Reflections on Reflections
Perpendicular Line Reflections
Perpendicular Line Reflections
The applet above shows that 2 reflections over perpendicular lines equal a 180 degree rotation and that 4 reflections over perpendicular lines equals a 360 degree rotation bringing you back the original object. A different proof shows why two reflections over intersecting lines returns a rotation about the point of intersection of twice the angle between the lines.
Composition of Isometrics Commutative?
Composition of Isometrics Commutative?
A composition of isometries is not commutative. Given an arbitrary point, perform a glide reflection, then perform the same two translation and reflection but in the opposite order. The two compositions will give you different ending results which proves isometries are not commutative. In the example above, polygon ABCDE is first translated to become polygon A'B'C'D'E' then reflected over line GH, and is not the same as when it is reflected over line GH then translated
Composition of Reflections is a Rotation
Composition of Reflections as a Rotation
Given triangle ABC and an arbitrary point E. Reflect polygon ABC over line DG then line FE. Then construct a circle with a radius the distance between the point the lines intersect and point A on the triangle. The angle created between vertex AEA'' is twice the angle of the angle created between FEG because it is the angle between the intersecting lines and is reflected twice. Polygon ABC rotated about the origin (in the same direction as the angle beta measured) with angle beta will land “on top” of polygon A''B''C''. This shows that the composition of reflections can be represented as a single rotation about the intersection of the lines based on the angles created between polygons.
Composition of Reflections as a Translation
Compositions of Reflections over 2 Parallel Lines as a Translation
A polygon reflected over two parallel lines returns a translation. As seen above, polygon ABC is reflected over line FG then line HD. With all angles and slide lengths being preserved, all the vertices of polygon ABC is translated twice the length of the distance between the parallel lines in the same direction making it a translation. The distance of triangle A'B'C' that has been reflected once is one distance from the first parallel line and one from the second. Each of these distances is doubled when reflected over the lines. This results in the entire distance between the two parallel lines doubled to get the total translation distance.
Reflections over Parallel Lines Pt.2
A reflection over a line is the shortest(perpendicular) distance from the point to the line in the same direction. However, a single reflection changes the orientation of the figure as we can see segment AB can't be changed into A'B' by a single translation. Although, if you compose another reflection over a line parallel to the first with the first reflection, the orientation is flipped back to the original. When you repeat the perpendicular distance to the line process, you get the final figure. Because you are reflecting by each distance from the point between the two lines, you end with a translation of twice the distance between the parallel lines.
Composition of Rotations
Composition of Rotations
A composition of two rotations about the same point forms a single rotation about that same point. Given an arbitrary point and rotate it about another arbitrary point. Perform another rotation about the same arbitrary point. The composition of the final rotation from the first point to the second rotation is the sum of the two rotations(clockwise in the positive direction and counter clockwise the negative direction in the example above). In the example above, CEC' plus C'EC'' equals the complete CEC''.
Rotation about different points
Composition of Rotations about different points
The above example shows how the composition of two rotations about different points can be represented by a single rotation about a different point. Take an arbitrary polygon ABC. Rotate the polygon by an arbitrary angle to get A'B'C'. Then, rotate the original polygon ABC about a different arbitrary point with a different arbitrary angle to get polygon A''B''C''. Construct a line between the two points of rotation along with two lines with angles half of that of each rotation(BDB' 22 degrees because of a rotation of 45 and B'EB'' 30 degrees because of the 60 degree rotation). Construct lines between D, E and the new point F. The new point F is the new point of rotation rotated the two angles of rotation added together. The image shows the overlapping of the two triangles because the rotation about point F of 105 degrees is equal to the rotation to create polygon A''B''C''
Note: I'm still slightly spotty on this proof but I can see how it works. I also understand how you can write it as four reflections which can be written as two reflections and two reflections can be written as a rotation about a point.
Every composition of isometry is at most 3 reflections
Every isometry of the plane is a composition of 1,2, or 3 reflections
Translations and rotations can be represented as a composition of 2 reflections. The composition of a rotation and a translation can be defined as a rotation of the same degree about a point found using the intersection of lines drawn between two corresponding vertices on the original and final polygon. As shown above, I performed a few random isometries and am able to get to the triangle A''B''C'' from triangle ABC in at most 3 reflections. With the first reflection, get one point correct(A->A'')reflect over the perpendicular bisector of the segment connecting the two points, then get the second point correct by reflecting it over line HA'' which goes through point A'/A'' and the midpoint between B' and B''. This second reflection achieves the final goal in this case but sometimes can require a third reflection to get the other points in the correct space given that the other points could be on the opposite side.