|Manifesto||More and more and more we are delegating our work to our tools. |
The “oral tradition” is now to be found on the internet. So, this is my contribution
to that cloud choir.
|Accuracy vs. Precision||But, they're the same thing, right? Not exactly. . . . .|
|The 3-4-5 Right Triangle||A good quality steel tape is the most precise square; you can carry in your tool pouch; |
that doesn't require batteries (not included) or come with the warning
“Do not gaze into laser aperture with remaining eye.”
|Cumbersome Division||What do you do when you have to equally divide a line with an awkward (39-5/16”) or unknown length|
into a difficult number of parts; say 7?
|Geometry||Links to animations of the Straightedge and Compass Constructions used in these examples.|
|Ellipses||The Trammel Method of creating precisely accurate, repeatable (and concentric) ellipses on the cheap|
|Draw an Arc Without Access|
to its Center
|Hm-m-m-m. . . .How do you do that?|
|Other Constructions||The Stove Pipe Problem, Compound Rake, Determining the Counter Rake for Work That Lies |
Athwart the Plane of the Rake
Unlike English, where gag is mostly a verb; in the Theater gag is often a noun. A gag is an artifice by which an action may be made to happen onstage without apparent human intervention – as if by magic or by coincidence or for effect (eight times a week.)
In the shops that build the furnishings for plays, and such, it came to mean any particularly clever procedure or gizmo; or the explanation of how some feat was attained. “How was that telephone gagged?”
What follows are the gags I have personally used and found to be effective. For the most part they come down to us from the Ancient Greeks.
Some are part of the “oral tradition” of the Theatrical Carpentry shops where I learned my trade; and others are from the related crafts.
I did not invent or discovery any of these methods. They were shown to me by my colleagues and mentors that I might become a better craftsman. To pay that debt forward I have (sort-of) organized what I know, together with some simple illustrations, that others may partake of the benefits I obtained.
More and more and more we are delegating this work to our tools. They do a fantastic job of it; much faster than we. Tireless perfectionists; they do exactly what we tell them to; even if that turns out not to be what we meant.
I'm only partly Luddite. They bring other blessings. Everyone working simultaneously from a common document even in near real-time. Once you've drawn something (or somewhere) once; you always have it. Re-using it is but a few clicks instead of hours of transcription-error prone transcribing. These have a cumulative effect. The inverse of “To err is human; but to really F/U requires a computer.” is “With these gizmos; not only can we create Spectacle and Illusion, but, sometimes even magic.
The “oral tradition” is now to be found on the internet. So, this is my contribution to that cloud choir.
There are those who believe that whoever has the most stuff when he dies; wins. My experience has inclined me to believe that: “He who shares, what he has learned; the most; wins-before he dies. As do those who come after”
Accuracy vs. Precision
But, they're the same thing, right? Not exactly (Eli, is that your idea of a joke?. . . more or less.) You can only build as accurately as the tolerances of your measuring and marking tools will permit. For the overwhelming majority of tasks, careful use of the usual critters is quite sufficient. However, there inevitably arise situations where you need more precision than that.
For example, to encourage my work to come out square, my best practice was to cut all the parts that wanted the same dimension; from the same fence setting or saw stop - at the same time. A case where “the same” has greater precision than the most highly accurate repeated measurement. Much faster too.
Another example is, when making measurements to mark out cuts for interior parts, using scrap stock of the actual exterior parts rather than relying on measurement or calculation.
More pointed examples include:
When the size of the object is beyond the margin of error of your square
If your framing square is 1/8 deg out of true that's a 1/2” error at 20'
Where the calculation of a distance or a division is cumbersome or will produce an unmeasurable number
39-5/16” into seven equal parts
When you must lay out all, or part of a large regular polygon that isn't rectangular
Where you require an ellipse or elliptical arc.
When you must fit together objects too large to be worked on the available bench tools.
The 3,4,5 Right Triangle
You would think from all my pontificating about accuracy and precision; that I never use a steel tape.
