11/8/2023 0 Comments Defintion of glide reflection![]() Rotation - If O is any point in the plane and is a real number, then the rotation about O as a center through, denoted by, if a function from the plane to the plane that maps O onto itself and any other point P onto point P' such that OP = OP' and. That is, it has both length and direction. If P is on line AB, the P' is the point P' for which ABPP' is a degenerate parallelogram. If P is ont on line AB, then P' is the point in the plane for which ABPP' is a parallelogram. Translation - A translation is a transformation of the plane from A to B that assigns every point in the plane point P'. In some contexts, d(P,Q) denotes the distance between two points P and Q in the plane. Or if we use the notation that A' is the image of A and B' is the image of B, the transformation is an isometry if and only if AB = A'B'. An isometry is a transformation of the plane such that for every two points A and B the distance between A and B equals the distance between T(A) and T(B). Isometry - A transformation that preserves distance. Inverse of T- (Q) = P if and only if Q = T(P). We write ( T 2 º T 1) = T 2( T 1(P) for all P in the plane. Identity - I is the identify function for the plane if I(P) = P for all P in the planeĬomposition of transformations T 1 with T 2 - We write T 2 º T 1 meaning that first T 1 acts on ap point, and then T 2 acts on its image. P is the preimage of T(P).Ī transformation is a function whose range is the same as the domain. T(P) is the image of point P under the transformation. Transformation - a one-to-one onto mapping of the plane to the plane. So a reflection is a transformation of the plane. ![]() Therefore M l maps the plane onto the entire plane.Īlso, if A ≠ B then M l(A) ≠ M l(B). ![]() Because every point in the plane has a preimage, the range M l is the whole plane and the mapping is onto. (I will use M l or M k as substitute symbols.)įrom the definition, if we have a point B in the plane we can find its preimage A - that is, given a line l, M l(A) = B. Notation: will designate a reflection (think 'mirror') of the plane in line L. If P is on line l, then P is paired with itself. Mapping - a function that assigns elements of a domain to elements of a range.Ī reflection in a line l is a correspondence that pairs each point P in the plane and not on line l with a point P' such that l is the perpendicular bisector of PP'. Overview of Section 5.1 Reflections, Translations, and Rotations
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