. Plane and solid analytic geometry . quation of an equilateralhyperbola. 3. Rotate the parabola y^ = 2 mx through — 90° about theorigin. 4. Prove analytically that, if an arbitrary straight line berotated about the origin through 90°, the new line will be per-pendicular to the old one. 5. Prove that, if an arbitrary line be rotated about theorigin through the angle 0, the angle from this line to the newline will be B. 3. Transformations of Similitude. Let the plane be stretched,like an elastic membrane, uniformly in all directions away from the origin. This tranformationis evidently represent
. Plane and solid analytic geometry . quation of an equilateralhyperbola. 3. Rotate the parabola y^ = 2 mx through — 90° about theorigin. 4. Prove analytically that, if an arbitrary straight line berotated about the origin through 90°, the new line will be per-pendicular to the old one. 5. Prove that, if an arbitrary line be rotated about theorigin through the angle 0, the angle from this line to the newline will be B. 3. Transformations of Similitude. Let the plane be stretched,like an elastic membrane, uniformly in all directions away from the origin. This tranformationis evidently represented analyti-cally by the equations : where A; is a constant greaterthan unity. If k is positive, butless than unity, the transforma-tion represents a shrinking toward the origin. The stretchingsand shrinkings defined by (1) are known as transformations ofsimilitude. These transformations, like the translations and the rota-tions, preserve the shapes of all figures; but, unlike thosetransformations, they alter the sizes of Fig. 6 TRANSFORMATIONS OF THE PLANE 335 Example. The equilateral hyperbola _ 2^2 _ ^2 is carried by (1), if k is taken equal to -: x = -, 2/ = ^, or x=ax, y = ay\a a into the curve a2a;2 - a?y^ = a% or x^ — y^ = 1. Thus all equilateral hyperbolas are seen to be similar to oneanother, since each can be transformed by (1) into the particu-lar equilateral hyperbola a;2 — ?/2 = 1. Inverse of a Transformation. The transformation, (2) x y y = —, obtained by solving the formulas (1) for x, y, is called the in-verse of the transformation (1). In general, if a given trans-formation carries (x, y) into (x\ y), the transformation carry-ing (x, y) into (x, y) is known as the inverse of the giventransformation. Thus, the rotation (2), § 2, is the inverse ofthe rotation (1), § 2. It is clear that the effect of the inverse transformation, ifperformed after the given one, is to nullify the given (1), § 2, rotates all figures through the angl
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