Notation, Q Conventions and Defining Data for Isometries

Notation, Question Conventions, and 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.
See the examples below for examples of questions that do and do not expect an answer of this kind. (Examples 4 and 5 require defining data.)
The notation below will be used for isometries. You are urged to use it also in your answer.Subscripts will be used in the notation except when for emails, etc, when sometime a subscript such as Rm will be denoted by R_m, using an underscore to indicate a subscript.

(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.

Questions: What Transformation? vs. What kind of Transformation?

Example 1

Question: Suppose that S is a glide reflection and T is a rotation. What kind of transformation is the product ST?
Answer: ST is either a glide reflection or a line reflection. (Reason: ST is the product of 3 + 2 = 5 line reflections.)

Example 2

Question: Suppose that S is a translation and T is a rotation. What kind of transformation is the product ST?
Answer: ST is a rotation. (Reason: ST is the product of 2 + 2 = 4 line reflections. It can't be a translation because no direction stays the same.)
Notice that the answer "rotation or translation" would be pretty good, but not as good as the given answer, since translation is not actually possible.

Example 3

Question: Suppose that S is a rotation by 90 degrees and T is a rotation by 90 degrees. What kind of transformation is the product ST?
Answer: ST is a point symmetry (also called half-turn or rotation by 180 degrees).
. (Reason: ST is a rotation by angle = 90 + 90 = 180.)

Example 4

Question: Draw points A and B on your paper. Suppose that S is a rotation by 90 degrees with center A and T is a rotation by 90 degrees with center B. What transformation is the product ST?
Answer: ST is a rotation by 180 degrees with center C so that triangle ABC has angles 45-45-90 degrees, with angle ACB = + 90 degrees (counterclockwise).
Note: On a test or problem, you will likely construct C by the "kite method" and thus show exactly where it is.

Example 5

[Assume that you are given a figure with points A, B, and C already drawn.]
Question: Draw points A, B and C on your paper. Suppose that S is a rotation by 90 degrees with center A, T is a rotation by 90 degrees with center B and U is rotation by 180 degrees with center C. What transformation is the product STU?
Answer: STU is a translation T_CD, where to complete the answer you must construct or describe exactly the point D. (Or it could be T_PQ, where you construct both points P and Q.)
Note: Just saying that STU is a translation is only a fraction of the correct answer.

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. For a specific list of defining data, see below.