Not so. A good quality steel tape is the most precise square; you can carry in your tool pouch; that doesn't require batteries (not included) or come with the warning “Do not gaze into laser aperture with remaining eye.”
The quickest and most accurate way to check if a rectangle is “square” (four “true” right angles) is to compare the diagonal measurements. If they agree exactly – “square.” Not exactly -”out of square”. A typical carpenter's steel tape is calibrated in 1/16” increments; meaning that you can discern this agreement to within 1/32” of an inch (exactly between the 1/16” hash-marks.) For a 2' x 3' rectangle each 1/32nd inch of disagreement represents a difference of .065 of one degree from 90. The longer the limbs; the longer the diagonal. The longer the diagonal; the smaller the divisions of the angle it can parse.
Pythagoras discovered that, of all the triangles which contain a right (90 deg.) angle; the ones where one limb was exactly three units and the other exactly four units had a diagonal of exactly five units; a nice easy to remember group. This fact coupled with the principle outlined above puts an extremely precise tool for laying out a right angle or perpendicular (in your pouch), particularly over distances greater than 5' where the tolerance of a regular square becomes iffy.
In my work 4d finish nails were always my most faithful assistants. One always begins with a straight line; for which a chalk-box (or chalk-line) is the most portable and precise. Centered on the thickness of this line, where I wished to erect my perpendicular, I would drive one of my assistants leaving the head a scant 3/8” proud. From there, along the line, I would measure three units and drive a 2nd nail; same centered, the same proud. From the first nail, using my tape, I would strike an arc with a radius of four units where my eye guesstimated the sought after point might be found. Then from the 2nd nail I would strike an arc of five units. If my eye was good, that day, the arcs weren't too large and most importantly they crossed one another. A chalk line from the first nail through that intersection is a very precise perpendicular. That done, very very important to send the assistants out for recycling; so they don't bite you.
You can also do this with two tapes but the “especial point” lies beneath the centers of the two tapes where it is hard to see and mark.
To strike the distances as arcs, lock the tape about 2” long of your measure; and, grasping the tape between your thumb (held on the mark) and your folded forefinger, (the tape case held against the palm with the other three fingers); place the pencil against your thumb at the mark and while pulling the tape taut against the assistant, let the tape guide your arc strike.
Now, I've been deliberately cagey using “units” rather than, say, feet. This is to drive home that point that any units which have this relationship to one another work just as well. 3', 4' & 5' may be the most simply stated; but 6', 8' & 10' is even better because of the longer diagonal. So does 1'-6”, 2'-0” and 2'-6”. However, this is only precise to about 1/10 of a degree. A 30' diagonal (18',24',30') by contrast is precise to within less than 1/100 of a degree.
All in all, not bad for a FatMax™
In my earlier post “Accuracy vs Precision” I mentioned an issue that crops up from time to time. What do you do when you have to equally divide a line with an awkward (39-5/16”) or unknown length into a difficult number of parts (7)? No problem in this modren age, right? Just whip out the old Construction Master™ tap a few keys and bingo there's your number 5-5/8”. In order not to have to calculate (in your head_ 10-10/8” = 11-2/8” = 10-1/4”. . . 15-15/8” = 17-7/8. . . . . . you set a compass/divider to exactly 5-5/8” and set about stepping off your seven divisions. Except when you get to the last one you discover you're almost 1/8” out. WTF?
Don't get me wrong. I LOVE my Construction Master™. Could not live without it. But, it is lying to you (for your own good.) In actuality, seven EQUAL divisions of 39-5/16” is precisely 5-81/132”. In order not to make you crazy with a number you can't measure; it rounds to the nearest 1/16”. The more divisions you have, the worse the problem gets; because the rounding error accumulates with each repetition.
There is a gag to beat this problem. You can use it to precisely mark out any number of equal divisions of your odd length line. You can also mark out unequal divisions as long as they're all multiples of the smallest one. Let me step you through it using the example right.
At one end of the line erect a perpendicular of a suitable length.
From the other end look for a diagonal that is easily divisible by the number of divisions you want; as it passes through the perpendicular.
Tick off the the chosen increments along the diagonal.
