Isometry Transformations in Math

1. Math 310 Sections 12.1-2 Isometry

2. Transformations Def A transformation is a map from the plane to itself that takes each point in the plane to exactly one other point. We call this a one-to-one correspondence or a bijection.

3. Isometry Def An isometry is a transformation that preserves distance. In terms of how we will use it, an isometry is a transformation that preserves congruence between figures.

4. Types of Isometries • Translation • Rotation • Reflection

5. Translation Def A translation is a motion of a plane that moves every point of the plane a specified distance in a specified direction along a straight line.

6. Properties of Translation • Its an isometry so it preserves congruence. • The image of a line, is a line parallel to it.

7. Translation Construction A translation can be constructed on paper with the use of a compass and a straight edge. To translate any object we simple translate one point at a time, or, for polygons, simply their vertices. This is done by constructing a parallelogram from the point you are translating with an arrow, indicating the translation direction and length.

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9. Translation in the Coordinate Plane A translation in the coordinate plane can be modeled using the following function: (x, y) → (x + a, y + b)

10. Rotation Def A rotation is an isometry of the plane determined by holding one point – the center – fixed and rotating the plane about this point by a certain amount in a certain direction.

11. Rotation Construction A rotation can be constructed with the use of a compass and straightedge if the center, angle and direction of the rotation are given. This is accomplished by copying the angle of the rotation onto the ray created by the center and the point and then marking of a congruent segment on the new ray, equal to the distance between the center and the point being rotated.

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13. Reflection Def A reflection in a line l is an isometry of a plane that pairs each point P of the plane with a point P’ in such a way that l is the perpendicular bisector of segment PP’, as long as P is not on l. If P is on l then P = P’.

14. Reflection Construction The key for constructing a reflection of a point is given in the definition. Since the reflection of a point P over some line l causes l to be the perpendicular bisector of a segment connecting P with its image, simply construct a perpendicular line to l through P and then mark off a congruent segment from the intersection of the two lines, equal to the distance from l to P.

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