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Compound-Angle Joinery
Donald L. Snyder Bill Gottesman^1
1 Introduction
In a previous note, one of us (DLS) wrote about compound-angle joinery and stat- ed, without explanation, the mathematical expressions which specify the blade tilt and miter-gauge setup angles on a table saw used to cut the parts of a compound- angle joint [1]. The same expressions apply for other tools used to cut the joints, such as miter saws, scroll saws, and even hand saws. For completeness, the math- ematical basis for the expressions is developed in this note. In addition, the earlier Frink computer-procedure [1] for calculating the setup angles is updated to include correct formulas for setup angles and for not just compound miter-joints but also compound butt-joints, and the visual display of results on portable devices such as smart phones and tablets is improved. The mathematical basis for compound-angle joinery is developed in two ways. For the first, in Section 3, plane trigonometry, vector notation and rotation matrices are used. For the second, in Section 4, spherical trigonometry is used following insights of Bill Gottesman. In Section 1.1 we give some examples of objects made using compound- angle joinery. Several angles that occur in the joinery are identified in Section 2. The development of the setup angles by using plane trigonometry, vectors and ro- tation matrices is in Section 3. The development using spherical trigonometry is in Section 4. Results are summarized in Section 5, and computer implementations are in Section 6. Section 7 has four examples, and 8 lists cited references.
1.1 Examples of objects exhibiting compound-angle joinery
Compound-angle joinery is an integral feature of diverse objects. Examples of closed forms shown in Figs. 1-4 illustrate the wide range of objects where this join- ery is encountered. By a ‘closed form’ is meant a multisided object in which the multiple sides are connected to form an enclosure, such as a box.
(^1) Bill Gottesman of Burlington, Vermont, contributed the ideas and material for the section in which spherical trigonometry is used to develop expressions for the setup angles needed for compound-angle joinery.
Figure 1. Octagonal jewelry box with removable shelf insert. Made by Vic Barr using maple and cherry woods. The eight sides are at 90° to the base.
Figure 2. Four-, seven- and six-sided closed-form objects involving compound-angle joinery.
Figure 3. Hexagonal bowl involving compound-angle joinery. Each of the six sides slope outward at 60° from the base.
2.1 Angles that characterize a compound-angle joints
Illustrated in Fig. 6 are two components of a closed or open form that come togeth- er in a compound-angle joint. Both components are assumed to rest on the XY - plane of a three-dimensional coordinate system^2 , with the resting edge of one com- ponent ( component 1 ) aligned with Y -axis and having a slope of S degrees meas- ured (counterclockwise) from the XY -plane. The origin of the coordinate system is located at the point in the XY -plane where the two components come together. The resting edges of the two components meet an angle π in the XY -plane. For a closed form having the shape of an N -sided regular polygon in the XY -plane, π < (^) ∋ N, 2 *180( ν^ /N degrees. For example, a four-sided square box will have π < 2 *180 / 4ν^ < 90 ν^ regardless of any slope the sides may have, the six-sided bowl
of Fig. 3 has a hexagonally shaped base with π < 4 *180 / 6ν^ < 120 ν^ , and the seven-
sided heptagon-vase of Fig. 2 has π < 5 *180 / 7ν^ ≡128.6ν^. Generally, 0 ′ π′ 180 ν^. There is another angle that is important in describing the joined components of Fig. 6. It is called the “dihedral angle.” The dihedral angle can be measured by constructing two lines, one in the face of each component. The line in a face is constructed so that it is perpendicular to the mating line where the two faces join. The two constructed lines are positioned to meet at common point anywhere along the mating line. The smallest angle between the two lines constructed in this way is the dihedral angle (also called the plane angle [11]).
Figure 6. Compound-angle joint connecting two components of a closed or open form
(^2) A coordinate system with a right-hand convention is used. Angles measured counter- clockwise are positive and clockwise negative.
If S < 90 ν^ , the two components in Fig. 6 are perpendicular to the XY -plane, and the dihedral angle equals π. Generally, (^0) ′ S′ 180 ν^ , and the dihedral angle is smaller than π if S ÷ 90 ν^. The dihedral angle is determined in the following way using plane trigonome- try, vectors and rotation matrices. Define unit vectors along the coordinate axes as
1 0 0
e X
<^ ^
θ ,
e Y
<^
θ , and
e Z
<^ ^
θ .
