# Translations

# Translations

In Euclidean geometry, a translation is the act of moving all points within a group in a constant distance and in a specified direction. It is one of the three transformations that move a figure in the plane without changing its size or shape. (The other two are rotations and reflections.) In a translation, the figure is moved in a single direction without turning it or flipping it over (Figure 1).

A translation can, of course, be combined with the two other rigid motions (as transformations that preserve a figure’s size and shape are called), and it can, in particular, be combined with another translation. The product of two translations is also a translation, as illustrated in Figure 2.

If a set of points is drawn on a coordinate plane, it is a simple matter to write equations that will connect a point (x,y) with its translated image (x^{1},y^{1}). If a point has been moved a units to the right or left and b units up or down, a will be added to its x-coordinate and b to its y-coordinate (Figure 3). (If a< 0, the motion will be to the left; and if b< 0, down.) Therefore,

*x* ^{1} = *x* + *a x* = *x* ^{1} – *a*

or

*y* ^{1} = *y* + *b y* = *y* ^{1} – *b*

In these equations, the axes are fixed and the points are moved. If one wishes, the points can be kept fixed and the axes moved. This is called a translation of axes. If the axes are moved so that the new origin is at the former point (a,b), then the new coordinates, (x^{1},y^{1}) of a point (x,y) will be (x - a, y - b).

If one has two translations:

*x* ^{1} = *x* + 3 *y* ^{1} = *y* – 6

*x* ^{11} = *x* ’– 2 *y* ^{11} = *y* ^{1} + 1

they can be combined into a single transformation. By substitution, one has

*x* ^{11} = *x* + 1 *y* ^{11} = *y* – 5,

which is another translation. This arrangement illustrates that the product of two translations is itself a translation, as claimed earlier.

## If one has two translations

The idea of a translation is a very common one in the practical world. Many machines are translational

in their operation. A machinist that cranks the cutting-tool holder up and down the bed of the lathe, is translating it. The piston of an automobile engine is translated up and down in its cylinder. The chain of a bicycle is translated from one sprocket wheel to another as the cyclist pedals, and so on.

The bicycle chain is not only translated, it works because it has translational symmetry. After a translation of one link, it looks exactly as it did before. Because of this symmetry, it continues to fit over the teeth of the sprocket wheel (which itself has rotational symmetry) and to turn it.

One important use of translations is to simplify an equation that represents a set of points. The equation xy - 2x + 3y -13 = 0 can be written in factored form (x + 3)(y - 2) = 7. Then, letting x^{1} =x+3andy^{1} =y - 2, the equation is simply x^{1} y^{1} = 7, which is a much simpler and more easily recognized form.

Such transformations are useful in drawing graphs where many points have to be plotted. The graph of x^{1} y^{1} = 7 is a hyperbola whose branches lie entirely in the first and third quadrants with the axes as asymptotes. It is readily sketched. The graph of the original equation is also a hyperbola, but that fact may not be immediately apparent, and it will have points in all four quadrants. Many points may have to be plotted before the shape takes form.

If one has an equation of the form ax^{2} +by^{2} +cx+ dy + e = 0, it is always possible to find a translation that will simplify it to an equation of the form ax^{2} + by^{2} +E=0.

For example, the transformation x = x^{1} - 2 and y=y^{1} + 1 will transform x^{2} +3y^{2} +4x -6y-2=0 into: x^{2} + 3y^{2} - 9 = 0, which is recognizable as an ellipse with its center at the origin.

Transformations are particularly helpful in integrating functions such as integral (x + 5)^{4} dx because integral x^{4} dx is very easy to integrate, while the original is not. After the translated integral has been figured out, the result can be translated back, substituting x + 5 for x.

Translational symmetry is sometimes the result of the way in which things are made; it is sometimes the goal. Newspapers, coming off a web press, have translational symmetry because the press prints the same page over and over again. Picket fences have translational symmetry because they are made from pickets all cut in the same shape. Ornamental borders, however, have translational symmetry because such symmetry adds to their attractiveness. The gardener could as easily space the plants irregularly, or use random

### KEY TERMS

**Rigid motion** —A transformation of a plane figure that does not alter the size or shape of the figure.

varieties, as to make the border symmetric, but a symmetric border is often viewed as aesthetically pleasing.

*See also* Rotation.

## Resources

### BOOKS

Coxeter, H. S.M., and S. L. Greitzer. *Geometry Revisited. Washington,* DC: Mathematical Association of America, 1967.

Hilbert, David, and S. Cohn-Vossen. *Geometry and the Imagination.* Providence, RI: AMS Chelsea, 1999. Newman, James, ed. The World of Mathematics. New York: Simon and Schuster, 1956.

Noronha, Maria Helena. *Euclidean and Non-Euclidean Geometries.* Upper Saddle River, NJ: Prentice Hall, 2002.

Pettofrezzo, Anthony. *Matrices and Transformations.* New York: Dover, 1966.

Savchev, Svetoslav, and Titu Andreescu. *Mathematical Miniatures.* Washington, DC: Mathematical Association of America, 2003.

Slavin, Stephen L. *Geometry: A Self-teaching Guide.* Hoboken, NJ: John Wiley & Sons, 2005.

Yaglom, I. M. *Geometric Transformations.* Washington, DC: Mathematical Association of America, 1962.

J. Paul Moulton

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#### NEARBY TERMS

**Translations**