«Abstract We give a systematic presentation of the mathematics behind the classic miter joint and variants, like the skew miter joint and the (skew) ...»
The Mathematics of Mitering and Its Artful Application
Tom Verhoeff Koos Verhoeff
Faculty of Mathematics and CS Valkenswaard, Netherlands
Eindhoven University of Technology
Den Dolech 2
5612 AZ Eindhoven, Netherlands
We give a systematic presentation of the mathematics behind the classic miter joint and variants, like the skew miter
joint and the (skew) fold joint. The latter is especially useful for connecting strips at an angle. We also address the problems that arise from constructing a closed 3D path from beams by using miter joints all the way round. We illustrate the possibilities with artwork making use of various miter joints.
1 Introduction The miter joint is well-known in the Arts, if only as a way of making ﬁne frames for pictures and paintings.
In its everyday application, a common problem with miter joints occurs when cutting a baseboard for walls meeting at an angle other than exactly 90 degrees. However, there is much more to the miter joint than meets the eye. In this paper, we will explore variations and related mathematical challenges, and show some artwork that this provoked.
In Section 2 we introduce the problem domain and its terminology. A systematic mathematical treatment is presented in Section 3. Section 4 shows some artwork based on various miter joints. We conclude the paper in Section 5 with some pointers to further work.
2 Problem Domain and Terminology We will now describe how we encountered new problems related to the miter joint. To avoid misunderstandings, we ﬁrst introduce some terminology.
Figure 1: Polygon knot with six edges (left) and thickened with circular cylinders (right)
2.1 Cylinders, single and double beveling, planar and spatial mitering Let K be a one-dimensional curve in space, having ﬁnite length. Such a curve is inﬁnitely thin and thus difﬁcult to realize faithfully in the physical world. To make the curve more tangible, one can search for ways to thicken it.
Let us assume that K is a ﬁnite chain of line segments. The knotted hexagon with sixfold symmetry depicted left in Figure 1 serves as an example. Hard to see, isn’t it?
A simple thickening is obtained by blowing up the line segments into circular cylinders with diameter d.
The original line segments are the center lines of these cylinders. Some care is needed where the line segments meet. Cutting the cylinders along the interior bisector plane of the angle where they meet, results in an elliptic cut face. The cylinders of t
Figure 2: Top view of cylinder being beveled (left) and resulting miter joint (right) Cutting off a cylinder at an angle is called beveling (see Figure 2). This is usually done in a miter box. The cylinder is clamped horizontally inside the box. The cutting blade moves vertically down, and can rotate (yaw) around the vertical axis to set the bevel angle α between blade and center line of the beam, or equivalently between the blade normal (perpendicular to the blade) and the plane perpendicular to the beam.
The complement of α is the angle α between blade normal and center line, or equivalently between blade and plane perpendicular to the beam. The joint angle after mitering equals 2α, that is, twice the bevel angle.
Originally, the term miter joint applied only to right-angled joints, with α = α = 45◦. The classic picture frame is an example.
The cylinder can be rotated in the miter box around its center line. Because of the full rotational symmetry of a cylinder, this rotation has no effect when beveling the ﬁrst end. When beveling the second end, however, the cylinder must be rotated over an appropriate angle β in order to follow the planned path. Referring to Figure 3 (left and middle), consider a line segment to be thickened, connecting to two other line segments − and +. Segments and − span a plane, and so do and +. The angle between these two planes is the desired rotation angle β. In the case of β = 0◦, the three segments −,, and + lie in one plane (are coplanar), and we call this planar mitering. For 0◦ β 180◦ we speak of spatial mitering.
In practice, the rotation of a circular cylinder is not so easy to carry out accurately, because it lacks a clear reference position for measuring angles due to its roundness.
Note that our distinction between planar and spatial mitering involves both ends of the cylinder. Some authors use the terms planar and spatial mitering differently, viz. when beveling a single end . This confusion can be explained as follows. Instead of rotating the cylinder (around its center line), one can equivalently rotate (roll) the blade away from its vertical position and adjust its yaw angle α appropriately.
This could be called double beveling, in contrast to single beveling, where the blade remains vertical and has only one degree of freedom as opposed to two. Whether double beveling of one end yields a spatial Figure 3: Cylinder with two beveled ends (left), spatial mitering with angle-spanning planes at β = 90◦ (middle), piece with square cross section beveled at both ends resulting in rectangular cut faces (right) structure (with non-coplanar segments) depends, however, also on what happens at the other end. Single and double beveling say something about the working method, but not about the planar or spatial effect of multiple joints.
2.2 Polygon as Cross Section The circular cross section used above for thickening can be replaced by any other shape. Common choices are simple polygons, like squares, rectangles, and even rhombuses, parallelograms, or various triangles. In mathematics, a cylinder is any set of parallel lines, not necessarily circular in cross section. We will also use the term beam to refer to a cylinder with polygonal cross section. These cross sections work almost the same way as circles. The main difference occurs at the joints. It is more pleasing if the edges of both beams connect properly across the joint, as is usually the case in a picture frame. We say that the beams match, to distinguish it from the situation where edges do not connect properly.
