Question

The vector \( [5,\ 3] \) in a right-handed planar coordinate system was transformed to a different coordinate system resulting in the vector \( [6.66,\ 12.20] \) using a four-parameter similarity transformation.

Select the CORRECT set of transformation parameters from the choices given below. The transformation parameters are given in the following order: scale, rotation angle, translation vector.

Hint: For a right-handed system, the rotation matrix \( R(\theta) \) in two dimensions is given as
\[ R(\theta) = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}. \]

• A. \( 2,\ 30^\circ,\ [1,\ 2] \)
• B. \( \dfrac{1}{2},\ -30^\circ,\ [2,\ 1] \)
• C. \( \sqrt{2},\ -30^\circ,\ [1,\ 2] \)
• D. \( \dfrac{1}{\sqrt{2}},\ 30^\circ,\ [2,\ 1] \)

Hint:

Substitute each option's scale, rotation angle and translation into x' = s(x cosθ - y sinθ) + tx, y' = s(x sinθ + y cosθ) + ty and check which one reproduces [6.66, 12.20].

Answer 1

The Correct Option is A

Step 1: Write the general four-parameter (Helmert) similarity transformation.
A two-dimensional four-parameter similarity transformation maps \( (x, y) \) to \( (x', y') \) using a scale \( s \), a rotation \( R(\theta) \), and a translation \( (t_x, t_y) \):\[ \begin{bmatrix} x' \\ y' \end{bmatrix} = s \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} t_x \\ t_y \end{bmatrix} \]Here \( (x, y) = (5, 3) \) and the target is \( (x', y') = (6.66, 12.20) \).
Step 2: Test option (A): \( s = 2 \), \( \theta = 30^\circ \), \( [t_x, t_y] = [1, 2] \).
With \( \cos 30^\circ = 0.8660 \) and \( \sin 30^\circ = 0.5 \):\[ R(30^\circ)\begin{bmatrix}5\\3\end{bmatrix} = \begin{bmatrix}5(0.8660) - 3(0.5)\\5(0.5) + 3(0.8660)\end{bmatrix} = \begin{bmatrix}2.830\\5.098\end{bmatrix} \]
Step 3: Apply the scale and the translation.
\[ 2\begin{bmatrix}2.830\\5.098\end{bmatrix} + \begin{bmatrix}1\\2\end{bmatrix} = \begin{bmatrix}5.660\\10.196\end{bmatrix} + \begin{bmatrix}1\\2\end{bmatrix} = \begin{bmatrix}6.660\\12.196\end{bmatrix} \approx \begin{bmatrix}6.66\\12.20\end{bmatrix} \]This exactly matches the given transformed vector, so option (A) is consistent.
Step 4: Check that the remaining options fail.
Repeating the same substitution: option (B) \( (s=0.5, \theta=-30^\circ, T=[2,1]) \) gives \( (4.92, 1.05) \); option (C) \( (s=\sqrt{2}, \theta=-30^\circ, T=[1,2]) \) gives \( (9.25, 2.14) \); option (D) \( (s=1/\sqrt{2}, \theta=30^\circ, T=[2,1]) \) gives \( (4.00, 4.61) \). None reproduce \( (6.66, 12.20) \), so all three are incorrect.
Step 5: Conclude.
Only option (A) reproduces the given transformed vector exactly.\[ \boxed{s = 2,\ \theta = 30^\circ,\ [t_x, t_y] = [1, 2]} \]