Responses and suggested excercises for Chapter 9 opf BTD: Non-Euclidean Geometry and nonorientable surfaces.
One of the most interesting parts of this chapter presented itself in the very beginning. The section on non commutative algebra focusses on a square whose vertices can be exchaged in the plane via particular transformations. For example, its interesting that you should mention that a square can be reflected about its diagonal or rotated about a line which lies in the xy plane. It is also intereting that the composition of these transformations is not commutative.
1. In thge study of computer graphics, this problem presents itself in numerous ways. We have all taken a course called "Introduction to Computer Graphics" in the CS department in which we learned how to apply transformations on objects in the 2-D and 3-D plane. We learned the rule that SRT, is the order in which transformations should be applied. Incidently SRT is short for Scale,Rotate,Translate. Tranformations in any plane can be applied to an object via a transformation matrix. There exists matrices which scale an object by a given factor, rotate it about a particular axis or line by a given angle, or translate it in a particular direction, with a given unit. The beauty of this process is that linear algebra meets euclidean geometry. Transformation matrices, as they are called, can be composed in a particular order and the resulting matrix can be applied to the vertices of the objects in its space to produce the desired transformations.
We quickly learned that in order to preserve the relative orientation of the object, that the matrices must be composed in the same order in which they would be applied to the object.
We also learned that following the above rules, easily lends itself to the creation of an inverse transformation matrix which will undo all tranformations to a given object.
We just thought that it was very interesting how one can view a process by its application (ie tranformation matrices) and not think about the reasons why the algorithm works. While in the computer graphics course, we thought of this process as purely mathematical, but seeing it again here, and thinking about it in terms of non commutative algebra, lets us appreciate it even more.
2. In BTD, the section on the development of non-euclidean geometry presents a case where one is to envision a flatworm who is confined to the surface of a sphere. This worm essentially lives in a two dimensional world. Consider the following puzzling question: 2a. A triangle is composed of three line segments whose intersections occure only at the end of each distinct segment. If a worm lived on a sphere, then we can assume that a all line drawn parallel to the surface of the sphere (in essence, on the sphere) will eventually circumnavigate the entire sphere to arrive at its beginning. Consider one such line which comes very close to touching itself, but does not. How could this be explained to the worm, who percieves that his world is essentially flat. He/She would be asked to believe that a two lines which are not connected, are indeed part of the same line. Would this be a way to prove to the two-dimesional creature that her world is sphere in shape.> Also, it would be impossible to draw a triangle on this sphere in which all of the lines are of this same length. The lines would inevitably intersect at some point along its path. Four angles would appear here, two interior and two exterior.? How would it be possible to draw such a triangle, without adding an extra dimension? 3. Again consider the method used in the graphics computer science course, which applies a series of matrices to perform tranformations on an objects vertices in any space. The definition of inverse matrix, by linear algebra is one where the product of a matrix and its inverse produces 1, or the identity matrix. M x M ^-1 = I, where I is the identity. There was a method develop by Friedrich Gauss, called Gaussian elimination. This process allows us to find the solution, or set of solutions to a series of linear equations. This method was also used to produce a method called the Gauss-Jordan method, which is used to find the inverse matrix, of any given matrix, provided that elimination, does not break down, i.e the matrix is non-singular. If we took one of the transformation matrices from our cs course and applied the Gauss-Jordan method to it to pruduce its invers would the following still be true: a. Taking the product of the inverse and the tranformation matrix produces the identity matrix. b. Applying the inverse matrix to the object vertices, subsequent to applying the original transformation matrix, produces the untransformed, orginal object?
These are just some of the ideas which this chapter made us think about. IT would be a nice computer science exercise to code the algorithm in problem three to see if the transformations actually work, and if not, what result occurs.