Plane Transformations

For example, to generate a rhombus with vertices (0, 0), (1, 1), (0, 2), and (1, −1), we store the pairs as columns of a matrix: \[ {\bf T} = \begin{bmatrix} 0 & 1 & 0 & 1 & 0 \\ 0 & 1 & 2 & -1 & 0 \end{bmatrix} . \] An additional copy of the vertex (0, 0) is stored in the last column of T so that the previous point (1, −1) will be connected back to (0, 0) [see Figure 4.2.3(a)].

Leon, page 205, Figure 4.2.3

We can transform a figure by changing the positions of the vertices and then redrawing the figure. If the transformation is linear, it can be carried out as a matrix multiplication. Viewing a succession of such drawings will produce the effect of animation. The four primary geometric transformations that are used in computer graphics are as follows:

1. Dilations and contractions. A linear operator of the form \[ T({\bf x}) = c\,{\bf x} \] is a dilation if c > 1 and a contraction if 0 < c < 1. The operator T is represented by the matrix cI, where I is the 2 × 2 identity matrix. A dilation increases the size of the figure by a factor c > 1, and a contraction shrinks the figure by a factor c < 1. Figure 4.2 shows a dilation by a factor of 1.5 of the rhombus stored in the matrix T.
2. Reflections about an axis. If T_x is a transformation that reflects a vector x about the x-axis, then T_x is a linear operator and hence it can be represented by a 2 × 2 matrix A. Since \[ T_x ({\bf e}_1 ) = {\bf e}_1 \qquad \mbox{and} \qquad T_x ({\bf e}_2 ) = -{\bf e}_2 , \] it follows that \[ {\bf A} = \begin{bmatrix} 1 & \phantom{-}0 \\ 0 & -1 \end{bmatrix} . \]

   Leon, page 205, Figure 4.2.3

   Similarly, if T_y is the linear operator that reflects a vector about the y-axis, then T_y is represented by the matrix \[ \left[ T_y \right] = \begin{bmatrix} -1 & 0 \\ \phantom{-}0 & 1 \end{bmatrix} . \] Figure 4.2.3(c) shows the image of the rhombus after a reflection about the y-axis.

   Leon, page 205, Figure 4.2.3

3. Rotations. Let T be a transformation that rotates a vector about the origin by an angle θ in the counterclockwise direction. We saw in Example ???? that T is a linear operator and that T(x) = A x, where \[ {\bf A} = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \phantom{-}\cos\theta \end{bmatrix} . \] Figure 4.2.3(d) shows the result of rotating the triangle T by 60 ◦ in the counterclockwise direction.

   Leon, page 205, Figure 4.2.3

4. Translations. A translation by a vector a is a transformation of the form \[ T({\bf x}) = {\bf x} + {\bf a} . \] If a ≠ = 0, then T is not a linear transformation and hence T cannot be represented by a 2 × 2 matrix. However, in computer graphics it is desirable to do all transformations as matrix multiplications. The way around the problem is to introduce a new system of coordinates called homogeneous coordinates.

\[ \begin{bmatrix} 1 & 0 \\ \frac{3}{2} & 1 \end{bmatrix} \]

These three successive row operations can be performed by multiplyingA on the left successively by \[ {\bf E}_1 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} , \qquad {\bf E}_2 = \begin{bmatrix} \frac{2}{3} & 0 \\ 0 & 1 \end{bmatrix} , \qquad {\bf E}_3 = \begin{bmatrix} 1 & -\frac{2}{3} \\ 0 & \phantom{-}1 \end{bmatrix} . \] Inverting these matrices, we get

1. shearing by factor of ⅔ in the abscissa direction;
2. expanding by a factor of 3/3 in the x-direction;
3. reflecting about the line y = x.

1. Find the standard matrix for a linear transformation T: ℝ² ↦ ℝ² that first reflects points through the horizontal x₁-axis and then reflects points through the line x₁ = x₂.
2. Find the standard matrix for a linear transformation T: ℝ² ↦ ℝ² that first rotates points through -3π/4 radian (clockwise) and then reflects points the vertical x₂-axis.
3. Find the standard matrix for a linear transformation T: ℝ² ↦ ℝ² that maps i=(1,0) into 2i-3j but leaves the vector j=(0,1) unchanged.
4. Find the standard matrix for a linear transformation T: ℝ² ↦ ℝ² that rotates points (about the origin) through 3π/2 radians (counterclockwise).
5. In ℝ², clearly R(θ+φ) = R(θ) R(φ). By writing out these matrices and performing matrix multiplication, derive the laws for the sine and cosine of the sum of two angles.
6. If you need the formulas for sin(θ + π/2) and cos(θ + π/2) and don't remember them, what is a simple way to find them ?
7. Find all 2 × 2 rotation matrices that are also diagonal.
8. In ℝ², if the list of vertices of a square starts with (0, 0) and (𝑎, b) going counterclockwise, what are the remaining two vertices? (Hint: The vertex opposite (𝑎, b) can be obtained by rotating (𝑎, b) by 90° about the origin.)
9. Find the standard matrix for a linear transformation T: ℝ² ↦ ℝ² that rotates points (about the origin) through -π/4 radians (clockwise).
10. If \( {\bf A} = \begin{bmatrix} 1&2 \\ -1&-2 \end{bmatrix} , \) find two matrices B ≠ C such that AB = AC.
11. Suppose that numbers 𝑎 and b in matrix \( \displaystyle \quad \begin{bmatrix} \phantom{-}a&b \\ -b&a \end{\bmatrix} \quad \) are not both zero. Find the entries of rotation matrix that takes (1, 0) to a unit vector in the direction of (𝑎, b). (You don't need to express the angle of the rotation.)
12. Show that a matrix \( \displaystyle \quad {\bf A} = \begin{bmatrix} \phantom{-}a&b \\ -b&a \end{\bmatrix} \quad \) is equal to a rotation matrix times a scalar matrix rI with r > 0. (Hence, x ← x A preserves shapes and orientation while expanding or contracting the size uniformly.)

1. Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International
2. Dunn, F. and Parberry, I. (2002). 3D math primer for graphics and game development. Plano, Tex.: Wordware Pub.
3. Foley, James D.; van Dam, Andries; Feiner, Steven K.; Hughes, John F. (1991), Computer Graphics: Principles and Practice (2nd ed.), Reading: Addison-Wesley, ISBN 0-201-12110-7
4. Matrices and Linear Transformations