A graphics programmer wishes to draw objects that have been stretched in a particular direction. They know how to stretch an object along the X or Y axes, but they need help to stretch the object in an arbitrary direction.
Here are two figures that they provided to help explain what they mean by stretching an object. Figure B-1 shows (a) the original object, which is a circle of radius 1, (b) the circle stretched by a factor of two along the X axis, (c) the original circle stretched by a factor of 2 along the Y axis, and (d) the original circle stretched by a factor of 2 in a 45 degree direction. Figure B-2 shows (a) an original unit square and (b) the result of stretching it by a factor of 2 in a 45 degree direction. Note that stretching is about the origin of the axes, not the centre of the object being stretched
B1. What well-known transformation corresponds to (, 0) – i.e. stretching in the X axis direction by a factor of s? Your answer should include the parameters of the transformation, and will include at least one mathematical expression that depends on s.
St(s,0) corresponds to scaling in the x-axis by a factor of s.
B2. Use the “classic way” to derive a sequence of well known elementary transformations that are equivalent to (, ). Write the transformations using the abbreviated notation presented in lectures.
Rotate(),Scale(s,1)
B3. Expand each of the elementary transformations in your solution of question B2 to homogeneous matrices. The elements of each matrix will be mathematical expressions involving and/or .
() \begin{bmatrix} &− &\\ & &\\&&\end{bmatrix}
(,) \begin{bmatrix}&&\\&&\\&&\end{bmatrix}
B4. Multiply the homogeneous matrices together to obtain the combined transformation as a single matrix containing mathematical expressions. \begin{bmatrix} &− &\\ & &\\&&\end{bmatrix}
B5. Prove that (1, ) is the identity transformation for all values of .
B6. (−1, ) is clearly not an identity transformation. Intuitively, what is it? It may help to sketch the result of transforming the unit square B-2(a), or the flag A-1, for some simple values of . Mathematically, it may help to consider your answers to questions B1 and B2.
B7. Apply (2,45) to the four points of the unit square. Show whether the labels on the points in figure B-2(b) are correct or not.
B5 and beyond is where I need help. I dont understand how St(1,) is the identity. Wouldnt that just rotate the object by ? Im afraid I mightve done B1-B4 wrong, so if someone could comment on that Id really appreciate it.