Perpendiculars, from the base line, which pass through these ticks gives you a precise division of the line into seven exactly equal parts.
I had thought I'd gotten this gag out of the Backtage Handbook; but The Backstage Handbook shows a different construction method for the equal division of a line; involving multiple parallel line constructions.
I've been tossing around terms like; erect a perpendicular, strike an arc; geometry. Unlike what you may have thought in Middle School; Geometry is your friend.
The most frequently used method for erecting a perpendicular is by reference to a precise copy of a right angle - Framing/Rafter, Speed, Combination, Try, Drywall, Miter, (the Sliding Bevel 1) and the Stanley 46-101 Center Square.
The 3,4,5 right triangle and comparing diagonals? Definite force multipliers. However, all (except the Try, Miter and Sliding Bevel) involve measurement. There is another family of techniques - the “straightedge and compass constructions.” Very useful; and often involve no measurement or calculation at all.
There's a nice compilation at http://www.mathopenref.com/worksheetlist.html with nifty animations and clear explanations.
These are the ones I used most often. Simplest first.
Bisect a line.
Perpendicular to a line from a point
Erecting a perpendicular from a line at a stated point
The above is the one we used; but I like this one below (just stumbled across) better
Quick and dirty method (4+ tangents - Euclid would not approve; but he might wink.)
When striking an arc you can feel the scribing end move away from and back towards you. If you know the distance from the baseline to the parallel; set your compass to that distance and strike 4-5 arcs from along the baseline; where you feel both the push and the pull. A straight line that kisses the extreme extents of all of those arcs (NOT THE POINTS WHERE THEY MAY, BY HAPPENSTANCE, INTERSECT) is parallel to the baseline at the stated distance.
Perpendicular gives you 90 degrees. You can also construct 60 and 30 degree angles
You can bisect an angle, very useful if you have to start from a given angle
You can add or subtract angles from one another to derive a whole assortment of useful angles.
Also useful to remember:
Finding the center of a circle or an arc.
The perpendicular bisector of a chord lies along the radius of that arc. The point at which two or more of such radius lines (radii) intersect is the center of that arc.
The center head on a combination square or the 46-101 Center Square both operate via the principle of the perpendicular bisector of a chord.
The points at which the two limbs of any right angle (rafter square e.g.) placed against an arc both intersect that arc, describes a diameter of that arc. The point at which two diameters intersect one another is the center of that arc.
All of the references give a method, of drawing an ellipse, using the two foci and some string. I have found this method suitable only for drawing jelly bean or kidney shapes. Keeping the tension and hence the length of the string constant is problematic.
This method, based on the principle of the trammel, is IMHO much preferable. It requires a little more prep than the kidney method; but the results are uniformly true, exactly and rapidly repeatable, and the apparatus (constructed from scrap) can be used to draw any number of exactly concentric ellipses. There are two parts; the ellipstick and the track block.
Since I was doing this fairly often; I kept a stock of ellipstick blanks on hand. 8”- 24” lengths of scrap, with a scant 1/4” hole centered on the stock, at 1/2” from one end, with the center lines chased around the stick; and the mid-lines down its length. Best from the drill press. You'll be holding the pencil 3/4” away from the mark and out of plumb affects precision.
You can do the same thing with a notch in the end that will let you position the center of the pencil line at the exact end of the ellipstick. The object is to make the transfer of the critical measurements as simple and reliable as possible.
Next, a little 'rithmatic (ambition, distraction, uglification and derision). An ellipse is defined by its greatest extents, its major and minor axes (1 axis, 2 axes). An ellipse also has two foci (1 focus, 2 foci) Foci are the result of a mystical union between ½ the major axis; as an arc, struck from either terminus of the minor axis. Luckily we don't have to foc with either of them.
All we need to know is “what is ½ the major axis” and “what is ½ the minor axis.” If you've got numbers that are easy to measure, cool. If the numbers are wild; use the bisector gag to obtain precise measurements.)
Transfer these to the ellipstick (each measured from the center of the pencil hole - why the hole is exactly 1/2" from the end.)
