Defining Data for Isometries

Notation and Defining Data for an Isometry

When you are asked in a problem what isometry T is, your answer should supply the defining data for the isometry T in order to spell out precisely what T is.

(a) Identity = I.	State that all points are fixed.
(b) Line Reflection = R_m (or R_m).	Specify the mirror line m.
(c) Point Reflection (also called halfturn) = R_A or A₁₈₀ or H_A (or R_A, etc.).	Specify the center point A.
(d) Translation = T_v or T_PQ (or T_v or T_PQ).	Specify the translation vector v or the points P and Q defining vector PQ.
(e) Rotation = A_t (or A_t).	Specify the center A and the angle measure t of the counterclockwise rotation.
(f) Glide Reflection = G_AB (or G_AB).	Specify the invariant line AB and the glide vector AB.

Summary of Question Examples

If you are asked "What kind of transformation?", then the answer is a kind or type, namely something from the list of identity, rotation (special case of point symmetry), translation, line reflection, glide reflection).

If you are asked "What transformation?", then the answer is a specific transformation. This means that you have to give the geometric defining data that tells what the transformation is exactly. The specific list of defining data is in the table above.