The 4 isometries-in-color

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

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## 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.

### Page1 / 4

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online