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.
PLATE 14.
PLATE XIV.
LEVEL LANDING WREATH, OR HALF TWIST.
Fig. 1 shows the plan of the rail, with the top riser placed in the springing and the level rail 4 inches above the landing, so that when the rail is raised to its proper height, 2 feet 8 inches above the treads, measured vertically over the face of the risers to the top of rail, it will be 3 feet from the landing to top of level rail. To get the radius of the centre line of rail, make B H, Fig. 3, equal 4 inches; then A B will be the required radius. At Fig. 2 set up one step and landing. Draw the centre falling line resting on the corners, also draw the level rail 4 inches above the landing. Make N C equal the stretch-out of the centre line of rail, Fig. 1, and complete the falling line from A to C, as shown.