## A Parallel Universe…

**Centering Photographs on a Matt**

A common task in photography or graphics is to center and attach a photograph or document to a matt or tag-board background. The technique often used to accomplish this is to place the photograph on the matt and align it with one of the corners (i.e. as in the image below, placed in the lower left-hand corner).

In this example, measurements are made and marks are placed half-way between the picture and the opposite edges of the matt (in this case, the right and top edges of the matt). The problem often encountered in this process is that the distance from the edge of the picture to the opposite edge of the matt is seldom a distance that is conveniently divided in two when finding the half-way points.

**A slick trick…**

Photographers and graphic artists have developed a simple procedure to overcome this problem by placing the ruler at some angle other than 90 degrees to the edges of the picture and matt. The angle of the ruler is adjusted so that the ruler crosses the edge of the picture on a whole-inch mark and at the same time crosses the edge of the matt on a different whole-inch mark – preferably with an even number of inches between those intersections. The mid-point or halfway point is now more easily calculated and marked. This is done twice along one side of the picture (right side in our example) and once along the remaining side (top in this case). Three small marks are placed to establish the center locations and the picture is aligned with them. In practice, these marks are made to be barely noticeable so that they do not show when the picture edge is aligned with them. Trying this technique a few times, you will find it a very quick and accurate solution to centering a picture.

**The mathematics behind this technique: Applied geometry…**

Mathematically, the task is to center a smaller rectangle inside a larger rectangle.

Establishing two marks on the right side of the picture:

In this picture centering exercise, once the picture is aligned with a corner of the matt, the opposite sides of the picture and the opposite sides of the matt form parallel lines (technically, parallel line segments: lines by definition extend infinitely in both directions, and since the edge of the picture has a fixed length with two end points, by definition it becomes a “line segment”). The edge of the ruler represents a “transversal” (a third line) intersecting the two parallel line segments. Calculating the midpoint of the transversal is referred to mathematically as finding the transversal bi-sector. Furthermore, one of the well-known geometric postulates is the fact that the bi-sector of a transversal is also the mid point between the two parallel lines.

This task requires establishing two transversals (AB and CD) and two bisectors (M1 and M2) between the parallel line segments of the right sides of the picture (AC) and matt (BD). In geometry, two points are all that are required to establish a line, and moving the edge of the picture to align with the two bi-sector points (points M1 and M2 in the figure above) essentially makes the edge of the picture a line segment passing through the two points. Additionally, since the two points M1 and M2 are equidistant from the original parallel line segments, the edge of the picture is now parallel to the outside edge of the matt.

Finally, sliding the picture up to the third mark: Why is only one mark needed on the top?

The top of the matt and the top of the picture are now also parallel to each other. This can be proven by the fact that the top of the matt and the top of the picture are “perpendicular” or at right angles (90 degrees – by definition of a rectangle) to the two original parallel line segments. And lines that are perpendicular to parallel lines will in fact be parallel to each other. Since the top of the picture and the top of the matt are already parallel to each other (equidistant from each other), it is only necessary to slide the picture up to a single mark (M3) that will leave an equal distance between the top and bottom edges of the picture and matt.

One last note of practicality:

It is always wise to minimize the impact of errors that might occur when measuring. By selecting whole inch marks on the ruler to serve as the intersection points for the transversals, this technique reduces the frequency of division mistakes by allowing the photographer/graphic artist to select numbers (preferably even whole numbers of inches) that are easily mentally divided in half. Even so, the process of transferring the measured location to a mark on the matt includes the potential for a small amount of error. Therefore the marks on the matt (M1 and M2) should be made as far apart as possible. That will minimize the degree to which the edges of the picture and matt will be out-of-parallel if a mark is slightly off of its correct position.