# Subdividing Lines Without a Calculator

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## Introduction: Subdividing Lines Without a Calculator

When working on CAD and drafting projects there are often times when a line segment needs to be divided into N segments. If you are fortunate to own a commercial copy of one of the major programs on the market then there is usually a tool designed for this specific task. However, not all free programs have such capabilities, and when board drafting I don't know of any tool designed to handle this task barring the use of dividers, a calculator, and a scale (ruler).

This instructable seeks to provide a method by which a line segment may be easily divided into as many equal segments as required using little more than parallel lines. The technique relies on properties of similar triangles. The technique can also be adapted to subdivide into unequal parts, but requires an additional step.

This method works well in all CAD packages tested, though may feel cumbersome for board drafting, as it requires apx. 2N construction lines.

This is not a lesson in how to use CAD or how to do manual drafting, so there won't be specific information on how to accomplish each task. Rather the general idea will be conveyed.

-Copy/Array Functionality
-End Point and Line Intersection Snapping

Board Drafting Required List:
-Parallel Ruler
-Straight Edge
-Pencil
-Inking Pen
-Eraser
-Scale (Ruler, Optional needed for unequal subdivisions)
-Rubber Band, Elastic Material (Optional, Cheating Apx. Method)

**UPDATE**
See page 10 for other suggested techniques.

## Step 1: Locate Line to Subdivide

The original line for reference purposes will be Black. From this point forward each new step will add a new color line.

## Step 2: Add First Reference Line

Draw a line (Red) connecting to one end of the line to be divided. Mathematically, it can be in any orientation relative to the original line (Black) and any length longer than zero.

## Step 3: Copy Reference Line

For this example the number of subdivisions required was six. So six copies (Magenta) of the reference line (Red) were copied such that their relative spacings were identical.

Copy and/or Array functionality should be used for CAD
Parallel Rulers should be used for Board Drafting

The only requirements for the spacing are that the lines do not remain co-linear and the end of the last line must not be on the Black line.

**Note**
If you wish to create unequal spacings go to step 9.

## Step 4: Connect Opposing Corners of Reference Array

Draw a line (Yellow) connecting the intersection of the Red and the Black line, with the opposite corner of the reference line array indicated by the end of the last reference line (Magenta).

## Step 5: Draw Base Subdivision Line

Connect the free end of the Black line to the intersection at the end points of the Yellow and Magenta lines, creating a line (Blue).

## Step 6: Array the Subdividing Lines

From the intersection point of the Blue, Yellow, and Magenta lines, copy/array the Blue line to each intersection point of Magenta and Yellow. This creates the Blue lines shown below.

## Step 7: Mark Subdivisions

Now at each each intersection point between the Black and Cyan, you can place a node (Green).

## Step 8: Clean Up and Finished

Delete or Erase all the construction marks (Everything except Black and Green) and congratulate yourself on perfectly subdividing your target line.

## Step 9: Dealing With Unequal Spacing

If you need to have unequal spacing the method still works, the only change needs to occur at Step 3, instead giving the reference lines equal spacing, instead space them such that they are proportional to the desired final spacing. As a simple example, the images shown represent how to subdivide a line into segments of 2/3 and 1/3. this was done by drawing the first Magenta line 2 units from the Red and the second line 1 unit from the previous Magenta line. If the proportions needed to be a percentage of the total, then the separation could be the %/100 for the first line and 1 - %/100 for the second gap or just 1 from the initial reference.

Elastic Method:

If board drafting you can cheat by using an elastic material. Mark a start point on the material then measure out the reference lines and mark them on the elastic just like you would on the paper. When the elastic is then stretched, for the most part the proportions are maintained and they can be transferred directly to the line, though it won't be perfect do to inherent material flaws, it will be good enough most of the time.

Ruler Method:

snoyes recommended the following technique found in a woodworking magazine.

This method actually requires only half as many steps in the case of board drafting. Though, it requires a scale and a drafting square be used. However, for CAD the technique may require more work.

- Draw a line perpendicular to one end-point of your segment
- Place one end of your ruler at the other end-point.
- Rotate the ruler until the number crosses the perpendicular line. So if you want 5 segments, put the 5 (inch or cm or whatever) on the perpendicular line.
- Mark a point at each number along your ruler.
- Drop a perpendicular from each mark to your segment.

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## Questions

I don't know if it's cheating, or any easier, but another method:

- Draw a line perpendicular to one end-point of your segment
- Place one end of your ruler at the other end-point.
- Rotate the ruler until the number crosses the perpendicular line. So if you want 5 segments, put the 5 (inch or cm or whatever) on the perpendicular line.
- Mark a point at each number along your ruler.
- Drop a perpendicular from each mark to your segment.

2 replies

Good idea, your method is actually quite a bit easier for board drafting. Would you mind if I added a step describing it, with credit to you of course.

Not at all; I picked it up from some wood working magazine, like American Handyman or something.

Huh. Wish I'd known this method for my drafting courses (I'm an old hand at CAD programs, but hadn't done hand-drafting in a long time). Normally I just use parametric metering in CADs to divide lines/curves into segments (a thought - do you have any simple methods like this for arcs? I've seen some nice locus-based methods, but they're cumbersome to to use in Pro-E or similar packages), but for an initial to-scale sketch, this would work very well. Kudos.

2 replies

I am actually working on developing a method to accomplish the subdivision of arcs, right now. As of yet however, I have not yet discovered a simple method that which does not require a computational tool. If I figure out a method I will be sure to post an I'ble about it.

However, for QCAD (the only Package I currently have access to) there is a tool called Bisectors under the Lines Menu. If you know the center point of the Arc, you can draw lines from the center to the end points. Then using Bisector, it will prompt you for a length and a number, the number is N-1 as described in the instructable. The length can be any value >0, but if you select a number greater than the radius of the arc you can skip a step (the lines won't need to be extended. See the attached *.gif for an example with six subdivisions.

Hope this helps/that there is a similar tool in Pro-E, etc. If I had access to a parametric program, I would divide by setting all segments equal, then defining a total length for the group.

Yeah, that's pretty much what I have to do now - define how many nodes I want at any point, split the curve, and define all wraparound dimensions equal. It's...not the most fun...especially when the sheer number of notations overloads the program (my laptop does not enjoy running Pro-E anymore...desktop handles it okay though).

Huh I haven't seen QCAD for a few years now - didn't realize it was still kicking around out there. I'd played with one of the Windows builds, liked the interface a lot (simplicity is something most of these programs severely lack), but I do mostly 3D builds and needed Pro-E/Solidworks/AutoCAD for college courses.

Neat ideas. I especially liked the applet in motion in the Introduction. It did a lot to make you process clear, even without the text.

2 replies

Thanks, it isn't the most exciting Instructable for sure but I thought people might find it useful. Since, it was my first, I wanted to make sure I put out a quality piece and set the bar high for my future entries.

I am always surprised by what members of this web site find exciting. I now have 148 Instructables on the site. Some that I think are the best and most helpful or solve a real problem go virtually unnoticed. Some that I think are barely worth publishing get all sorts of acclamation. And, some Instructables languish unnoticed for months on end, but one day someone sees one of those and it gets linked in some obscure place that brings sudden heavy viewer traffic. Just do what you are passionate about and explain it to the best of your ability. It all sifts out in the end where it belongs. One thing to remember, also, is that many of the members here are young high school and junior high kids who are fascinated by slimy, gory things made from duct tape and old Altoids candy tins. Getting a lot of views from them is not necessarily a compliment.