Bear with me. It's going to take a bit of prelude to get to the point of this

article. If you already understand all there is to know about angle notation,

skip to the end of the article.

The ancient Babylonians counted using a base sixty system. Unfortunately,

this system has survived to today in the numerics we use to count time and

angle. We write time as h:m:s where sixty seconds (s) = one minute (m) and

sixty minutes = one hour (h). Similarly, angles are written as d:m:s with the

same relationships (60 arcseconds = 1 arcminute, 60 arcminutes = 1 degree).

Mathematicians normally add the prefix 'arc' to distinguish the fact that

they're talking about angle and not time. (There really ought to be a special

circle in hell for anyone who uses the same term for two completely disparate

units or, like the American gallon, redefines an existing unit.)

For most practical applications it's much more convenient to express angles as

decimal numbers. This raises the problem of converting between the two

notations.

Going from d:m:s to decimal notation is straightforward. Consider converting

12:34:56 (12 degrees, 34 arcmin, 56 arcsec) to decimal degrees. We know that

34 arcmin is 34/60 of a degree. We also know that there are 60*60 = 3600

arcsec in a degree. So the 56 arcsec is 56/3600 degrees. Adding them, we

have:

12 deg + 34/60 deg + 56/3600 deg = 12.582221... degrees

or, in general form:

d:m:s = d + m/60 + s/3600 decimal degrees

If your print calls out 12:34:56 d:m:s and you need the tangent of that angle

you'll need to perform the above calculation to get the decimal degrees

to feed to the tangent function. (Better scientific calculators have this

conversion built-in but the less expensive ones often lack it.)

Converting from decimal to d:m:s isn't very difficult. Using 12.582221

decimal degrees as an example:

Extract the integer degrees:

12.582221 = 12 + 0.58222112 degrees

Multiply the remainder by 60 (arcmin/deg):

60 * 0.582221 = 34.93326

Extract the integer arcmins:

34.93326 = 34 + 0.9332634 arcminutes

Multiply the remainder by 60 (arcsec/arcmin):

60 * 0.93326 = 55.9956~56 arcseconds

Again, better scientific calculators have a single key to do this conversion.

However, if yours lacks it, no worry. You won't be doing it frequently and

the procedure above is straightforward.

Most scientific calculators can deal with angles in decimal degree notation,

radian notation and grad notation. So, the question arises:

What the hell are radians and why do we need them? Isn't d:m:s notation

confusing enough? Now you're telling me that we need two more ways of

expressing angles?

When doing mathematics, it's much more useful to express angles in a

notation such that the angle so expressed, when multiplied by the radius of a

circle, yields the length of the arc on the circle subtended by that angle.

Consider a 90 deg angle. It subtends one-quarter of the circumference of a

circle or an arc length of 2*pi*r/4 (2*pi*r = the circumference of a circle

whose radius is 'r'). We want this angle (we'll call it 'A') expressed in

radians to satisfy:

A (rad) * r = 2 * pi * r / 4

That is, the angle in this radian notation, multiplied by the radius of the

circle, equals the length of the arc on said circle subtended by this angle.

Cancelling the 'r's, we have:

A (rad) = pi/2 radians

Since we assumed that A=90 deg, we now have a relationship between degrees and

radians.

90 deg = pi/2 radians

or:

1 deg = pi/180 radians =~ 0.017453 rad

or:

1 radian = 180/pi degrees =~ 57.295831 deg

Which makes things pretty simple. If we have degrees and want radians,

multiply degrees by 180/pi. If we have radians and want degrees, multiply

radians by pi/180. Rather than trying to memorize that, simply remember that

a full rotation, 360 deg, equals 2*pi radians.

-------------------------------

For completeness, a brief note about grads.

The French are never happy with any measurement system they didn't personally

invent. They thought that 90 degrees was an awkward number for a right angle

so they 'metricized' it to be 100 grads. I don't remember the details but

their argument for this aberration revolved around the fact that slopes

expressed in percent (as we express the slope of hills in road-building) would

then convert directly to grads without the need to do any calculation.

Apparently the French are as intimidated by arithmetic as they are by warfare.

Don't worry about grads. They're only used by the French and a few other

equally mis-guided Europeans. In 30+ years of doing mathematics for a living,

I *never* had to convert any angles to grads. Should you ever need to do so,

the relationships are:

100 grads = 90 degrees = pi/2 radians

-------------------------------

Back to radians and the mathematicians. If you ask a mathematician to

calculate the sine of an angle, he'll write down something like this:

sin (x) = x - x^3 / 6 + x^5 / 120 - ...

This is the 'series expansion' for the sine of x and it's only a valid

equation if x is expressed in *radians*. By using enough terms in this series,

you can calculate the sine to whatever precision you desire. In fact, early

calculators used a series similar to this to calculate trig functions.

(Today, with cheaper memory, they use a table lookup scheme.)

