Tech Tip

Find the Center of a Circle Using Only a Pencil and Straight Edge

Posted by Justin A. Parr on February 18, 2021
Posted in: Math and Science, Tech Tip. 1 comment

Find the Center of a Circle Using Only a Pencil and Straight Edge

I bought a wood lathe recently, and the first time I tried to center some wood, I went through the painful process of remembering how to find the center of a circle.  This, despite having had three years of mechanical drawing classes.

Starting with a pencil and straight edge:

  1. Mark a line on the straight edge that’s a little more than 3/4 of the diameter (height) of the circle.  It doesn’t have to be exact.
  2. Make a mark (M) at the top of the circle.  The location of the mark need not be exact.
  3. Align the straight edge so that the corner is on mark M, and the line on the straight edge touches the right edge of the circle.  Draw a line (A) from somewhere above the center of the circle out to it’s edge.
  4. Repeat on the left side, drawing line B.
  5. From where line B ends, line up the straight edge so that its corner sits on the end of line B, and its line touches the lower-right-hand edge of the circle.  Draw line C along the straight edge, crossing the vertical center line.
  6. Repeat from the end of line A to the lower-left-hand edge of the circle, and draw line D
  7. At this point, lines C and D should form an “X” near the bottom of the circle.
  8. Draw a vertical center line (E) by aligning the straight edge with mark M and the intersection of lines C and D.  Start the line above the center, and go all the way to the bottom edge.
  9. At this point, line E passes through the center vertically, but we need to find the horizontal center.
  10. From where line E touches the bottom of the circle, align the straight edge so that it crosses line A, and the line on the straight edge touches the upper-right-hand edge of the circle.  Draw line F, crossing line A.
  11. Repeat this on the left side, crossing line B, and drawing line G.
  12. At this point, there should be an “X” on the left formed by lines B and G, and one on the right formed by lines A and F.
  13. Align the straight edge across the intersections of the left (B,G) and right (A,F) intersections, and draw the horizontal center line H.

The center of the circle is at the intersection of the vertical (E) and horizontal (H) center lines.

 

How This Works

Let’s call the distance from the corner of the straight edge to its mark S.

Let’s call the points where the horizontal center line meets the circle’s edges H1 (left) and H2  (right), and the point where the vertical center line meets the bottom V.

If we start at M and walk clockwise around the circle, we would pass through the points as follows:  M (start) -> H2 -> V -> H1 –> M (back to start).

The circle’s radius, R, is 1/2 the distance of M -> V or H1 -> H2.  The distance from M -> H2 would be SQUARE_ROOT( R^2 + R^2), and the same is true for H2 -> V, V -> H1, and H1 -> M.

If we assume that the radius R is “1 unit” (the actual length is irrelevant), then SQUARE_ROOT(1^2 + 1^2) = SQUARE_ROOT(2), or about 1.41.

If the radius R = “1 unit” then the diameter = 2 * R = “2 units”.

By selecting S such that S is 3/4 of the diameter, 3/4 * 2 = 1.5.  Since S>1.41, and we make two consecutive S-length marks around the circle, we know for a fact that 2*S > (M -> H2 -> V).

By doing this twice, once clockwise, and once counterclockwise, we get lines C and D.  We can call its intersection point T.

It’s easy to prove that this forms two congruent triangles:  (M->A-> T) and (M->B->T).  If the line M -> T is the base of both, then because they are congruent, the height of both are the same, which means that M -> T perfectly bisects (A -> B).

Since this line (M -> T) also bisects the equilateral triangle M -> A -> B which is inscribed within the circle, that line must pass through the center of the circle as well.

Carrying M ->T out to the edge of the circle gives us point V.

By drawing a line from V of length S to the edge, we get line F, which crosses line A.  We can call this intersection point U.

If we knew where the center of the circle (point O) was located, we could make the congruent triangle argument, that M -> U -> O is congruent to V -> U -> O, and just as with the vertical center line bisecting M->A->B, U -> O would perfectly bisect the larger equilateral triangle M -> U -> V.

