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Chapter 4: Problem 2

Determine the effect of the following transformations. a. Rotation through \(\frac{\pi}{2},\) followed by projection on the \(y\) axis, followed by reflection in the line \(y=x\). b. Projection on the line \(y=x\) followed by projection on the line \(y=-x\). c. Projection on the \(x\) axis followed by reflection in the line \(y=x\).

Short Answer

Expert verified
a) Transforms \((x, y)\) to \((x, 0)\). b) Transforms any point to \((0, 0)\). c) Transforms \((x, y)\) to \((0, x)\).

Step by step solution

01

Rotation Through \( \frac{\pi}{2} \)

A rotation through \( \frac{\pi}{2} \) counterclockwise about the origin transforms a point \((x, y)\) to \((-y, x)\). This changes the coordinates by swapping them and negating the first coordinate.
02

Projection on the \( y \)-axis

Projection on the \( y \)-axis keeps the \( y \)-coordinate the same and sets the \( x \)-coordinate to zero. After rotating through \( \frac{\pi}{2} \), this transforms \((-y, x)\) to \((0, x)\).
03

Reflection in the Line \( y=x \)

Reflecting a point across the line \( y=x \) involves swapping its coordinates. This transforms \((0, x)\) to \((x, 0)\). Thus, the overall effect of the transformations in sequence (a) is to swap the original \( y \)-coordinate with the \( x \)-coordinate, resulting in \((x, 0)\) when starting from \((x, y)\).
04

Projection on the Line \( y=x \)

Projection on the line \( y=x \) transforms any point \((x, y)\) to the point that lies on the line, which is \(\left(\frac{x + y}{2}, \frac{x + y}{2}\right)\). This effectively maps the original point onto \( y=x \).
05

Projection on the Line \( y=-x \)

Projection on the line \( y=-x \) transforms a point to \(\left(\frac{x - y}{2}, \frac{y - x}{2}\right)\), thus effectively mapping the result onto \( y=-x \). For sequence (b), this converts \(\left(\frac{x + y}{2}, \frac{x + y}{2}\right)\) to the origin \((0,0)\).
06

Projection on the \( x \)-axis

Projection on the \( x \)-axis transforms a point \((x, y)\) by setting \( y \) to zero, resulting in \((x, 0)\).
07

Reflection in the Line \( y=x \)

Reflection of the point \((x, 0)\) in the line \( y=x \) swaps the coordinates to \((0, x)\). For sequence (c), starting from \((x, y)\) leads to \((0, x)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rotation Matrix
A rotation matrix is a fundamental tool in geometry that helps rotate points around an origin. The matrix for a rotation through \( \frac{\pi}{2} \) counterclockwise looks like this:
  • \[ \begin{bmatrix} 0 & -1 \ 1 & 0 \end{bmatrix} \]
This matrix takes a point \((x, y)\) and transforms it to \((-y, x)\). Essentially, it swaps the coordinates and negates the first one. This transformation is widely used in computer graphics to rotate objects in 2D space.
If you visualize the process, think of moving a point 90 degrees around a circle centered at the origin.
Projection Transformation
Projection transformations reduce dimensions by flattening points onto a specific line or axis. Imagine shining a light straight down on a shape; the shadow it casts is a projection.
  • For instance, projecting onto the \( y \)-axis changes \((x, y)\) to \((0, y)\).
  • When projecting onto the line \( y=x \), it becomes \( \left(\frac{x + y}{2}, \frac{x + y}{2}\right) \), where both coordinates are equal.
Projection transformations are helpful in various fields including physics and engineering where analyzing only one dimension is necessary. They simplify 3D objects into 2D views.
Reflection Across Lines
Reflection across a line switches coordinates in a specific way, like looking in a mirror. The line \( y=x \) is a common line for reflection:
  • The reflection of \((x, y)\) over \( y=x \) swaps the coordinates to \((y, x)\).
  • In the case of reflection over \( y=-x \), it turns \((x, y)\) into \((-y, -x)\).
These transformations are used in designing symmetrical structures and graphics where mirroring effects are needed.
Coordinate Transformation
Coordinate transformations are methods to change from one coordinate system to another. They can include operations like rotations, translations, and reflections.
  • They are used to reposition objects in graphics and navigation systems.
  • A simple coordinate swap or alteration, as seen in the exercises, is often the result of multiple transformations working together.
Understanding these transformations can simplify problems in physics and mathematics by adapting coordinates to a more convenient form.
Matrices in Graphics
Matrices are essential in computer graphics for executing transformations efficiently. They enable multiple operations like scaling, rotating, and translating shapes with precision and speed.
  • Each transformation, such as a rotation or projection, is represented by a specific matrix.
  • Matrices allow combination of transformations in sequence, creating complex effects by matrix multiplication.
In graphics software, these concepts help render 3D models onto 2D screens, manage animations, and simulate realistic environments.

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Most popular questions from this chapter

Given the cube with vertices \(P(x, y, z)\) where each of \(x, y,\) and \(z\) is either 0 or \(2,\) consider the plane perpendicular to the diagonal through \(P(0,0,0)\) and \(P(2,2,2)\) and bisecting it. a. Show that the plane meets six of the edges of the cube and bisects them. b. Show that the six points in (a) are the vertices of a regular hexagon.

Simplify \((a \mathbf{u}+b \mathbf{v}) \times(c \mathbf{u}+d \mathbf{v})\).

Show that points \(A, B,\) and \(C\) are all on one line if and only if \(\overrightarrow{A B} \times \overrightarrow{A C}=0\).

In each case, write \(\mathbf{u}=\mathbf{u}_{1}+\mathbf{u}_{2},\) where \(\mathbf{u}_{1}\) is parallel to \(\mathbf{v}\) and \(\mathbf{u}_{2}\) is orthogonal to \(\mathbf{v}\). a. \(\mathbf{u}=\left[\begin{array}{r}2 \\ -1 \\ 1\end{array}\right], \mathbf{v}=\left[\begin{array}{r}1 \\ -1 \\ 3\end{array}\right]\) b. \(\mathbf{u}=\left[\begin{array}{l}3 \\ 1 \\ 0\end{array}\right], \mathbf{v}=\left[\begin{array}{r}-2 \\ 1 \\ 4\end{array}\right]\) c. \(\mathbf{u}=\left[\begin{array}{r}2 \\ -1 \\ 0\end{array}\right], \mathbf{v}=\left[\begin{array}{r}3 \\ 1 \\ -1\end{array}\right]\) d. \(\mathbf{u}=\left[\begin{array}{r}3 \\ -2 \\ 1\end{array}\right], \mathbf{v}=\left[\begin{array}{r}-6 \\ 4 \\ -1\end{array}\right]\)

In each case, verify that the points \(P\) and \(Q\) lie on the line. $$ \begin{array}{ll} \text { a. } & x=3-4 t \quad P(-1,3,0), Q(11,0,3) \\ & y=2+t \\ & z=1-t \\ \text { b. } & x=4-t \quad P(2,3,-3), Q(-1,3,-9) \\ & y=3 \\ & z=1-2 t \end{array} $$
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