If one has two translations
A translation 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.
A translation can, of course, be combined with the two other rigid motions (as transformations which 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 which will connect a point (x,y) with its translated "image" (x1,y1). 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-cordinate and b to its y-coordinate. (If a < 0, the motion will be to the left; and if b <0, down.) Therefore
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, (x1, y1) of a point (x, y) will be (x - a, y - b).
If one has two translations:
they can be combined into a single transformation. By substitution one has
which is another translation. This illustrates that the "product" of two translations is itself a translation, as claimed earlier.
The idea of a translation is a very common one in the practical world. Many machines are translational in their operation. The machinist who 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 which 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 x1 = x + 3 and y1 = y - 2, the equation is simply x1y1 = 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 x1y1 = 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 ax2 + by2 + cx + dy + e = 0 it is always possible to find a translation which will simplify it to an equation of the form ax2 + by2 + E = 0.
For example, the transformation x = x1 - 2 and y = y1 + 1 will transform x2+ 3y2 + 4x - 6y - 2 = 0
into x2 + 3y2 - 9 = 0 which is recognizable as an ellipse with its center at the origin.
Transformations are particularly helpful in integrating functions such as intergral (x + 5)4 dx because intergral x4 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 varieties, as to make the border symmetric, but a symmetric border is often viewed as esthetically pleasing.
See also Rotation.
Coxeter, H.S.M., and S.L. Greitzer. Geometry Revisited. Washington, DC: The Mathematical Association of America, 1967.
Hilbert, D., and S. Cohn-Vossen. Geometry and the Imagination. New York: Chelsea, 1952.
Newman, James, ed. The World of Mathematics. New York: Simon and Schuster, 1956.
Pettofrezzo, Anthony. Matrices and Transformations. New York: Dover, 1966.
Yaglom, I.M. Geometric Transformations. Washington, DC: The Mathematical Association of America, 1962.
J. Paul Moulton