## Cutting crown molding

This post captures my notes on how to determine the miter and bevel angles for cutting crown molding with a compound miter saw.  There are plenty of web sites with tables of these angles, and even formulas for calculating them, but I thought it would be useful to be more explicit about sign conventions, orientation of the finished cut pieces, etc., as well as to provide a slightly different version of the formulas that doesn’t involve a discontinuity right in the middle of the region of interest, as typically seems to be the case.

Instructions

Let $s$ be the measure of the spring angle, i.e., the angle made by the flat back side of the crown molding with the wall (typically 38 or 45 degrees).  Let $w$ be the measure of the wall angle (e.g., 90 degrees for an inside corner, 270 degrees for an outside corner, etc.).

To cut the piece on the left-hand wall (facing the corner), set the bevel angle $b$ and miter angle $m$ to $b = \arcsin(-\cos\frac{w}{2}\cos s)$ $m = \arcsin(-\tan b \tan s)$

where positive angles are to the right (i.e., positive miter angle is counter-clockwise).  Cut with the ceiling contact edge against the fence, and the finished piece on the left side of the blade.

To cut the piece on the right-hand wall (facing the corner), reverse the miter angle, $m' = -m = \arcsin(\tan b \tan s)$

and cut with the wall contact edge against the fence, and the finished piece still on the left side of the blade.

Derivation

Let’s start by focusing on the crown molding piece on the left-hand wall as we face the corner.  Consider a coordinate frame with the ceiling corner at the origin, the positive x-axis running along the crown molding to be cut, the negative z-axis running down to the floor, and the y-axis completing the right-handed frame, as shown in the figure below.  In this example of an inside 90-degree corner, the positive y-axis runs along the opposite wall. Cutting crown molding for left-hand wall. Example shows an inside corner (w=90 degrees).

The desired axis of rotation of the saw blade is normal to the triangular cross section at the corner, which may be computed as the cross product of unit vectors from the origin to the vertices of this cross section: $\mathbf{u} = (0, 0, -1) \times (\cos\frac{w}{2}, \sin\frac{w}{2}, 0)$

To cut with the back of the crown molding flat on the saw table (the xz-plane), with the ceiling contact edge against the fence (the xy-plane), rotate this vector by angle $s$ about the x-axis: $\mathbf{v} = \left(\begin{array}{ccc}1&0&0\\0&\cos s&-\sin s\\0&\sin s&\cos s\end{array}\right) \mathbf{u}$

It remains to compute the bevel and miter rotations that transform the axis of rotation of the saw blade from its initial $(1,0,0)$ to $\mathbf{v}$.  With the finished piece on the left side of the blade, the bevel is a rotation by angle $b$ about the z-axis, followed by the miter rotation by angle $m$ about the y-axis: $\left(\begin{array}{ccc}\cos m&0&\sin m\\0&1&0\\-\sin m&0&\cos m\end{array}\right) \left(\begin{array}{ccc}\cos b&-\sin b&0\\ \sin b&\cos b&0\\0&0&1\end{array}\right) \left(\begin{array}{c}1\\0\\0\end{array}\right) = \mathbf{v}$

Solving yields the bevel and miter angles above.  For the crown molding piece on the right-hand wall, we can simply change the sign of both $s$ and $w$, assuming that the wall contact edge is against the fence (still with the finished piece on the left side of the blade).  The result is no change to the bevel angle, and a sign change in the miter angle.

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