This section is from the book "Cyclopedia Of Architecture, Carpentry, And Building", by James C. et al. Also available from Amazon: Cyclopedia Of Architecture, Carpentry And Building.
A form of hip rafter which is sometimes a source of considerable trouble is one which occurs in a curved roof, such as an ogee roof over a bay window, or a curved tower roof. The slope of the curve to which the top edges of the common rafters must be cut, is determined from the shape of the section of the curved roof surface, but the curve at the top of the hip rafter is entirely different and must be determined in another way. The principle used in finding this curve is the same as was employed in the case of the valley rafter, namely, that any line drawn in the roof surface parallel to the wall plate must be horizontal, or that it must be exactly the same elevation throughout its entire length.
Fig. 206 shows how this may be applied. At A is shown a plan of an ogee roof over a bay window with a hip rafter D E and common rafters. At B is shown an elevation of one of the common rafters cut to coincide with the curve of the roof surface. The shape of the curve may be varied to suit the fancy of the designer. At C is shown an elevation of the hip rafter D E, showing the curve to which it must be cut in order to fit into the roof.
To determine this curve we draw on the roof plan at A any number of lines, parallel to the wall plate. These must be horizontal, so that any point in either of the lines is at the same height above the top of the plate as in every other point in the same line. The lines F G and H I in the elevation, shown at B and C, represent the level of the top of the plate. By projection we find that the line K O X L, for example, is at a distance M N above the top of the plate at the point where it crosses the common rafter shown at B. Every other point in this line is at the same elevation, including the point 0, in which it intersects the center line of the hip rafter D E. By projection we can locate the point 0 in the elevation shown at C, making the distance O P equal to the distance M N.
In the same way we can obtain as many points in the curve of the hip rafter as we have lines drawn on the roof plan. The lines may be drawn as close together as we wish, and the number of points obtained may thus be increased indefinitely. When a sufficient number of points have been located, the curve can be drawn through them, and a pattern for the hip rafter is thus obtained. The shape of the curve for a valley rafter is found in the same way as explained for a hip rafter.