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.

CAD Required List:
-Copy/Array Functionality
-End Point and Line Intersection Snapping

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

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.

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.

Step 10: Additional Methods

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.