Mirror-Constructions - Mirror Constructions and Isometry...

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Mirror Constructions and Isometry Type Identifications Translations Given only a figure F and its image figure F ' under the translation T PQ , appropriate mirror lines L 1 and L 2 can be constructed as follows: 1) Locate a point A in the figure F and its image A' in the image figure F ' . 2) Construct vector AA' . 3) Perform the construction of L 1 and L 2 using AA' rather than PQ . This suffices because PQ and AA' both serve as defining vectors for the same translation: T PQ = T AA' . In fact, any two lines L 1 and L 2 will suffice as long as: 1) they are parallel lines which are perpendicular to PQ , and 2) the distance between them is d = (1/2) (PQ), and 3) the direction from L 1 to L 2 is the same as the direction from P to Q . L 2 L 1 d Construct: 1) The midpoint M of PQ . 2) Line L 1 through P and PQ 3) Line L 2 through M and PQ Then, T PQ = R L 2 ° R L 1 . Given: Construction: Given the defining vector PQ , construct two parallel mirror lines L 1 and L 2 such that T PQ = R L 2 ° R L 1 . Note: The distance d between the two lines is d = (1/2) (PQ) . Every translation T PQ is a direct isometry which is the composition of two reflections with parallel mirror lines. M Q P Q P
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Mirror Constructions and Isometry Type Identifications Rotations α If θ is undetermined, set θ = APA' , viewed as a directed angle. L 2 L 1 n m Given only a figure F and its image figure F ' under the rotation R P , θ , the points A and A' can easily be located. However, we must first construct the center of rotation P . Construction: Given a figure
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Mirror-Constructions - Mirror Constructions and Isometry...

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