Also, define rotation matrices
∋ (
0 cos sin 0 sin cos
X X X X X X
R ι ι ι ι ι
< ^ ,
, (^) ∋ (
cos 0 sin 0 1 0 sin 0 cos
Y Y Y Y Y Y
R
ι ι ι ι ι
< ^
and
∋ (
cos sin 0 sin cos 0 0 0 1
Z Z RZ Z Z Z
ι ι ι ι ι
< ^
For example, the operation R X (^) ∋ ιX (^) (v
θ rotates the vector v < v (^) X vY vZ^ T
θ through
an angleι X around the X -axis to become the vector
∋ (
0 cos sin cos sin 0 sin cos sin cos
X X X X X X Y Y X Z X X X Z Y X Z X
v v R v v v v v v v
ι ι ι ι ι ι ι ι ι
^ ^ ^ ^
< ^ , ^ ^ < ^ ,
θ (^). (1)
Now, consider unit vectors that are perpendicular to each of the two components in Fig. 6. A unit vector that is perpendicular to the showing face of the component aligned along the Y -axis (labeled component 1 ) results by a counterclockwise rota- tion around the Y -axis of the unit vector along the Z -axis,e Z
θ , through the angle of
the slope, S , of the face
1 ∋^ (
sin 0 cos
Y Z
S u R S e S
< <^ ^
θ θ
. (2)
For example, if S < 90 ν^ , component 1 is perpendicular to the XY -plane, and u 1 < eX
θ θ . A unit vector that is perpendicular to the showing face of the other compo- nent (labeled component 2 ) results by a counterclockwise rotation of u 1
θ through an
angle of 180 ν^ , π around the Z -axis:
It will also be convenient to have an expression forπ dihed in addition to the one in
Eqn. (8) for the half angle. One such expression is obtained by squaring both sides
of Eqn. (8) and using the half-angle identity cos^2 ∋ ε/ 2( < (^) ∋ 1 ∗ cos ε(/ 2 twice to obtain
cos (^) ∋ π dihed( < cos πsin^2 S , cos^2 S. (10)
2.2 Cut planes
Compound-angle joints are commonly formed using either a miter joint or a butt joint. The appearance of these configurations as seen in the XY -plane is shown in Fig. 7.
Figure 7. Miter and butt joints viewed in the XY -plane
The two components of the compound-angle joint are cut to form these configura- tions. The complexity when cutting them arises because cutting involves planes that are not parallel to the XY -plane. The principal change is in the angle appearing to join the components. It is here that the dihedral angle becomes important.
Shown in Figs. 8 and 9 are the planes that a saw blade^4 must occupy to form the mating parts.
(^4) Here, we think of the saw blade as having a very narrow kerf. Otherwise, the cut plane contains only one side of the blade.
Figure 8. Cutting plane to form a miter joint
Figure 9. Cutting plane for a butt joint
The cutting plane for a miter joint divides in half both the angle π and the dihedral angleπ dihe of the compound-angle joint, whereas the cutting plane for a butt joint is oriented so as to contain the show face of component 2 of the compound-angle joint.
Figure 10. Blade and blade-tilt angles
is useful because this is the angle displayed on the angle scale that is built into saws made by many manufacturers. However, these scales are coarse, making precise settings difficult. Instead of using the saw’s angle scale when more precise settings are needed, it is convenient to set an auxiliary tool, such as a bevel gauge,
to the desired blade angle BA^ o using an accurate protractor. The bevel gauge is then placed on the table and the blade adjusted to match its angle. An alternative that can be even more precise is to print onto regular printer paper a right triangle
with one of the acute angles being BA^ o. The printer paper is then glued to a heavy card stock backing, which is cut to form a triangular template that is used to set the
blade to angle BA^ o.
A rotation of the combined component 1 and its cut plane around the Z-axis is also needed to bring the cut plane into the plane of the tilted saw blade. That angle is the required miter-gauge setting. This we identify in the following way. A unit vec- tor is constructed along the line where the two components meet. This unit vector is rotated clockwise through an angle S around the Y -axis along with component 1 and its attached cut plane. The resulting vector is then rotated around the Z -axis through an angle that yields a vector with an X component of zero. The angle re- quired to accomplish yields the miter-gauge angle.