First consider a square cross section; that is, cutting the cylinder perpendicular to its center line (α = 90◦ ) yields a square cut face. When beveled at an angle 0◦ α 90◦, the resulting cut face is in general a parallelogram: opposite sides remain parallel, regardless of cutting angles α and β. In the special case where the beam lies ﬂush in the miter box, beveling with a vertical blade yields a rectangular cross section (see on the right in Figure 3).
Just like an ellipse, a parallelogram has two (rotational) symmetries. Hence, after beveling the square beam, the two pieces can be matched in two ways. In the ﬁrst way, the pieces remain together as before cutting, which is uninteresting (why cut at all?). The second way is obtained by rotating one beam over 180◦ around the perpendicular center line of the parallelogram. This way we obtain an angle between the beams, such that the common cut face bisects the angle and is perpendicular to the plane spanned by the two beams.
A practical advantage of using a polygon as cross section is that beveling is easier when the beam needs to be rotated around its center line over angle β 0◦. The edges on the beam can serve as a natural reference for such rotation. Ideally, the beam lies ﬂush in the miter box. In the case of a square beam, that leaves only two interesting rotation angles: β = 0◦ and β = 90◦. By the way, a disadvantage of polygons as cross section is that it is even harder to determine at what thickness d non-neighboring beams start to intersect.
The complicating factor is that now the rotational ‘phase’ of the beams matters, and not just the distance between their center lines.
2.3 Surprising Twist One might think that, by using matched miter joints, every spatial polygon K can be thickened to polygonal cylinders having the same cross section. For planar polygons K, like the picture frame, this is indeed the case. However, for nonplanar polygons K, like the hexagonal knot of Figure 1, it generally does not work out. The game starts to become mathematically interesting, because of a surprising twist.
We pick a line segment to start the thickening process. Blow up one half of it to, say, a square cylinder.
Cut it off at the next vertex along the interior bisector plane. The resulting cut face determines the matched thickening of the next line segment. Finding appropriate values of α and β and carrying out the cutting accordingly, is a matter of craftsmanship.
Continue along the entire polygon, and then see what happens after the last vertex, when the remaining half of the initial line segment is thickened. Figure 4 (left) shows that, in the case of our hexagonal knot, the edges fail to match. In general, the edges can be made to match everywhere, except possibly halfway the initial segment. There, a twist may occur. The mitering does not match properly.
Figure 4: Square cross section fails to match (left), equitriangular cross section does match (middle), and circular cross section with yellow seam revealing the 120◦ rotation (right) Of course, one can try and use a different polygonal cross section. It does not have to be a square.
Figure 4 (middle) uses an equilateral triangle as cross section, which does result in a completely matched mitering. Actually, the positions of the vertices of the hexagonal knot were designed to yield a matched mitering for this triangular cross section! Note that the resulting object has only two symmetries, whereas the knotted hexagon has six. In general, the thickened object inherits a subgroup of the symmetries of the original polygon.
Because of the way we have set up the thickening process, the two ends of the ‘cylindriﬁed’ polygonal space walk K meet in the middle of the initial segment. The polygonal cross sections at these two ends are congruent, but they need not coincide. One end may be rotated with respect to the other: the twist. The mitering matches if, and only if, the amount of rotation happens to be a symmetry of the polygonal cross section. Figure 4 (right) shows again the hexagonal knot thickened to circular cylinders. But this time a seam is drawn on the cylinders. This seam runs parallel to the center line, and matches nicely at all joints.
Again we see that it does not match all the way round. The amount of rotation is readily visible. In the case of this particular hexagonal knot, the amount of rotation was designed to equal 120◦.
We have the following Miter Joint Rotation Invariance Theorem: The total amount of rotation does not depend on
• the choice of initial segment to start the process;
• how much the cross section is initially rotated around the center line;
• the shape of the cross section.
Thus, if the mitering does not match, then rotating the cross sections will not improve the situation. And if all miters happen to match, then after rotating all cross sections ‘synchronously’, they still all match. This freedom can be exploited to obtain a conﬁguration which can be put down stably on the side of a beam.
Given a polygon K in space, it is a mathematical challenge to determine its amount of rotation, also referred to as its torsion. When designing objects, the inverse problem is, however, more relevant: How to locate the vertices in space, such that the resulting polygonal curve has the desired properties. These properties include such things as symmetries, knot class, and aesthetic appeal, but also a suitable torsion, which needs to be a symmetry of the beam’s cross section. The inverse problem is usually more difﬁcult than just determining the torsion. In most cases, it cannot be solved analytically and requires that one resorts to numerical approximation. The locations of the vertices offer many degrees of freedom, which also makes solving the inverse problems computationally costly.
2.4 Strip as Cross Section The polygonal cross section can degenerate into a line segment. In that case, the beam becomes inﬁnitely thin and turns into a (2-dimensional) strip. Besides via the regular miter joint, strips can be ‘joined’ at an angle in another way, viz. by folding. When folding a strip, the fold line lies in the exterior angle bisector plane of the extended strips (see Figure 5, left). To our surprise, the PostScript Language offers a miter joint to join two line segments, and also beveled and rounded joints, but not a fold joint.