1982 | OriginalPaper | Chapter
Equations for Isometries
Author : George E. Martin
Published in: Transformation Geometry
Publisher: Springer New York
Included in: Professional Book Archive
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The equations for a general translation were incorporated in the definition of a translation. Equations for a reflection were determined in Theorem 4.2. We now turn to rotations. Equations for the rotation about the origin through a directed angle of Θº are considered first. Let ρ(0,),Θ =where l is the X-axis. Then one directed angle from l to m has directed measure Θ/2. From the definition of the trigonometric functions, we know (cos(Θ/2), sin(Θ/2)) is a point on m. So line m has equation (sin(Θ/2))X - (cos(Θ/2))Y+ 0 = 0. Hence σm has equations $$\begin{gathered} x' = x - \frac{{2\left( {\sin \left( {\Theta /2} \right)} \right)x - \left( {\cos \left( {\Theta /2} \right)} \right)y]}}{{{{\sin }^2}\left( {\Theta /2} \right) + {{\cos }^2}\left( {\Theta /2} \right)}} \hfill \\ = \left[ {1 - 2{{\sin }^2}\left( {\Theta /2} \right)} \right]x + \left[ {2\sin \left( {\Theta /2} \right)\cos \left( {\Theta /2} \right)} \right]y \hfill \\ = \left( {\cos \Theta } \right)x + \left( {\sin \Theta } \right)y, \hfill \\ y' = y + \frac{{2\left( {\cos \left( {\Theta /2} \right)} \right)\left[ {\left( {\sin \left( {\Theta /2} \right)x - \left( {\cos } \right)\left( {\Theta /2} \right)} \right)y} \right]}}{{{{\sin }^2}\left( {\Theta /2} \right) + {{\cos }^2}\left( {\Theta /2} \right)}} = \left( {\sin \Theta } \right)x - \left( {\cos \Theta } \right)y \hfill \\ \end{gathered} $$