COMPOUND MITRE PROBLEM
David J. Murrow
Compound Mitre Problem—D.J. Murrow
Problem: as posed by Samuel C. Murrow to David J. Murrow on 3/2/2001
Two identical pieces of wood are to be cut with a circular saw on their ends such that, when the pieces are placed at a 90 degree angle to each other, and then each is tilted back away from their respective planes by angle , the planar surfaces of the two cut ends mate. This is illustrated in Figure DJM-1.
Figure DJM-1 Compound Mitre Cut
Each block of wood must be laid horizontally for cutting with the circular saw. The blade of the circular saw can be adjusted by rotating it in the horizontal plane ( and by tilting it away from the vertical plane ( . The problem is to determine the proper horizontal and vertical angle settings for the circular saw.
This problem can be solved in two steps. First the horizontal saw setting angle is found. wThis is the angle above, which is the angle between a vector in plane 1 perpendicular to the x-axis and the cut line. The cut line is the intersection of the two planes 1 and 2 . Second, the vertical saw setting is found by matching up the plane of the saw blade to the required cut plane.
The unit normal vector to plane 1 can be found by rotating the unit normal y-vector counter-clockwise(CCW) about the x-axis by angle .
.
The unit normal vector to plane 2 can be found by rotating the unit normal x-vector counter-clockwise(CCW) about the y-axis by angle .
.
The equations for the two planes are
.
The intersection of these two planes is x=y and z=-ycot()=-zcot(). A unit vector in the direction of the cut-line intersection is
.
A vertical unit vector tilted back by angle into plane 1 would be given by
.
The angle between the vectors e1 and e12 is the angle , which is the required horizontal saw setting angle.
,
from which
Referring back to Figure DJM-1, the plane of the cut must be he x=y plane with the two pieces as shown. However, when plane 1 is rotated back to a horizontal position for cutting, the x=y cut plane rotates back with it. The saw blade plane must be aligned with this required cut plane.
The normal to the u=y-x cut plane is
When the first wood block, represented by plane 1, is rotated back to the horizontal(by angle /2-), this vector normal to the required cut plane becomes
.
In general, after horizontal rotation by and vertical tilting by the saw blade unit normal vector is
.
Equating the saw blade normal vector and the the required cut plane gives the required vertical angle saw setting
For example, if =450, then the required saw settings are =35.260 and =300. The required compound mitre saw settings for other tilt back angles are given in Table DJM-1 below. This table is an imbedded EXCEL spreadsheet. Therefore it can be used to find the settings for other tilt angles by replacing any of the given tilt angles in the table with the desired tilt angle. The required saw setting angles are also shown in Figure DJM-2 vs tilt angle.
Table DJM-1 Required Saw Settings for Compound Mitre
Figure DJM-2 Compound Mitre Angles(red=horizontal, blue=vertical)
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