Describe The Transformation That Rotates An Object Point Q

Describe the transformation that rotates an object point, Q(x, y), ï± degrees about a fixed center of rotation P(h, k). [3 Marks]

In two-dimensional graphics, rotating a point around a fixed center involves applying a rotation transformation that moves the point in a circular path by a specified angle. When rotating a point Q(x, y) about a fixed point P(h, k) by an angle ï±, the process typically involves translating the point so that P becomes the origin, performing the rotation, and then translating back to the original position. Mathematically, this transformation can be expressed through matrix operations using homogeneous coordinates, which facilitate combining translation and rotation into a single matrix operation. The process preserves the distance from the rotation center and changes the point's position based on the rotation angle.

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The rotation of an object point Q(x, y) by a specific angle ï± around a fixed center P(h, k) is a fundamental operation in computer graphics and geometric transformations. This operation is essential in manipulating objects within a coordinate system, enabling effects such as animation, object alignment, and simulation of real-world movements. The precise mathematical description of this transformation, including the use of matrix algebra, provides a robust and flexible method for performing complex graphical operations efficiently and accurately.

Introduction to Rotation about an Arbitrary Center

In standard coordinate geometry, a rotation about the origin is straightforward; however, in practical applications, objects are often rotated about points other than the origin. For this reason, the concept of rotating a point about a fixed center P(h, k) becomes critical. This process involves a combination of translation and rotation steps, where the object is first translated so that the rotation center becomes the origin, then rotated, and finally translated back to its original position. These transformations can be expressed succinctly using transformation matrices in homogeneous coordinates, allowing for easy chaining of multiple operations.

Mathematical Derivation of Rotation About a Fixed Point

The key to understanding the rotation about a point P(h, k) is the combination of translation and rotation matrices. To rotate a point Q(x, y) around P(h, k), one can follow these steps:

  1. Translate the point so that P becomes the origin: Q' = (x - h, y - k)
  2. Rotate the translated point by ï± (counterclockwise) about the origin:

The rotation matrix in 2D for an angle ï± is:

R(ï±) = \[\begin{bmatrix} \cos ï± & -\sin ï± \\ \sin ï± & \cos ï± \end{bmatrix}\]

Applying this matrix to (x - h, y - k):

(x'’, y’’) = R(ï±) * (x - h, y - k)

Finally, translate the point back by adding h and k:

(x_{new}, y_{new}) = (x'' + h, y'' + k)

Homogeneous Coordinate Matrix Representation

Using homogeneous coordinates, the entire transformation can be represented as a single matrix multiplication. The transformation matrix for rotation about point P(h, k) is obtained by combining translation and rotation matrices as:

T(-h, -k)    R(ï±)    T(h, k)

Where T(a, b) is the translation matrix that moves points by (a, b), and R(ï±) is the rotation matrix about the origin. The combined matrix becomes:

M = \[\begin{bmatrix}


\cos ï± & -\sin ï± & h(1 - \cos ï±) + k\sin ï±) \\


\sin ï± & \cos ï± & k(1 - \cos ï±) - h\sin ï±) \\


0 & 0 & 1


\end{bmatrix}\]

This matrix can be applied directly to the homogeneous coordinate vector of Q(x, y, 1) to obtain the rotated position.

Conclusion

The rotation of a point about a fixed center involves translating to the origin, rotating, and translating back, which can be efficiently expressed using homogeneous transformation matrices. This approach simplifies complex graphical transformations and allows for chaining multiple operations seamlessly. Understanding and deriving these matrices are fundamental skills required for advanced computer graphics, animation, and geometric modeling.

References

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  • Huang, J., & Li, H. (2014). Transformation Matrix for Rotation About Arbitrary Axis. International Journal of Computer Mathematics, 91(4), 795-808.

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