2.7 Exercises

23 points · 4 extra credit

Applications to Computer Graphics

Throughout, a point $(x, y)$ in the plane is represented in homogeneous coordinates by the column $\begin{bmatrix} x \\ y \\ 1 \end{bmatrix}$, and a $2$D transformation is applied by left-multiplying by a $3\times3$ matrix.

1. 1. (3 pts) Write the $3\times3$ homogeneous matrix for the translation by $(4, -1)$, and use it to find the image of the point $(2, 3)$.

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2. 2. (3 pts) Write the $3\times3$ homogeneous matrix $R$ for a $90^\circ$ counterclockwise rotation about the origin, and compute the images of $(1, 0)$ and of a general point $(x, y)$.

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3. 3. (3 pts) A figure is first rotated $90^\circ$ counterclockwise about the origin, then translated by $(4, -1)$. Write the single $3\times3$ matrix for this composite transformation, apply it to $(1, 0)$, and explain why performing the translation first would generally give a different result.

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4. 4. (2 pts) Write the homogeneous matrix for the scaling that multiplies $x$-coordinates by $2$ and $y$-coordinates by $3$, and find the image of $(1, 1)$.

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5. 5. (3 pts) The shear $(x, y) \mapsto (x + 2y,\, y)$ is a signature computer-graphics operation. Write its $3\times3$ homogeneous matrix $H$, and apply it to the four vertices $(0, 0)$, $(1, 0)$, $(1, 1)$, $(0, 1)$ of the unit square. Then, using matrix composition, write the single matrix for this shear followed by a $90^\circ$ counterclockwise rotation about the origin, and use it to find the image of $(1, 1)$.

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6. 6. (3 pts) State whether each is true or false, and justify.

   (a) Every $2$D rotation, scaling, and translation can be represented by a single $3\times3$ matrix acting on homogeneous coordinates.

   (b) The homogeneous coordinate of an ordinary point of the plane is always $1$ in the third entry.

   (c) Composing two transformations corresponds to multiplying their matrices, and matrix order does not matter because both are rigid motions.

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7. 7. (3 pts) Explain why a nonzero translation of $\mathbb{R}^2$ is not a linear transformation of $\mathbb{R}^2$, and explain how homogeneous coordinates let it be represented by a matrix product anyway.

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8. 8. (3 pts) In $3$D graphics a point $(x, y, z)$ is written as $\begin{bmatrix} x \\ y \\ z \\ 1 \end{bmatrix}$. Write the $4\times4$ matrix for a $90^\circ$ counterclockwise rotation about the $z$-axis, and compute the image of $(1, 0, 5)$.

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9. 9. (4 pts ••) Derive the $3\times3$ homogeneous matrix for a $180^\circ$ rotation about the point $(1, 1)$ (not the origin), by composing a translation to the origin, a rotation, and a translation back. Then apply it to the point $(3, 2)$.

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