Drive a 4d finish nail straight into the ellipstick at each of those marks and clip off the heads leaving the pins just scant of 3/8” proud. Sweeten the ends with a kiss of a mill file.
Mark out the major axis on the work or layout. Bisect that distance with a perpendicular on which you locate the minor axis.
Next you need to construct the track block. This wants to begin as a true square piece of ¾ scrap, with no voids, that is twice the difference between ½ the major and ½ the minor axes +1/4”.
This done, lower the tablesaw blade so it only takes a 3/8”deep kerf and re-set the fence so the blade splits the two mid-lines of the square. You should end up with something that looks like this.
With your best try square mark the centers of the kerfs at the bottom of the block.
To set the block, line up the marks, under the kerf exits; with the major and minor axis layout lines. When all four are aligned; you're positioned perfectly.
Fasten the block in place on the layout. With one ellipstick "pin" in each kerf (and a pencil in the hole) strike the ellipse using the saw kerfs to guide the nails on the ellipstick.
This simple square block will work just fine where the ratio between the major and minor axes is 2:1 or less. (You can see it on the ellipstick or divide the lesser into the greater to get the exact ratio.)
As this ratio increases (the ellipse gets “flatter”) the corners of the square begin to intrude on your intended line. It is a simple matter to trim these to suit.
When the ratio climbs beyond 3:1, the difference becomes greater than ½ the minor axis; and more drastic action is required.
Trace the path of the curve on the track block using the ellipstick.
You then have two choices. You can trim to within 1/4” of the exit of the kerf from the block; leaving you a 5/8” gap (leap) at both ends of the minor axis (the flat side.)
This can be readily faired with some pins and a flexible strip.
Or, you can make the offending bits of track removable/replaceable.
Fit them back when needed and remove them when they're in the way.
You can also use the ellipstick to strike ¼ of an ellipse against a flat square; and by repetition obtain a full ellipse. There are situations where this is simpler.
Shorten the nail heads to just scant the thickness of the square (no scratches)
Lay the outside edges along one of the four right angles produced by the intersection of Maj and Min. The ellipstick with the nail-heads riding against the square will describe ¼ of an ellipse.
The orbits of concentric ellipses are parallel. The same block in the same position will guide any concentric ellipse.
With this device, as you increase the distance between the guide pins and the marking point you also change the ratio; making the ellipse less flat.
This technique can be scaled up very effectively using trammel points, or a custom built track and stick for very large radii.
You can also trammel a router or bayonet saw. Due to the forces involved it is best to add some sort of shoe to the pins as things get larger and electrified. There are a number of commercially made router jigs that work on this same principle.
I built this whiz-bang out of basswood as a kid's toy to sell on eBay. The shoes and track are loose sliding dovetails. It draws the largest ellipse that will fit on a piece of letter sized paper leaving a half inch margin. But mostly it's soothing fun to just fiddle with it.
Draw an Arc Without Access to its Center
Saw this in a print copy of “Fine Woodworking On Proven Shop Tips” (in the previous millennium); but have not been able to find it on their website. It was sent in by Thomas Baird of Woodland, CA; attributed only to a Danish carpenter of his acquaintance. I've snootied the language up a bit.
Method to draw a circular arc of known displacement and chord length; where there is no access to the center.
Place a nail at the beginning, the end, and the point of maximum displacement of the curve (defining the chord of the arc.)
Erect a parallel to the chord at the point of maximum displacement. Place a fourth nail on that line.
Fix together two straight pieces of rigid scrap so that one piece lies against the point of max. displacement (at one end) and on the end of the curve at the other. While the other lies on the parallel line; at the point of max. displ.
Remove the fourth nail.
A pencil/scribe held at the crook between the two pieces of scrap. . .
. . .will describe a uniform curve (circular arc segment) when the edges are guided against the nails.
Flip the sticks over to mark the other half of the arc.
If the displacement is greater than the radius of the arc; the technique draws a bubble (and may require the use of automatic self-lengthening sticks.)