The vector that results by forming the cross product between u 1
θ and u 2
θ ,
1 2 ∋^ ( 2
sin sin cos cos sin cos sin cos sin sin sin
dihed
S S u u S S S S n S
π π π π
≥ < ^ , , <
θ θ θ , (13)
lies in the cutting plane (for both miter and butt joints) along the mating line of components 1 and 2; n
θ is a unit vector that lies along the mating line. Clockwise rotation of this vector about the Y -axis through angle S yields
∋ ( (^1 )
2
2 3
2 2
cos 0 sin sin sin cos 0 1 0 cos sin cos sin cos sin 0 cos sin sin
sin sin cos sin sin cos sin cos sin cos sin sin cos sin sin cos
sin sin cos
w R Y S u u
S S S S
S S S S
S S S
S S S
S S S S
S S S S
S
π π π
π π π π π
π
< ^ ^ , ,
< ^ , ,
,^ ∗
θ θ θ
2 2
sin cos sin cos. sin sin cos sin sin cos
S S S S
S S S S
π π π
,^ ∗
The vector w^ θ^ is in the XY -plane aligned along the mating line of the two compo- nents, as shown in Figure 11 for component 1.
Figure 11. Component 1 positioned flat on the XY-plane
We now use the half-angle formulas sin π < 2 sin ∋ π/ 2 cos( ∋ π/ 2(and
1 ∗ cos π< 2 cos^2 ∋ π/ 2( to obtain
tan cos z tan / 2 ε S π
The miter-gauge angle^5 , shown as MG o in Fig. 12, is given in terms ofε Z by
MG^ o < 90 o^ ∗ εZ, which can be less or greater than 90° depending on the sign ofε Z.
Thus, from Eqn. (17)
tan tan 90 1 tan^ ∋^ / 2(
tan cos
o o Z Z
MG
S
π ε ε
Care is needed with the inverse tangent-function when determining the miter- gauge angle with this expression. The version of the inverse tangent-function to be used is:
MG^ o < arctan 2 tan∋ ∋ π / 2 ,cos( S(, (19)
where
arctan / , 0 arctan / 180 , 0, 0 arctan2 , arctan^ /^ 180 ,^ 0,^0 90 , 0, 0 90 , 0, 0 undefined, 0, 0
o o o o
y x x y x y x y x y^ x y^ x y x y x y x
Expressions for the blade and miter setup angles for compound-angle joints are de- veloped in this section using plane trigonometry, vectors and rotation matrices. Eqn. (12) is the expression for the blade-tilt angle, and Eqn. (18) is for the miter- gauge angle. In each of these expressions, the angle π is the angle in the XY - plane between the two components forming the compound-angle joint, as shown in
(^5) Note here that the miter-gauge angle MG o is measured as an angle about the Z-axis, with
90 o^ corresponding to the miter gauge set perpendicular to the plane of the cutting blade. The scale on miter-gauge tools are often marked with 90 o^ corresponding to perpendicular to the cutting blade and, further, the scale is marked symmetrically either side of 90 o^. Care is therefore needed when converting MG o into scale readings.
Figure 6. Expressions for the setup angles are developed in another way in the next section.
4 Development of setup angles using spherical trigonometry
Bill Gottesman formulated the ideas for this alternative development by using spherical trigonometry. Bill routinely uses spherical trigonometry for his designs of novel sundials, and the motivation for pursuing this development originated from a conversation between the authors at the Annual Conference of the North American Sundial Society held in Indianapolis, IN, in August 2014. We offer both the devel- opment of the previous section based on plane trigonometry and the development of this section using spherical trigonometry because each conceptualization pro- vides its own distinct insights that may be helpful to others interested in compound- angle joinery.
Spherical trigonometry plays an important role in many applications, including nav- igation, astronomy, and surveying. We will indicate its use in compound-angle joinery. First some terminology. A circle is formed at the intersection of a plane with the surface of a sphere. Such a circle is called a great circle when the inter- secting plane passes through the center of the sphere; otherwise it is called a small circle. For example, the equator is a great circle on the earth’s surface (when the earth is approximated as a sphere), dividing the earth into its northern and south- ern hemispheres. Great circles partition the earth into zones of longitude, and small circles partition it into zones of latitude. Spherical trigonometry deals with polygonal shapes that occur on the surface of a sphere when multiple great circles intersect. As seen in Fig. 10, three great circles can intersect to form a spherical
Figure 13. Spherical triangle
cos cos cos sin sin cos
cos cos cos sin sin cos
cos cos cos sin sin cos.
a b c b c A
b a c a c B
c a b a b C
Six angles are associated with any spherical triangle. These fundamental equations permit the determination of all six by knowing any three of them. Manipulation of these fundamental equations yields:
cos cos cos sin sin cos
cos cos cos sin sin cos
cos cos cos sin sin cos.