On the other side, we draw line G of length S, which crosses line B at point W.  Same congruency argument:  M -> W -> O is congruent to V -> W -> O, and W -> O bisects equilateral triangle M -> W -> V.

Because M -> U = V -> U we know that M -> U -> V is equilateral, and the same for M -> W -> V.  Further, because of the congruency argument, we know that M -> U = M –> W and V -> U = V -> W, and therefore (M -> U -> V) is congruent to (M -> W -> V).

If line U-> O bisects triangle M->U->V, then it must intersect M->V exactly at its center.  Same for W->O and triangle M->W->V.

Therefore, W->U passes through the center of both triangles at point O, where W->U intersects M->V.

And therefore, point O must be at the center of the circle, because V->O = M->O, and we know that M->V passes through the center of the circle.

Note that this trick ONLY works if the the mark on the ruler (distance S) is > 1.41 times the radius.

Reduce PDF Size of a Scanned Document

Posted by Justin A. Parr on March 21, 2020
Posted in: Tech Support, Tech Tip. Leave a Comment

Reduce PDF Size of a Scanned Document

I recently had to e-mail a multi-page document – e-mail is that awesome 1990’s technology for submitting documents when the recipient doesn’t have a secure file transfer server – only to find that the attachment size was too large.

I scanned in three separate multi-page documents and then used pdfSAM to combine them all in to a single, 9-page PDF document.

Are you with me so far?

The final file size was over 80 meg!

I didn’t think anything of it, until my e-mail server complained about the attachment size.

After a little bit of thinking…

  • I had used xsane, a GUI front-end for SANE.
  • Where Windows has TWAIN, Linux has SANE. 

    Connecting a scanner to a PC pre-TWAIN used to be a kludge, and TWAIN was the answer: a manufacturer-independent hardware interface and driver standard that provides a scanner user interface, and also allows third-party programs to access any TWAIN-compliant scanner via a standard Application Programming Interface (API).

    SANE works like TWAIN, but separated in to two pieces:  front-end and back-end.  The back-end drivers allow the OS to talk to the scanner hardware, and provides generic access to other applications (like the TWAIN API).  This allows any of several SANE front-end user interfaces to access any SANE back-end, even across a network.  Very cool stuff, and there is even a SANE shim for Windows that provides TWAIN services, while using SANE on the back-end.
  • As my print server is a Raspberry Pi (Linux) and my main PC is Linux, it made the most sense to set the Pi up as a SANE back-end server.
  • I scanned all three multi-page documents using xsane, as mentioned, which allows you to scan a multi-page document and then save it as a single file.
  • While poking around a bit, I noticed that the intermediate format for xsane’s multipage applet is TIFF.
  • Like most other computing standards from the 1980’s, TIFF just isn’t very good, especially compared to modern file formats that use much better compression algorithms. 

    Unfortunately, TIFF was developed as a standard file format for scanners, which means that any time you’re dealing with scanners (or, oh lord, fax machines), be prepared to deal with TIFF, despite the fact that it’s a completely obsolete standard.  I mean, while we’re at it, let’s just fire up the 1200 baud modem and a copy of Telix, and we can XMODEM some TIFF files, shall we??

So it became clear that xsane / sane was just saving a bunch of TIFF files, and then copying them straight in to the PDF.

It turns out that Linux has a built-in command to split a PDF apart:

pdftoppm file.pdf imagefilename -png

This will create an image file for each page of the PDF, stored in PNG format.  For example, if file.pdf is 5 pages, you will get:

imagefile-1.png
imagefile-2.png
...
imagefile-5.png

These can then be recombined using ImageMagick’s convert:

convert imagefile-*.png newfile.pdf

As predicted, the resulting file was around 20 meg – 25% of the original file size.

Internally, PDF probably stores the images as TIFF, but in either case, the round trip through the PNG format is what affected the compression.

