Tip: Using Trangulation For Proportional Accuracy

Triangulation is defined as a "method of determining the relative positions of points in space by measuring the distances, and sometimes angles, between those points and other reference points whose positions are known." (American Heritage Science Dictionary). This concept is used in navigation, GPS positioning, and even in cell phone technology (as in the illustration to the left where triangulation locates a particular phone).

In drawing, the term triangulation is used to describe several different techniques, but one of them can be used to capture accurate proportions when drawing from life. It can also be used to scale any image up or down without the need for grids or boxes. Because it is so versatile we have tried to describe it here as a tip for you to use.

This technique works because it is based on a principle of geometry that states "if two corresponding angles of two triangles have the same measure, the triangles are similar." In other words, if you draw two non-parallel lines that have the same angle with respect to your reference, then they will have the same proportions as your reference regardless of any differences in size.

Here is how it works:

1. Find three points on the reference image that are somewhat central.
   
   
   
   
   
2. Draw a line that is parallel to an imaginary line that connects two of the three points.
   
   
   
   
   
3. Then draw two points on the line to represent the two points from the reference. The distance between these two points will dictate the size of the final drawing. Make the distance between them larger than the original and you will be scaling up, make the distance smaller and you will be scaling down.
   
   
   
   
   
4. Through either of these points, draw another line that is parallel to an imaginary line that connects the point to the third point on the reference.
   
   
   
   
   
5. Through the other point draw a line that is parallel to an imaginary line that goes through it and the third point also used in the prior step. On your drawing, the third point will be located where the two new lines intersect.
   
   
   
   
   
6. Once you have established this first similar triangle, other points can be transferred from the reference image to the drawing using it and any two existing points on the drawing (repeating steps 4 and 5 described above).
   
   
   

Although I have made it sound as if a line can be drawn with a single stroke, it is more typical and more accurate to establish each line by making several light strokes, one over the other. Making multiple passes over each line will increase the likelihood that the composite of the lines is a better representation of the proportion in the reference.

If you have trouble seeing the angle between two points, you can imagine centering a clock face over one of the points. From the center of the imagined clock try to estimate where the hand on the clock would point to reach the second point. For example, it may point to 1:00 or 8:00 depending on the orientation. The time indicated on the clock shows the angle the line should take on your drawing.

You do not need to triangulate every point from your reference but enough to make the proportions realistic. Once the key proportions are accurately transferred, you can focus on completing the rest of the form or shape. In our example, you can see that the image has been scaled down slightly.