Plane transformations and isometries

Transcription

1 Plane transformations and isometries We have observed that Euclid developed the notion of congruence in the plane by moving one figure on top of the other so that corresponding parts coincided. This notion is made precise in the context of certain types of functions from the plane to itself. A transformation is a one-to-one correspondence f:p P of the plane P (if A,B P, then A B f (A ) f (B); to every A P we can find an A P so that A = f (A ) ). A transformation is linear if it takes collinear points to collinear points. Linear transformations are an important element of linear algebra. We often use coordinates for points in the plane: to represent a linear transformation, we write f (x, y) = ( x, y ) and find that a formula for f must have the form x = ax +by +u y = cx +dy +v for certain real numbers a, b, c, d, u, v. Theorem A linear transformation must take noncollinear points to noncollinear points. // Corollary The inverse of a linear transformation is a linear transformation. //

2 Corollary The composition (or product) of any two linear transformations is a linear transformation. // Corollary The identity map e( A ) = A is a linear transformation. // A plane transformation f:p P that preserves distances (if A,B P and A = f (A ), B = f (B), then AB = A B ) is called an isometry or rigid motion of the plane. Lemma An isometry preserves collinearity (hence is a linear transformation) as well as betweenness and angle measure. // Suppose l is a line in the plane; a transformation s l :P P with the property that if A = s l ( A), then the segment A A has l as perpendicular bisector, we say that s l is a reflection in the line l. A similar definition describes a reflection in a point P: such a transformation s P has the property that if A = s P ( A), then the segment A A has P as midpoint. Theorem [ABCD Property] Reflections, whether across a line or through a point, preserve Angle measure, Betweenness, Collinearity, and Distances. That is, reflections are isometries.

3 If the path from vertex A to vertex B to vertex C in ΔABC is said to have a positive orientation if proceeds counterclockwise around interior points of the triangle and a negative orientation if proceeds clockwise around the interior. We say that f is a direct transformation if it preserves the orientation of triangles, and is an opposite transformation is it reverses the orientation of triangles. Theorem A reflection, whether across a line or through a point, is an opposite transformation. // Theorem The product of an even number of opposite transformations is direct, and the product of an even number of opposite transformations is opposite. Further, the product of any number of direct transformations is direct. // Theorem The product of two line reflections s l and s m, where l m, is an isometry that sends any line n into a parallel line. Such a transformation is called a translation. // Corollary Suppose the translation f carries A to A and B to B. Then B B is congruent to and either collinear with or parallel to A A. Thus, f is completely determined by one point and its image (and we will denote it t A A ). //

4 Theorem The product of two line reflections s l and s m, where l meets m at a point C, is an isometry that fixes the point C and sends any circle centered at C to itself. Such a transformation is called a rotation and C is its center. // Corollary Suppose the rotation f carries A to A and B to B. Then BC B AC A (whose measure θ is the angle of the rotation f). That is, f is completely determined by its center and angle (so we will denote it r C,θ ). // A glide reflection is the product t A A s l of a reflection across a line l with a translation in a direction parallel to l ( A s A r l ). For instance, a glide reflection carries one footstep of a walking person into the next footstep (see Figure 5.25, p. 363). It is easy to see that glide reflections are opposite transformations. Theorem If an isometry has no fixed points, then it is a translation, if direct, or a glide reflection, if opposite. // Theorem If an isometry has exactly one fixed point, then it is a rotation. //

5 Theorem If an isometry has two fixed points P and Q, then it fixes the entire line P s Q r. If these are the only fixed points, then it is a reflection across this line. // Theorem If an isometry fixes three noncollinear points, then it fixes every point, that is, it is the identity map. // Corollary If two isometries, f and g, both map ΔABC to Δ A B C, then f = g. // Theorem Given any two congruent triangles ΔABC and ΔPQR, then there is a unique isometry that carries the first triangle onto the second. // Theorem Every plane isometry is the product of no more than three line reflections. //