![]() Orientation - Clockwise orientation and counterclockwise orientation.įigure orientation is Invariant Under a Transformation if for all triples of noncollinear points A, B, C on the figure the orientation of triangle ABC is the same as that of the orientation of image triangle A'B'C', it is called direct transformation otherwise it is opposite transformation. Reflections in the lines parallel to the sides through the center Notice that symmetry is about the figure, not the individual points.Įxample 5.2 lists the symmetries of a rectangle Symmetry of a Figure - Given a figure S, an isometry T is a symmetry of the figure is T(S) = S.Ī symmetry is simply a case where a figure remains unchanged under a transformation. Noteįixed Point - A point is invariant if it the image of itself. ![]() Half Turn - A rotation through 180 degrees is a half turn. 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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