PNG performs “filtering” or normalization of the image before it’s compressed, and thus, the resulting image is “more compressible”, and the artifact of this process is that going back to TIFF (or any other format) makes the older compression algorithms perform more efficiently with the normalized image data, resulting in a much smaller file size.

Moral of the story:

If you have a huge PDF (or any kind of source image), try converting to PNG and then back again, which might make the original compression more efficient.

Other options:

  • Subsample the document down to a lower resolution.  I scanned at 300 DPI, but my next option if that didn’t work was going to be to subsample the document down to 150 DPI, which would result in a 4x file size reduction.
  • Go from color to grayscale or black-and-white.  Going grayscale from color usually results in about a 4x reduction in file size, and black and white from color might result in readability issues, but can save 16x or greater from the file size.

Tech Tip: Navigating Tabs

Posted by Justin A. Parr on November 21, 2016
Posted in: Tech Support, Tech Tip. Leave a Comment

Tech Tip:  Navigating Tabs

Browsers and other applications use tabs as part of the user interface, allowing the user to multi-task within the application.

 

Here are some time-saving shortcut keys for navigating tabs within browsers and other applications.

Hotkey Action
CTRL+Tab Move to the next tab
CTRL+SHIFT+Tab Move to the previous tab
CTRL+F4 Close the current tab
CTRL+SHIFT+T (Firefox and Chrome) Open the previously-closed tab

 

Using hotkeys works especially well when you’re composing in one tab, while referencing material in a 2nd tab.  Hotkeys allow you to move back and forth without lifting your hands off the keyboard.

 

Tech Tip – Use USB Mouse and On-Screen Keyboard

Posted by Justin A. Parr on March 19, 2016
Posted in: Tech Tip. Leave a Comment

I had a unique situation occur while troubleshooting an XP machine – I had physical access to the machine, but not the keyboard and mouse.

I was able to use a USB mouse and On-Screen Keyboard (OSK) to navigate the user interface.

Here is how it works:

  1. Plug in the USB mouse.  After a couple of seconds, you should be able to move the mouse cursor.
  2. Click on “Start” (or whatever passes for Start, for your version of Windows)
    osk1
  3. Click on the computer name, to open an explorer window
    osk2
  4. Navigate to c:\windows
    osk3
  5. In the “windows” folder, navigate to “system32”
    osk4
  6. If you can’t find it, go to folder options, select “Show hidden files and folders” and make sure “hide protected system files” is NOT checked.
    osk5
  7. In “system32”, find a file called “osk” or “osk.exe”, and double-click to open it.
    osk6
  8. The OSK will be displayed:
    osk7

The OSK will remain in the foreground, and allows you to “type” using the mouse.

For example, click the Firefox Address bar, then click-type:

g – o – o – g – l – e – . – c – o – m – ent (enter)

You can manipulate most OS and application functions quickly and easily, even if the system’s keyboard and mouse are locked out for some reason.

 

Tech Tip – Unused Glue, No Cap

Posted by Justin A. Parr on January 3, 2016
Posted in: Tech Tip. Leave a Comment

Problem:  Unusued Glue, No Cap

You have unused glue, and no cap.

 

Solution:  Use a Wire Nut

A wire nut has a tiny metal spring that’s designed to create a secure mechanical connection between two or more wires, without requiring welding or soldering.  Usually, they are used in electrical applications to create a “splice” – a connection between a primary source and one or more secondary connections that end up wired in parallel.

Wire nuts come in various sizes, but the “Yellow” and “Red” wire nuts typically fit larger tubes, while the “Orange” and “Blue” wire nuts usually fit smaller glue tubes.  You can get a variety pack at your local hardware retailer for a couple of dollars.

When used on a tube of glue, the metal spiral forms a near-air-tight seal.

If you spent $5 on a tube of glue, why not protect that investment with a 30 cent wire nut??

 

Details

When this happens:

Glue_01

 

Try these:

Glue_02

 

On larger tubes, the red or yellow might fit.  Twist until tight:

Glue_03

 

Now, your glue is safe and sound, and won’t dry out.