[Note: This entry was actually written in April 2013. I changed the posting date to keep my blog entries in chronological order]
By George Taniwaki
This blog post describes how to calculate the compound curves and build the templates used in the breakfast bar project described in a separate Sept 2012 blog post. The explanation is a bit geeky and detailed so I separated it from that blog post.
Geometry, algebra, and trigonometry, oh my!
The breakfast bar project includes two countertops, the raised glass countertop (and the matching soffit) and the large granite countertop. Both surfaces are designed as compound curves. The edge of countertop facing the kitchen is straight. This is the edge that will be over the drawers of the cabinets. The edge facing the dining room is a large radius arc. This is edge over the bar stools. A tight radius arc joins these two edges at the end of the peninsula. A perpendicular line joins the straight edge and the large radius arc at the base of the peninsula. A schematic of the shape is shown in red in Figure 1 below.
Figure 1. Schematic of compound curve
We know the length (L), width, and tight radius (r) we want for the countertop. But we don’t know the radius of the large circle (R), the angle (Θ) where the two circles meet, or the distance (Δ) from the point where the large radius intersects the straight side of the countertop to the point where the small circle meets the straight side.
To solve for these dimensions, start by noticing that the center of the large circle (radius R) and the center of the small circle (radius r) and the left edge of the countertop form a right triangle. Label these sides A, B, and C as shown in Figure 1.
The length of the sides of the triangle are:
A = L – r
B = R – (W – r)
C = R – r
Since this is a right triangle, we also have:
A^2 + B^2 = C^2
Substituting and solving for R gives:
R = ((L – r)^2 + (W – r)^2 – r^2) / (2*(W – 2r))
The angle in degrees at which the two circles are tangent is:
Θ = 180 / (π∗arcsin(A/C))
The distance from the lower tangent to the point where the large radius intersects the base is:
Δ = r ∗ tan(Θ) =r ∗ (A/B)
I developed a spreadsheet to handle all the calculations. If you want to design your own countertops, a spreadsheet with instructions is available for download from SkyDrive.
Design specs for the actual countertops
For the breakfast bar project, there will be two countertops. The upper countertop is made of glass and is about two feet wide and four feet long. The base countertop is granite and we want it to have about 12″ of usable space beyond the glass countertop. We also want to make sure the flat edge is long enough to cover the cabinets underneath. Finally, we want the overall length to be as short as possible to give enough space for a person to walk around the peninsula. Another reason for keeping the peninsula short is that someday in the future we may want to replace the refrigerator with a bigger one that may require at least 42″ clearance.
Ultimately, I came up with the following dimensions for the raised glass countertop and matching soffit:
L = 52″, W =19.75″, and r= 5.5″.
Using the spreadsheet gives, R = 133.4″, Θ = 21.3°, and Δ = 2.1″
The granite countertop will have the following dimensions:
L = 78″, W =45.5″, and r= 16.75″.
Using the spreadsheet gives, R = 179.1″, Θ = 22.2°, and Δ = 6.8″
A pencil sketch is shown in Figure 2 below.
Figure 2. Pencil sketch of the layout of upper and lower countertops
Make the template
Once the dimensions are calculated, you can make the full-size template. Start with a piece of 1/4″ Masonite or similar material large enough to trace the template onto. Cut it into a rectangle just slightly larger than the dimensions of the finished template. Make sure the sides form a right angle by measuring the diagonals.
Pick one corner to be the right angle of the template. Starting from this point, measure the length (L) and width and mark them. Locate the center of the small radius (r) curve and draw it out using a circle guide or beam compass (Fig 3). Locate the point where the radius of the large radius (R) curve will intersect the straight edge (Fig 4). Draw a radius line from this point, through the center of the small curve to the tangent point of the two curves.
Figures 3 and 4. Laying out the small radius curve
Find an old metal tape rule. Drill a small hole into the 2″ mark and drive a finish nail through it. Hammer the finish nail into a weighted board (Fig 5). Measure out the length of the large radius (R) remembering to add two inches and align the template against the tape rule (Fig 6). Holding a pencil tight against the tape rule, trace out the curve until it is tangent with the small circle (Fig 7). Make sure the tape rule passes through the radius line drawn earlier.
Figures 5, 6, and 7. Laying out the large radius curve
Use a router with a circle cutting jig to cut out the small arc to the tangent points (Fig 8). Use a scroll saw to cut out the large arc. Be careful to stay outside the line. Sand or file the curve until it is smooth (Fig 9). Repeat for the other template. If the templates are used on cabinets, lay them in position to ensure they fit (Fig 10).
Figures 8, 9, 10. Cutting out the template and fitting it to the cabinets
All drawings and photographs by George Taniwaki