The Stove Pipe Problem
What are the dimensions of the opening required to pass a cylindrical object through an inclined plane?
This is a trammel problem. The opening in the rake is elliptical in shape. The value for 1/2 the minor axis is already known, it is the same as the radius of the cylinder. See the drawing below for a method of determining the value for 1/2 the major axis.
1- Draw two intersecting lines, the angle, between which, is that of the rake. One line represents the slope of the rake, and the other represents the "horizontal."
2- From the intersection of the two lines draw an arc whose radius is that of the cylinder and which crosses the "horizontal.”
3- From the point where the arc crosses the "horizontal" erect a perpendicular that crosses the line representing the slope of the rake. The distance between the center of the arc and this point is the value of 1/2 the major axis.
A trammel constructed using these two values will describe the ellipse which must be cut in the rake to pass a cylinder of the stated radius. (see drawing below)
The same method may be use to create a gauge to mark out the cuts required to miter together two cylinders of the same radius (at any angle) where the diameter of the material is too great to allow it to be cut with available bench tools.
Determine the angle at which the two cylinders want to be joined. 1/2 of this angle is the miter line. Use this angle as the slope of the rake in the above. The blank from which the ellipse has been removed can be used to mark out the necessary cuts. Care must be taken that the thickness of the gauge does not compromise the marked cut.
Where the rake is the combination of two angles the actual slope of the rake is a transcendent trigonometric function of those two angles. I ran off with the theater to avoid having to deal with algebra; and Trig is Algy's bigger, meaner older brother.
This construction derives the correct angle (true dip perpendicular to the strike line) using only a compass and straightedge. No calculation required.
If the slope angles are given as Run/Rise this is very straightforward (but you will have to measure.) Mark the two unit rises and draw the two angles. If the slope angles are given in degrees or you are dealing with wild numbers (or to fit an as-built); begin with the angles, copied with a bevel gauge or generated from a protractor, and extend them through the perpendicular to obtain AC and BC. To obtain ½ AC or BC use: http://www.mathopenref.com/constbisectline.html or, if you have easy numbers you can measure.
1- On a line of suitable length select an origin (O).
2- At one unit of run, erect a perpendicular.
3- From the origin erect the first angle (AOC).
4- From the origin strike an arc through AO with the radius ½ of AC.
5- From this intersection erect a perpendicular to AO a suitable distance.
6- Repeat this process for the second angle (BOC) using ½ BC for the radius struck through BO.
7- From this intersection erect a perpendicular to BO to where it intersects the perpendicular from AO at D.
8- Strike an arc with the radius DO from C to establish point E.
9- the angle between EO and CO is the “true dip”.
(Using this angle in the previous construction for the “Angle of Rake” gives the correct ellipse for the compound rake. )
Determining the Counter Rake for Work That Lies Athwart the Plane of the Rake
1- Horizontal reference
2- Vertical reference
3- Rake (2" in 1 foot- in this example)
4- Parallel to rake line (at the width of flat)
5- Parallel to the vertical Reference (at the width of flat)
6- From the rake line at A, draw a line at the known angle athwart the rake; of sufficient length to cross 5 or 4 – This distance AB is the length of the counter rake.
7- Strike an arc of length AB from C (an arbitrary point convenient usually at 1-1/4 times the width of the flat away from A) to intersect the rake line parallel.
8- Draw the line CD from center of arc to where it intersects the parallel.
9- Erect perpendicular from center of arc to the parallel. The angle between the perpendicular and CD is the angle of the counter rake.
The angle between AB and a perpendicular to the rake line at A gives the edge bevel for the flat.
Round numbers are given for this example, however, no calculation is required. The entire procedure may be performed, and the resulting length and angles, may be obtained and transferred to the work using only a beam compass, a straightedge and a bevel gauge. The procedure may be performed at any convenient scale. Remember, however, that at 1/4" - 1'-0" scale a 1/16” pencil line is nearly 1" wide; which can affect the results.
Note: If the rake you are dealing is a compound rake use the preceding construction to obtain the “true dip” for your counter rake calculation.