Verified Answer :
- Start with the initial point \(V_0 = (0, 1)\) and apply the operations in the order \(R_{O,90^{\circ}}\), then \(T_{(0,2)}\), then \(r_{y\text{-axis}}\).
- Rotate \(V_0\) \(90^{\circ}\) CCW about the origin to get \(V_1 = (-1, 0)\).
- Translate \(V_1\) by \((0, 2)\) to get \(V_2 = (-1, 2)\).
- Reflect \(V_2\) across the \(y\)-axis to get the final point \(V_3 = (1, 2)\).
- The final coordinates are \(\boxed{(1, 2)}\).
Explanation
- Analyze the transformation sequence.
The problem involves transforming an initial vertex \(V_0 = (0, 1)\) through a sequence of three geometric operations. The composition of these operations is defined as \(r_{y\text{-axis}} \circ T_{(0,2)} \circ R_{O,90^{\circ}}\). According to standard notation for the composition of functions, these operations are applied from right to left, starting with the rotation.
- Apply the rotation transformation.
The first transformation is a counter-clockwise rotation about the origin by \(90^{\circ}\). The rule for this rotation maps a point \((x, y)\) to \((-y, x)\). Applying this to our initial point \(V_0 = (0, 1)\):
\[V_1 = R_{O,90^{\circ}}(0, 1) = (-1, 0)\]
- Apply the translation transformation.
The second transformation is a translation by the vector \((0, 2)\). The rule for this translation maps a point \((x, y)\) to \((x + 0, y + 2)\). Applying this to the previously rotated point \(V_1 = (-1, 0)\):
\[V_2 = T_{(0,2)}(-1, 0) = (-1 + 0, 0 + 2) = (-1, 2)\]
- Apply the reflection transformation and find the final result.
The third and final transformation is a reflection across the \(y\)-axis. The rule for this reflection maps a point \((x, y)\) to \((-x, y)\). Applying this to the point \(V_2 = (-1, 2)\):
\[V_3 = r_{y\text{-axis}}(-1, 2) = (-(-1), 2) = (1, 2)\]
The final coordinates of the transformed vertex are \((1, 2)\).
Examples
Imagine you are programming a character in a 2D video game that moves across the screen. To move the character from its starting position at (0, 1), you might need to rotate it to face a new direction, move it to a different location on the map, and then reflect it so it faces the opposite way. By performing the transformations \(V_3 = (r_{y\text{-axis}} \circ T_{(0,2)} \circ R_{O,90^{\circ}})(0, 1)\), you can accurately track the character's new coordinates as \((1, 2)\) to ensure your game logic functions correctly. Understanding these transformations is foundational for animation, computer graphics, and engineering design.