TABLE OF CONTENTS The geometers of antiquity took two different paths toward understanding the shapes of objects and the figures they drew to represent them. One path led them to study properties that they could measure, such as angles, distance, area, and volume. The other led them to study logical relationships among shapes, such as equivalence, similarity, and constructability. Both paths led them to the conclusion that certain geometric properties of an object do not change when the object is translated or rotated, expanded or twisted. Geometers discovered that different kinds of changes, transformations, left different properties unchanged, or invariant, and this led them to identify different kinds of geometry.
This chapter explores geometric transformations and their invariant properties. It introduces systems of equations that produce linear transformations and develops the algebra and geometry of translations, rotations, reflection, and scaling, including how to combine them. Vectors and matrices are put to good use here, in ways that clearly demonstrate their effectiveness.
Here we will discover what geometric properties are preserved when we impose transformations on figures. Each transformation changes some properties and leaves other unchanged; for example, angles may be preserved but not distance.
Congruent geometry applies only to figures of identical size and shape. We can translate and rotate a figure without changing its congruent properties. Conformal geometry applies to similar figures, those whose size is different but whose corresponding angles are equal. Euclidean geometry studies the properties of both congruent and...
