Rotation and Translation of Axes
Introduction
In my previous blog post “2D Plane Transformation”, we have discussed how to do 2D plane transformation in a Cartesian coordinate system, i.e., how to convert one point from one plane to the other in a Cartesian coordinate system. In some other situations, the points are fixed, and we would like to compute the coordinates of the same point in different Cartesian coordinate systems whose relationships are rotation and translation.
In this blog post, I would like to quickly discuss the rotation and translation of axes primarily focused on the 2D Cartesian coordinate systems.
Rotation and Translation of 2D Axes
Rotation of Axes
Suppose the
Rotation of Axes
In the
Therefore, the coordinate transformation from the
or
To get the inverse transformation from the
Alternatively, the inverse transformation could also be derived as follows.
Add or subtract the two equations, we have
Therefore,
Translation of Axes
Suppose the
The coordinate transformation from the
or
The inverse coordinate transformation from the
or
Rotation and Translation of Axes
The rotation of axes counterclockwise around the origin through an angle
The coordinate transformation from the
or
The inverse coordinate transformation from the
or
Relationship with 2D Coordinate Mapping
From my previous blog post “2D Plane Transformation”, we learned that in a Cartesian coordinate system the rotation of a point
The mapping from the point
or
The inverse mapping from the point
or
Therefore, the point coordinates transformation from one Cartesian coordinate system to another Cartesian coordinate system that is rotated counterclockwise around the origin through an angle
References
Rotation and Translation of Axes