Now, if we look at this equation, we can see that, if x is a small number

(i.e., a lot less than one), x^3/6 is a lot less than x and x^5/120 is a lot

less still. In other words:

sin (x) ~= x for x << 1

A numerical example will verify this. Using my calculator:

sin (5 deg) = 0.087155742

5 deg * (pi/180) rad/deg = 0.087266462

In other words, 5 deg expressed in radians is pretty close to the sine of 5

deg. Some other examples:

angle = 1 deg: sine = 0.017452, radians = 0.017453, error = 0.005077 %

angle = 2 deg: sine = 0.034899, radians = 0.034907, error = 0.020311 %

angle = 3 deg: sine = 0.052336, radians = 0.052360, error = 0.045707 %

angle = 4 deg: sine = 0.069756, radians = 0.069813, error = 0.081278 %

angle = 5 deg: sine = 0.087156, radians = 0.087266, error = 0.127037 %

angle = 6 deg: sine = 0.104528, radians = 0.104720, error = 0.183005 %

angle = 7 deg: sine = 0.121869, radians = 0.122173, error = 0.249205 %

angle = 8 deg: sine = 0.139173, radians = 0.139626, error = 0.325666 %

angle = 9 deg: sine = 0.156434, radians = 0.157080, error = 0.412420 %

angle = 10 deg: sine = 0.173648, radians = 0.174533, error = 0.509506 %

-----------------------

Another aside. When physicists write the differential equation for a swinging

pendulum, they assume small angular motions of the pendulum and use this

sin(x) ~= x approximation. This allows them to obtain a simple equation for

the period of the pendulum. The terms of the series they throw away account

for an error that clock builders call the 'circular error' which will cause

timekeeping errors if the pendulum is allowed to swing through too wide an

arc. Ask yourself: Have you ever seen a pendulum clock where the pendulum

swings through a wide arc?

-----------------------

So, what good is all this to us as metalworkers? I recently bought a cheap

($10) laser level at Harbor Freight. The vials on it are adequate for

carpentry work but hardly anything one might use to align machinery.

Nevertheless, this 'instrument' consists of a flat base to which is attached a

laser capable of projecting a tiny dot or line over a considerable distance.

The light always travels in a straight line (Einstein will argue with that but

that's not anything we need to worry about here). In other words, this thing

is a very accurate, very long sine bar even if you never bother to look at the

bubbles in the vials.

I laid this 15" long level across the top of my 12" long surface plate such

that the projected laser line hit a vertical 6" scale on a bench 15' away on

the other side of the garage. I noted the reading on the scale. Then I

placed a 0.004" feeler gage under one end of the level. The reading on the

scale changed by 1/16".

Now for some arithmetic. Putting the feeler gage under the level turned it

into a sine bar. My 'stack height' is 0.004" and the length of the sine bar

is 12". What angle does that correspond to? We have, from trig:

sin x = 0.004/12

But the angle 'x' is tiny so sin x ~= x. So:

x ~= 0.004/12 = 0.000333 radians = 333 microradians (urad)

Does this make sense? Well, the end of the laser beam is swinging around

a circle with a radius of 15' = 180". Our angle is already in radians so the

arc length at that radius is:

333e-6 rad * 180" = 0.06"

which is pretty close to the 1/16" I measured with my bifocals while fighting

the laser speckle. (Ever wonder what causes that speckle effect in laser

light?)

Now, think about this for a moment. I detected (not measured) an angular

change of 333 urad with a $10 Chicom tool and a ruler! The Starrett

Machinist's Level is advertised to resolve (same as detect) 90 arcsec. An

arsec is about five urad (check that for yourself) so that's a resolution of

450 urad. I did better than that with a double sawbuck's worth of gear and

some math. The Starrett level costs $280. (The Starrett Master Precision

Level claims 10 arcsec (50 urad) resolution, but it costs $520).

The longer the path traveled by the beam, the more the deflection on the scale.

I put a first surface mirror where the scale was and moved the scale back on to

the surface plate. Now the light has to travel ~30' to reach the scale. Not

unsurprisingly, the 0.004" gage now moves the beam through about 1/8". So, by

bouncing the beam over a longer distance we can make detecting a small angle a

lot easier. Unfortunately, even with a quality first surface mirror, the laser

spot is smeared to a circle about 0.5" diameter by the time it reaches the

scale. This makes reading its location tricky. I picked out a particular

feature on the distorted spot and read its location on the scale. (I think I

have an idea about how to deal with this problem - see below.)

We all know that lathes don't really need to be level. Lathes produce

precision parts on ships that roll. (Lathes on submarines were sometimes

mounted with the bed vertical.) We're really using the level to detect any

twist in the lathe bed. What we're doing is trying to get one end ot the bed

to point the same direction in space as the other end. The laser level with a

long enough light path will do this with a resolution that approaches or

exceeds what you can do with an expensive level that you can't afford and will

use very seldom.

Some practical remarks here. The laser level packaging claims that it will

project its beam 1500'. Those must be 'Sears' feet. <g> Perhaps the

hyper-sensitive eye can detect the presence of the beam 1500' away but don't

count on doing any useful measurement at that distance. Projecting the beam

out of doors to get greater distance isn't going to work during daylight. The

red laser beam is washed out by sunlight. Better is to use mirrors to bounce

the beam around a darkened shop. You'll want to use first surface mirrors.

As the name suggests, these are mirrors where the reflective coating is on the

front of the mirror rather than the back, as most cheap mirrors have. If you

use a cheap mirror, the triple refraction (at the glass surface and at the

mirroring) will muddy the image and make it harder to read. I have some first

surface mirrors that I'll be happy to lend to anyone who wants to try this.

I found that, at 15', reading the beam location to better than 1/16" is

difficult. I have an idea, as yet untried, on how to deal with that. If I

build a simple circuit consisting of little more than a biased

photo-transistor powered by a 9v battery, I can use it as a light amplitude

detector. (The voltage across the biasing resistor will be proportional to

the light intensity falling on the photo-transistor.) If I attach this to a

height gage, I'll be able to measure the height of the maximum signal coming

from the laser light. This will take my bifocals and the laser speckle

problem out of the equation.

Now get out there and think of some other neat things one can do with a thirty

foot long, dead flat sine bar that'll fit in your toolbox.

Marv Klotz