance away from O N, equal to minor axis or radius of circle on
plan. Take the compasses, and for centre put one foot at O, and for
radius strike an arc just touching the tangent through S, bring it
around to cut A A in H, join O H for the required bevel. This bevel,
when applied across the cut, will be square to the level line S N.
[Illustration: PLATE 14.]