user: Rotation matrix for a 2d cartesian around a point [Timestamp: 2024/11/26 16:40:17]
ai: To create a rotation matrix for a 2D Cartesian coordinate system around a specific point ((h, k)), you can follow these steps:
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Translate the point of rotation to the origin: This involves subtracting the coordinates of the point ((h, k)) from the coordinates of the point you want to rotate.
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Apply the rotation: Use the standard rotation matrix for an angle (\theta):
[ R(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{bmatrix} ]
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Translate back: After rotating, you need to translate the point back to its original position.
The complete transformation can be expressed as follows:
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Translate the point ((x, y)) to the origin: [ x’ = x - h \ y’ = y - k ]
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Rotate the point: [ \begin{bmatrix} x” \ y” \end{bmatrix} = R(\theta) \cdot \begin{bmatrix} x’ \ y’ \end{bmatrix} = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{bmatrix} \cdot \begin{bmatrix} x - h \ y - k \end{bmatrix} ]
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Translate back: [ x_{\text{final}} = x” + h \ y_{\text{final}} = y” + k ]
Putting it all together, the final coordinates after rotation around the point ((h, k)) are:
[ \begin{bmatrix} x_{\text{final}} \ y_{\text{final}} \end{bmatrix}
\begin{bmatrix} \cos(\theta) & -\sin(\theta) \ \sin(\theta) & \cos(\theta) \end{bmatrix} \cdot \begin{bmatrix} x - h \ y - k \end{bmatrix} + \begin{bmatrix} h \ k \end{bmatrix} ]
This gives you the new coordinates ((x_{\text{final}}, y_{\text{final}})) after rotating the point ((x, y)) around the point ((h, k)) by an angle (\theta). [Timestamp: 2024/11/26 16:40:23]
user: That for a 90 degree clockwise? [Timestamp: 2024/11/26 16:40:37]
ai: To rotate a point ((x, y)) 90 degrees clockwise around a point ((h, k)), you can use the rotation matrix for 90 degrees clockwise, which is:
[ R(90^\circ) = \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix} ]
Here’s how to apply it step by step:
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Translate the point to the origin: [ x’ = x - h \ y’ = y - k ]
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Apply the 90-degree clockwise rotation: [ \begin{bmatrix} x” \ y” \end{bmatrix} = R(90^\circ) \cdot \begin{bmatrix} x’ \ y’ \end{bmatrix} = \begin{bmatrix} 0 & 1 \ -1 & 0 \end{bmatrix} \cdot \begin{bmatrix} x - h \ y - k \end{bmatrix} ]
This results in: [ x” = (y - k) \ y” = -(x - h) ]
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Translate back to the original position: [ x_{\text{final}} = x” + h = (y - k) + h \ y_{\text{final}} = y” + k = -(x - h) + k ]
Putting it all together, the final coordinates after rotating the point ((x, y)) 90 degrees clockwise around the point ((h, k)) are:
[ \begin{bmatrix} x_{\text{final}} \ y_{\text{final}} \end{bmatrix}
\begin{bmatrix} (y - k) + h \ -(x - h) + k \end{bmatrix} ]
So, the final coordinates are:
[ x_{\text{final}} = y - k + h \ y_{\text{final}} = -x + h + k ] [Timestamp: 2024/11/26 16:40:43]
user: What is k an h? [Timestamp: 2024/11/26 16:40:58]
ai: In the context of the rotation matrix and the transformation described, (h) and (k) represent the coordinates of the point around which you are rotating another point. Specifically:
- (h) is the x-coordinate of the center of rotation.
- (k) is the y-coordinate of the center of rotation.
For example, if you want to rotate a point around the point ((3, 2)), then (h = 3) and (k = 2).
So, when you perform the rotation, you are effectively rotating the point ((x, y)) around the point ((h, k)) by the specified angle (in this case, 90 degrees clockwise). [Timestamp: 2024/11/26 16:41:00]