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Three-directional box-splines: characterization and efficient evaluation Abstract: We propose a new characterization of three-directional box-splines, which are well adapted for interpolation and approximation on hexagonal lattices. Inspired by a construction already applied with success for exponential splines and hex-splines, we characterize a box-spline as a convolution of a generating function, which is a Green function of the spline's associated differential operator, and a discrete filter that plays the role of a localization operator.
This process leads to an elegant analytical expression of three-directional box-splines.
I used the rip fence to determine the length of each side in this case, all of the sides were the same length. To keep the workpiece from binding, which could be dangerous, I set up each cut with a block of scrap, then removed it while holding the piece firmly against the miter gauge.
My box sides were in effect the offcut. This made it even more critical to keep the workpiece firmly held against the miter gauge fence.
The block of mahogany is a temporary spacer to ensure consistent lengths for the sides in addition to a safe cut: After setting the workpiece in place against the miter gauge I slid it up to the block, held the workpiece in place with my left hand, then removed the block with my right. I turned the saw on with my right. Hold the workpiece firmly against the miter gauge fence while pushing it forward steadily to make the cut.
Full disclosure: The shots of mitering were set up after the fact.
An Easy Mitered Box Build with Splines | Popular Woodworking Magazine
The blade I used for the boxes was a Forrest Chopmaster, which made extremely clean mitered cuts. This step is not illustrated here. I presanded the interior faces of the box sides, then glued them up. There are different ways of clamping miters.
C4d spline thickness
I used lightweight clamps, applying them gently at first and gradually tightening alternating clamps until the miters were fully and evenly pulled up. Tip: I used Titebond Extend for a little extra open time. The next morning I removed the clamps and carefully gave the boxes a light sanding to clean up the corners.
To cut the grooves for splines I made a quick cradle that held the boxes at 45 degrees. I could easily push the cradle forward, holding the box in place while pushing the cradle forward, keeping it firmly against the table saw fence.
NOTE: If you make this kind of cradle and use screws or other metal fasteners, make sure that the fence is set so that the saw blade will not cut into them. Because all of my table saw blades leave a V-shaped point at the center of the kerf, I had to trim the ends of the notches.