The 4 isometries-in-color

The 4 isometries-in-color - on the circle C P PX such that...

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Given a vector AB : Given point X, X' = T AB (X) is the point so that XX' = AB, and vectors XX' and AB are "parallel" and point in the same direction. The Four Isometry Types I. Translations T AB Vectors CC' , DD' , EE' , and any XX' can be used as the translation vector to define the same translation isometry: T AB = T CC' = T DD' = T EE' = T XX' Inverse: ( T AB ) -1 = ( T BA ) The Translation through directed line segment (vector) AB is T AB . X' E ' D ' C ' A B C D E X
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Mirror Line m For any point X, X' = R m (X) is that point X' such that 1) If X is on line m , then X' = X , and 2) If X is not on line m , then the segment XX' has line m as its perpendicular bisector. Given a line m : The Four Isometry Types II. Reflections R m Inverse: ( R m ) -1 = R m The Reflection about Mirror line m is R m . Y' X' D ' E ' F ' G ' A ' B ' C ' C B A G'' = G F E D'' = D X Y
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θ θ θ Directed Angles WPW' , YPY' , ZPZ' , and any XPX' can be used as directed angle to define the same rotation isometry: R P , WPW' = R P , YPY' = R P , ZPZ' = R P , XPX' = R P , θ . Given point X, X' = R P , θ (X) is that point
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Unformatted text preview: on the circle C( P, PX ) such that ∠ XPX' ≅ ∠ ABC = θ . θ . ∠ WPW' ≅ ∠ XPX' ≅ ∠ YPY' ≅ ∠ ZPZ' ≅ ∠ ABC = θ The Four Isometry Types III. Rotations R P , θ Given directed angle θ = ∠ ABC and point P: Inverse: ( R P , θ )-1 = ( R P , - θ ) = ( R P , 360 °- θ ) The Rotation about point P through directed angle θ = ∠ ABC is R P , θ . Z ' Y ' W' X ' B C A X W Y Z P K ' Mirror line m C' = T AB ( Z ) Given a mirror line m and a vector AB which is parallel to m : The Four Isometry Types IV. Glide Reflections GL AB , m Z = R m ( C ) Inverse: ( GL AB , m )-1 = R m ° T BA Thus, for any point X, X' = GL AB , m (X) = T AB ( R m (X) ) = R m ( T AB (X) ) The Glide Reflection about mirror line m through vector AB (parallel to the mirror line) is GL AB , m , where GL AB , m = T AB ° R m . Also, GL AB , m = R m ° T AB . M J ' H ' G ' F ' E ' D ' C ' L' Z A L C D E F G H J K B...
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This note was uploaded on 11/02/2010 for the course MATH modern geo taught by Professor Shirley during the Spring '10 term at University of Texas.

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The 4 isometries-in-color - on the circle C P PX such that...

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