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Appendix: Transformation

# The Basics

Transformation exists in everywhere of people’s daily lives. The mirror is the perfect tool for reflection, which is one kind of transformation.  There is not much to discuss about the transformation. So we have listed the main concepts concerning the transformation:

In a plane, a reflection about line l is a transformation which maps each point p into a point p1 such that (1) if p is on l, then p1 = p; and (2) if p is not on l, then l is the perpendicular bisector of pp1.

A reflection is an isometry.

A translation in a plane from A to A1 is a transformation which maps any point p into a point p1 such that pp1 @ aa1 and pp1 || aa1.

A translation is an isometry.

The resultant image determined by two successive reflections about parallel lines is a translation.

A rotation Rp,m of point A about point P through an angle of measure m is a transformation which maps A into its rotation image A1 such that PA = PA1 and mÐ APA1 = m.

A rotation is an isometry.

The measure of the angle of rotation formed by two successive reflections (or by the product of two reflections) is twice the measure of the non-obtuse angle between the two lines of symmetry.

The transformation defined by adding a constant to the coordinates of each point is a translation.

A dilation is a transformation, which produces an image similar to the original figure.

The transformation defined by multiplying each coordinate of each point by a constant is dilation.

Please go to next page to see some examples of transformations.

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