When tested analytically, a figure will show symmetry if its equation after the reflection is, except for the primes, the same as before. Or, contrarily, artists are admonished to avoid too much symmetry because too much can make a picture dull. A reflecting pool enhances the scene of which it is a part. A building whose facade has reflective symmetry has a pleasing "balance" about it. While recognizing the reflective symmetries of letters may not be of great importance, there are situations where reflection is useful. Only the plainest styles are truly symmetrical.) If there is an axis of symmetry, a mirror held upright along the axis will reveal it. (This symmetry is highly dependent on the typeface. The letters A, M, and W have a vertical axis of symmetry the letters B, C, and E, a horizontal axis and the letters H, I, and O, both. Letters, for example, are in some instances symmetrical with respect to a line and sometimes not. One such use is to test a figure for reflective symmetry, to test whether or not there is a line of reflection, called the "axis of symmetry" which transforms the figure into itself. The idea behind a reflection can be used in many ways. When the line of reflection is the line x = -y, these equations will effect the reflection: x' = -y and y' = -x. When the line of reflection is the x-axis, the y-coordinates will be equal, but the x-coordinates will be opposites: x' = -x and y' = y. The effect of the reflection was to change the major axis of the ellipse from horizontal to vertical. The image of (3,1) is (1,3), and the image of the ellipse x 2 + 4y 2 = 10 is 4x 2 + y 2 = 10 (after dropping theįigure 4. These can be used the same way as before. When the line of reflection is the line x = y, as in Figure 5, the equations for the reflection will be x = y', and y = x'. If a set of points is described by an equation such as 3x-2y = 5, then the equation of the image, -3x'-2y' = 5, can be found, again by substitution. If a point such as (4,7) is given, then its image, (-4,7), can be figuredįigure 3. As the figure shows, the y-coordinates stay the same, but the x-coordinates are opposites: x' = -x and y' = y. In Figure 4 the line of reflection is the y-axis. Figures 4 and 5 show two such reflections. 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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