3. Estado de la cuestión
3.2. Sistemas Fiduciales Visuales
3.2.1. ARToolkit
We found four permutations that correspond to rotations of the square. In Problem 255 you found four permutations that correspond to flips of the square in space. One flip fixes the vertices in the places labeled 1 and 3 and interchanges the vertices in the places labeled 2 and 4. Let us denote it byϕ1|3. One flip fixes the vertices in the positions labeled 2 and 4 and interchanges those in the positions labeled 1 and 3. Let us denote it by ϕ2|4. One flip interchanges the vertices in the places labeled 1 and 2 and also interchanges those in the places labeled 3 and 4. Let us denote it by ϕ12|34. The fourth flip interchanges the vertices in the places labeled 1 and 4 and interchanges those in the places labeled 2 and 3. Let us denote it by ϕ14|23. Notice that ϕ1|3 is a permutation that takes the vertex in place 1 to the vertex in place 1 and the vertex in place 3 to the vertex in place 3, while ϕ12|34 is a permutation that takes the edge between places 1 and 2 to the edge between places 2 and 1 (which is the same edge) and takes the edge between places 3 and 4 to the edge between places 4 and 3 (which is the same edge). This should help to explain the similarity in the notation for the two different kinds of flips.
6.1. PERMUTATION GROUPS 119 •259. Write down the two row notation forρ3,ϕ2|4,ϕ12|34and ϕ2|4◦ϕ12|34.
260. (You may have already done this problem in Problem 255, in which case you need not do it again!) In Problem 255, if a rigid motion in three-dimensional space returns the square to its original location, in how many places can vertex number one land? Once the location of vertex number one is decided, how many possible locations are there for vertex two? Once the locations of vertex one and vertex two are decided, how many locations are there for vertex three? Answer the same question for vertex four. What does this say about the relation- ship between the four rotations and four flips described just before Problem 259 and the permutations you described in Problem 255? The four rotations and four flips of the square described before Problem 259 form a group called the dihedral group of the square. Sometimes the group is denotedD8because it has eight elements, and sometimes the group is denoted by D4 because it deals with four vertices! Let us agree to use the notationD4for the dihedral group of the square. There is a similardihedral group, denoted by Dn, of all the rigid motions of three-dimensional space
that return a regularn-gon to its original location (but might put the vertices in different places).
261. Another view of the dihedral group of the square is that it is the group of all distance preserving functions, also calledisometries, from a square to itself. Notice that an isometry must be a bijection. Any rigid motion of the square preserves the distances between all points of the square. However, it is conceivable that there might be some isometries that do not arise from rigid motions. (We will see some later on in the case of a cube.) Show that there are exactly eight isometries (distance preserving functions) from a square to itself. Online hint.
262. How many elements does the group Dn have? Prove that you are
correct.
•263. In Figure 6.3 we show a cube with the positions of its vertices and faces labeled. As with motions of the square, we let we letϕ(x) be the label of the place where vertex previously in position x is now.
(a) Write in two row notation the permutationρ of the vertices that corresponds to rotating the cube 90 degrees around a vertical axis through the facest (for top) andu (for underneath). (Rotate in
Figure 6.3: A cube with the positions of its vertices and faces labeled. The curved arrows point to the faces that are blocked by the cube.
u
b
f
l
t
r
3
4
5
1
6
7
8
2
a right-handed fashion around this axis, meaning that vertex 6 goes to the back and vertex 8 comes to the front.)
(b) Write in two row notation the permutation ϕ that rotates the cube 120 degrees around the diagonal from vertex 1 to vertex 7 and carries vertex 8 to vertex 6.
(c) Compute the two row notation forρ◦ϕ.
(d) Is the permutationρ◦ϕa rotation of the cube around some axis? If so, say what the axis is and how many degrees we rotate around the axis. If ρ◦ϕ is not a rotation, give a geometric description of it.
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264. How many permutations are in the group R? R is sometimes called the “rotation group” of the cube. Can you justify this? Online hint.265. As with a two-dimensional figure, it is possible to talk about isometries of a three-dimensional figure. These are distance preserving functions from the figure to itself. The function that reflects the cube in Figure 6.3 through a plane halfway between the bottom face and top face
6.1. PERMUTATION GROUPS 121 exchanges the vertices 1 and 5, 2 and 6, 3 and 7, and 4 and 8 of the cube. This function preserves distances between points in the cube. However, it cannot be achieved by a rigid motion of the cube because a rigid motion that takes vertex 1 to vertex 5, vertex 2 to vertex 6, vertex 3 to vertex 7, and vertex 4 to vertex 8 would not return the cube to its original location; rather it would put the bottom of the cube where its top previously was and would put the rest of the cube above that square rather than below it.
(a) How many elements are there in the group of permutations of [8] that correspond to isometries of the cube? Online hint.
(b) Is every permutation of [8] that corresponds to an isometry either a rotation or a reflection? Online hint.