By far the easiest lines to use for this purpose are the x-axis, the y-axis, the line x = y, and the line x = -y. Such equations will depend upon which line is used as the line of reflection. If a point is described by its coordinates on a Cartesian coordinate plane, then one can write equations which will connect a point (x, y) with its reflected image (x', y'). Reflections can also be accomplished algebraically. Some last adjustment in the slab's position is usually required.įigure 1. In moving the stone, however, one is limited to the lines of reflection that the edges of the stone provide. Someone who, instead of lifting a heavy slab of stone, moves it by turning it over and over uses this idea. Because a figure can be moved anywhere in the plane by a combination of a translation and a rotation and can be turned over, if necessary, by a reflection, the combination of four or five reflections will place a figure anywhere on the plane that one might wish. The angle of rotation will be twice the angle between the two lines and will be in a first-line to second-line direction. If the lines of reflection are not parallel, the effect will be to rotate the figure around the point where the two lines of reflection cross. If the lines of reflection are parallel, the effect is to slide the figure in a direction which is perpendicular to the two lines of reflection, and to leave the figure "right side up." This combined motion, which does not rotate the figure at all, is a "translation." The distance the figure is translated is twice the distance between the two lines of reflection and in the first-line to second-line direction. The position of the final image depends upon the position of the two lines of reflection and upon which reflection takes place first. One reflection can be followed by another.
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