A B C B C a
B C A C A b
C A B A B c
Daniel Wenger gives a derivation of the law of cosines by using the rotation matri- ces of Section 2.1. Many other relationships can be derived from the law of cosines by using trigonometric identities [8-11].
4.1 Placing a compound-angle joint in the geometry of spherical trigonome-
try
Figure 15. Initial geometry
Shown in Figure 15 are portions of planes containing two components that will form
a compound-angle joint once they are tilted through an angle S ν^. They are pres-
ently oriented 90 o^ to the XY -plane containing the equatorial great circle and making an angle π to one another. The center of the sphere is located at the point labeled O in the XY -plane where the two component planes intersect. A point labeled P lies 1 unit from O along the Z -axis. Now suppose the two components Figure 15 are tilted by an angle S measured from the XY -plane, as shown in Figure 16. The point P is split into two reference points, labeled P’ and P’’ , with each moving in
place on its respective component through 90 ν^ ,Sν^ degrees as the components are
tilted through S o degrees. The tilted components are then extended until they in-
tersect and form a dihedral angle equal toπ dihed.
Figure 16. Components tilted
Figure 17 shows component 1 resting on the surface of a table saw and resting
against a miter gauge. The gauge on the left is set at (^) MG o < 90 o, and the one on
the right is set at an angle MG o so that the line where the two components are to be joined is aligned with the cutting line of the saw blade.
2 2
cos / 2 sin , miter joint cos cos sin cos , butt joint
o
S BA S S
π
π
This is the same as Eqn. (11). To determine the arc P’ - Q , which we label p , we again use the cosine law, Eqn. (21), to obtain
cos P < , cos P ' cos Q ∗ sin P ' sin Q cosp.
Thus,
∋ ( ∋ ( ∋^ (^ ∋ ( ∋^ (
cos 90 / 2 cos 90 cos / 2 sin 90 sin / 2 cos sin / 2 cos.
dihed dihed dihed
p p
π π π π
ν ν ν (25)
The miter-gauge angle for a compound-angle joint is equal to the complement of the arc-angle p , so
sin / 2 cos 90 sin sin / 2
o o o dihed
MG MG
π π
This is an alternative expression for the miter-gauge angle from that in Eqn. (18). To see that the two expressions yield identical results requires some manipulations using trigonometric identities. Start by squaring Eqn. (26) and using Eqn. (23) to get
2 2 2 2 2 2 2 2 2
sin / 2 cos / 2 cos cos 1 sin 1 1 cos / 2 sin 1 cos / 2 sin
MG^ o MGo^ S S S
π π π π
Then,
2 2 2 2
sin (^) sin / 2 tan / 2 tan cos cos^ / 2 cos^ cos
o o o
MG
MG
MG S S
π π π
The square root then shows that Eqns. (26) and (18) are equivalent expressions for the miter-gauge angle for forming a compound-angle joint.
5 Summary of Results
Assume that the two components of a compound-angle joint meet at an angle π in the XY -plane. A special case arises when the components are part of a box whose shape in the XY-plane is a regular polygon (eg. square, hexagon) having N sides;
then π < 180 ν^ ∋ N , 2 (/N. The miter-gauge angle MG ν^ is referenced to be 90 ν^ when it
is perpendicular to the plane of the cutting blade, and the blade-tilt angle BT ν^ is referenced from the XY -plane (surface of the saw table). Assume that the compo-
nent parts slope S ν^ from the XY -plane. Then, the blade-tilt angle satisfies
2 2
cos / 2 sin , miter joint sin cos cos sin , butt joint.
o
S BT S S
π
π
The dihedral angleπ dihed of the compound-angle joint is equal to 2 * BAν^ for a miter
joint and BAν^ for a butt joint. The miter-gauge angle satisfies
tan tan^ ∋^ / 2(
cos
MG o S
π < , (28)
which is Eqn. (18). Alternatively, and equivalently, the miter-gauge angle also sat- isfies
sin / 2 sin sin / 2
o dihed
MG
π π
which is Eqn. (26). For regular polygonal boxes having N sides, π < 180 o^ ∋ N , 2 (/N,
these expressions become
2 2
cos 90 ( 2) / sin , miter joint sin cos cos 90 ( 2) / sin , butt joint
o o o
N N S
BT
S N N S
^
and
tan 90 ∋ 2 (/
tan cos
o MG o^ N^ N S
or, alternatively
sin 90 2 / sin sin / 2
o o dihed
N N
MG
π
where π dihed / 2< BA^ o for a miter joint in Eqn. (29).
