Lab 7 Composition of Line Reflections II

Answers to Some Questions

Double Reflections in Parallel Lines – Order is important

• In a new sketch, construct a line a = line OA and a parallel line b through B. Let R_A and R_B be reflections across these lines.
• Form a blob S and transform it by R_BR_A.  Draw point P and transform it to P'' = R_BR_A(P).  What kind of isometry is R_BR_A?
• This isometry is a translation. Specifically, it is a translation whose direction is perpendicular to the two lines and whose displacement is double the distance between the lines. The direction of the displacement is in the direction from a towards b. (In the language of vectors, if A' is the reflection of A in line b, the displacement vector of the translation is AA'.)
• Place line a to the left side of b, with the lines approximately vertical.  In this case, the point P'' should be to the RIGHT of P.  If not, you have probably found P'' = R_AR_B(P) instead of the other order.
• The notation FG means, as in calculus in linear algebra, the composition of the functions. So FG(P) = F(G(P)), which means G first, F second.
• Measure the distance from A to line b.  Measure the distance from P to P''.  How are they related?  How can you see this geometrically.
• The segment PP'' is perpendicular to the lines and the length is twice the distance between the lines. This is easy to see if P is positioned on the side of a opposite b, so that P', the reflection of P in a is between the lines. Then the distance from P to P' is 2 times the distance form P' to a, and the distance from P' to P'' is 2 times the distance from P' to b. But the sum of the distances from P' to a and to b is the distance from a to b, if P' is between the two lines. If P is located elsewhere, it is a bit more complicated because some distances will be subtracted instead of added.
• Now we will move a and b without changing the distance.  To do this, carefully make sure that everything is unselected.  Then select the two lines a and b (but not any points and not anything else).  Drag line of the lines.  They should both move without changing the distance.  In this case, does P' = reflection of P in a change?  Does P'' change?
• P' should move around a lot, but P and P'' should not change because the double reflection is still the same translation so long as the distance and direction of the lines is not changed.
• What isometry is R_AR_B and how is it related to R_BR_A?
• The isometry R_AR_B is also a translation, but with the opposite translation vector. In other words, the two isometries are inversed of each other.

Reflections in 3 Parallel Lines

• Add a line c to the figure, which is a line through a new point C and parallel to a and b.  Reflect P'' across OC.  You can do this with your blob also.
• What is the isometry R_CR_BR_A?
• This is a line reflection.
• For what line d is R_D = R_CR_BR_A?  How can you construct this line?
• One way is to find the image P''' of P and construct the perpendcular bisector of PP'''. Another way is given in the next answer.
• If you rewrite this equation as R_CR_D   = R_BR_A, how is it related to the double-reflection = translation fact from the first experiment?
• If d is the line parallel to a, b, and c so that the distance from c to d is the same as from a to b and finally that d is on the side of c so that the translation R_CR_D is in the same direction as  R_BR_A, then the translations are the same and R_CR_D  = R_BR_A. But multiply both sides on the left by RC to get R_CR_CR_D  = R_CR_BR_A. But R_CR_CR_D  = R_D   since R_CR_C = Identity. Thus d is the line given by the triple line reflection above..

Reflections in 2 Intersecting Lines – Order is important

• Draw lines OA and OB and a blob.  Let R_A and R_B be reflections across these lines.
• Measure angle AOB.  Drag B so that this angle is about +45 degrees.
• Form a blob S and transform it by R_BR_A.  Draw point P and transform it to P'' = R_BR_A(P).
• Measure POP''. This should be about 90 degrees, not –90.  If it is the latter you found P'' = R_AR_B(P).
• Read the discussion above about the meaning of FG and which comes first.
• What is the difference between R_BR_A and R_AR_B?  How are they related?
• They are inverses of each other as we can see by composing them. R_BR_A R_AR_B = R_BR_B = Identity. One is rotation by d degrees and one is rotation by -d degrees.
• What is angle AOB if R_AR_B is a point symmetry?
• If the composition if a point reflection (AKA point symmetry) then the rotation is 180 degrees, so the angle between the lines is 180/2 = 90 degrees. Notice the important point that in this case, the two transformations R_BR_A and R_AR_B are the same, since one is a rotation of +180 degrees and one is of -180 degrees.
• It is possible to change the Selection Arrow Tool to a Rotation Tool.  Double click on O to mark it the center.  Now select both A and B with the Rotation tool and move them.  The angle AOB should stay the same.  Does the reflection of P across OA stay the same?  How about P''.
• As for the parallel lines, if the lines meeting in O continue to make the same angle AOB, then the rotation which is the double reflection is the same. This means P and P'' stay the same (but the point P' will move).

Reflections in 3 Concurrent Lines

• Add a line OC to the figure.  Reflect P'' across OC.  You can do this with your blob also.
• What is the isometry R_CR_BR_A?
• A line reflection.
• For what line D is R_D = R_CR_BR_A?  How can you construct this line?
• As for the parallel lines, we can triple reflect P to get P''' and construct the perpendicular bisector, or use the method in the next answer.
• If you rewrite this equation as R_CR_D  = R_BR_A, how is it related to the double-reflection = rotation fact from the first experiment?
• The rewriting is the same as in the parallel case. What this means is that if the lines are OA, OB, OC, OD and angle AOB = angle DOC (where we measure angles with + or - as well as magnitude), then the rotations R_CR_D  = R_BR_A. This tells how to construct OD without triple reflecting anything.

Reflections in 3 General Lines

• Draw 3 lines A1A2, B1B2, C1C2.  Let the corresponding line Reflections be R_A, R_B, R_C.
• Do this experiment two ways.  Form a blob S and reflect S 3 times to get S''' = R_CR_BR_A(S).  Hide the intermediate steps and make a tool that will take S and the 3 lines to get S'''.  If we call T = R_CR_BR_A, then we want to find T(S), TT(S), TTT(S), etc.  What pattern emerges?
• If viewed with some imagination(!) the pattern will look like footprints -- left -right- left - right. See the Brown book in the Glide Reflection section.
• Now start with a point P and form P1 = T(P), P2 = TT(P), P3 = TTT(P), etc.  And connect P to P1 and P1 to P2, etc., with segments.  What pattern do these segments make?  What pattern to the midpoints of the segments make?
• For most P, the pattern will be a zig-zag. The midpoints of the zigzag are always on one special line (that does not depend on P) and the spacing and direction between the midpoints is always the same. The one exception to the zigzag is when P is on the special line.
• What happens if the 3 mirror lines are move so that they are (approximately) concurrent?
• The footprint pattern almost becomes two shapes forming a pattern with line symmetry. The zigzag segments almost overlay one single segment. The shape is almost what one would get for concurrent lines.

GSP Definition of Translations and Glide Reflections

• Translation:  Mark Vector and Translate
• Glide Reflection.  Draw line AB.  Mark Vector AB and Mark Mirror Line AB.  For any P, translate by the marked vector to get P'and then reflect to get P''.  Hide the point P'.
• Read Brown to see the Definition of a Glide Reflection and a Translation.