Two Dimensional Transformations and Viewing Questions and Answers

Q: In general pivot point rotation what is the correct sequence of transformations ?

The sequence is Translation, rotation, translation.Basic rotation formulas turn a point about the origin. To rotate about any other pivot point (xr, yr), the object is first translated so that the pivot moves to the origin, then rotated about the origin, then translated back so the pivot returns to its original position. The composite matrix is \(T(x_r, y_r) \cdot R(\theta) \cdot T(-x_r, -y_r)\).Scaling plays no part in a rotation, and rotating twice around a translation does not keep the pivot fixed, so the other three sequences are wrong.

Q: The transformation matrix \(\begin{bmatrix} 0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}\) is used to obtain the reflection about the.......... .

The matrix gives the reflection about the diagonal line Y = -x.Apply it to a point in homogeneous coordinates:\[ \begin{bmatrix} 0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} = \begin{bmatrix} -y \\ -x \\ 1 \end{bmatrix} \]The point (x, y) goes to (-y, -x), which is its mirror image in the line y = -x. For example (2, 0) goes to (0, -2).Reflection about y = x swaps the coordinates without changing signs (the same matrix with +1s). Reflection about the y-axis has diagonal (-1, 1, 1), and about the line y = 0 (the x-axis) has diagonal (1, -1, 1).

Q: A transformation that distorts the shape of an object such that the transformed shape appears as if the object was composed of internal layers that had be caused to slide over each other is called as ......... .

The transformation is shear.A shear shifts each point in proportion to its distance from a fixed axis, for example \(x' = x + sh_x \cdot y\). The effect is as if the object were made of thin layers sliding over one another, turning a square into a parallelogram.Reflection mirrors an object and translation moves it, both without distortion. Dither is a technique for simulating extra colours or shades with patterns of pixels, not a geometric transformation.

In general pivot point rotation what is the correct sequence of transformations ?

A.Rotation, translation, rotation
B.Rotation, scaling, rotation
C.Scaling, rotation, translation
D.Translation, rotation, translation

Exam Year(s)

MH SET - 2021, SET Exam - 2021,

Answer: D

The sequence is Translation, rotation, translation.

Basic rotation formulas turn a point about the origin. To rotate about any other pivot point (xr, yr), the object is first translated so that the pivot moves to the origin, then rotated about the origin, then translated back so the pivot returns to its original position. The composite matrix is \(T(x_r, y_r) \cdot R(\theta) \cdot T(-x_r, -y_r)\).

Scaling plays no part in a rotation, and rotating twice around a translation does not keep the pivot fixed, so the other three sequences are wrong.

Discuss in Forum

The transformation matrix \(\begin{bmatrix} 0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix}\) is used to obtain the reflection about the.......... .

A.Y-axis, flipping x-coordinates
B.line Y = 0
C.Diagonal line Y = -x
D.Diagonal line Y = x

Exam Year(s)

MH SET - 2023, SET Exam - 2023,

Answer: C

The matrix gives the reflection about the diagonal line Y = -x.

Apply it to a point in homogeneous coordinates:

\[ \begin{bmatrix} 0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix} = \begin{bmatrix} -y \\ -x \\ 1 \end{bmatrix} \]

The point (x, y) goes to (-y, -x), which is its mirror image in the line y = -x. For example (2, 0) goes to (0, -2).

Reflection about y = x swaps the coordinates without changing signs (the same matrix with +1s). Reflection about the y-axis has diagonal (-1, 1, 1), and about the line y = 0 (the x-axis) has diagonal (1, -1, 1).

Discuss in Forum

A transformation that distorts the shape of an object such that the transformed shape appears as if the object was composed of internal layers that had be caused to slide over each other is called as ......... .

A.Reflection
B.Shear
C.Dither
D.Translation

Exam Year(s)

MH SET - 2024, SET Exam - 2024,

Answer: B

The transformation is shear.

A shear shifts each point in proportion to its distance from a fixed axis, for example \(x' = x + sh_x \cdot y\). The effect is as if the object were made of thin layers sliding over one another, turning a square into a parallelogram.

Reflection mirrors an object and translation moves it, both without distortion. Dither is a technique for simulating extra colours or shades with patterns of pixels, not a geometric transformation.

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