The finite groups generated in this way are examples of Coxeter groups. In general, a group generated by reflections in affine hyperplanes is known as a reflection group. Examples of Reflection Over the X Axis and Y Axis: Notice how the reflection rules for reflecting across the x axis and across the y axis are applied in each example. Similarly the Euclidean group, which consists of all isometries of Euclidean space, is generated by reflections in affine hyperplanes. Thus reflections generate the orthogonal group, and this result is known as the Cartan–Dieudonné theorem. Every rotation is the result of reflecting in an even number of reflections in hyperplanes through the origin, and every improper rotation is the result of reflecting in an odd number. The product of two such matrices is a special orthogonal matrix that represents a rotation. The matrix for a reflection is orthogonal with determinant −1 and eigenvalues −1, 1, 1. Properties A reflection across an axis followed by a reflection in a second axis not parallel to the first one results in a total motion that is a rotation around the point of intersection of the axes, by an angle twice the angle between the axes. Point Q is then the reflection of point P through line AB. P and Q will be the points of intersection of these two circles.
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