Coordinate Transformations

Representing a Point in Space

A point $P$ in 3D space is represented using a vector (sometimes). Although, the coordinates of this vector depend entirely on the coordinate frame we are using as a reference.

Let's consider two different coordinate frames, {A} and {B}.

A point $P$ described in frame {A} is denoted by the vector $^{A} P$ .
The same point $P$ described in frame {B} is denoted by the vector $^{B} P$ .

$^{A} P = [\begin{matrix} p_{x} \\ p_{y} \\ p_{z} \end{matrix}]$

Each vector is just a set of three coordinates that locate the point $P$ along the axes of that particular frame.

Rigid Body Transformations

When we talk about transformations, we're often dealing with rigid bodies. This means the object itself doesn't bend or deform; it only changes its position (translation) and orientation (rotation) in space.

Translation: Describes the movement of a point from one location to another. We can represent this by adding a translation vector.
Rotation: Describes the change in orientation of a frame. This is more complex and requires a rotation matrix.

Rotation Matrix

A rotation matrix describes the orientation of one coordinate frame with respect to another. ( This statement might not always hold btw)

The notation $_{B}^{A} R$ represents the rotation matrix that maps the coordinates of a vector from frame {B} to frame {A}.

^{A} P =_{B}^{A} R^{B} P

This equation takes a point described in frame {B} ( $^{B} P$ ) and, by pre-multiplying it with the rotation matrix $_{B}^{A} R$ , gives us the coordinates of that same point as described in frame {A} ( $^{A} P$ ).

Deriving the Rotation Matrix

How do we find the values inside $_{B}^{A} R$ ? The matrix is constructed from the dot products of the unit vectors of the two frames.

Let the unit vectors for frame {A} be ${\hat{X}}_{A}, {\hat{Y}}_{A}, {\hat{Z}}_{A}$ and for frame {B} be ${\hat{X}}_{B}, {\hat{Y}}_{B}, {\hat{Z}}_{B}$ .

The matrix $_{B}^{A} R$ is a 3x3 matrix whose columns are the unit vectors of frame {B} expressed in frame {A}'s coordinates.

_{B}^{A} R = [\begin{matrix} ^{A} {\hat{X}}_{B} & ^{A} {\hat{Y}}_{B} & ^{A} {\hat{Z}}_{B} \end{matrix}]

Each of those column vectors can be found by projecting the {B} unit vectors onto the {A} axes:

Coordinate Transformations

Representing a Point in Space

A point $P$ in 3D space is represented using a vector. Although, the coordinates of this vector depend entirely on the coordinate frame we are using as a reference

Let's consider two different coordinate frames, {A} and {B}.

A point $P$ described in frame {A} is denoted by the vector $^{A} P$ .
The same point $P$ described in frame {B} is denoted by the vector $^{B} P$ .

$^{A} P = [\begin{matrix} p_{x} \\ p_{y} \\ p_{z} \end{matrix}]$

Each vector is just a set of three coordinates that locate the point $P$ along the axes of that particular frame.

Rigid Body Transformations

Translation: Describes the movement of a point from one location to another.
Rotation: Describes the change in orientation of a frame.

Rotation Matrix

The primary tool for handling rotations is the rotation matrix. A rotation matrix describes the orientation of one coordinate frame with respect to another.

Pasted image 20250919004914.png

The notation $_{B}^{A} R$ represents the rotation matrix that maps the coordinates of a vector from frame {B} to frame {A}. This mapping is expressed as:

^{A} P =_{B}^{A} R^{B} P

(Craig, 2014, Eq. 2.13)

Deriving the Rotation Matrix

The matrix $_{B}^{A} R$ is a 3x3 matrix whose columns are the unit vectors of frame {B} expressed in frame {A}'s coordinates

Let the unit vectors for frame {A} be ${\hat{X}}_{A}, {\hat{Y}}_{A}, {\hat{Z}}_{A}$ and for frame {B} be ${\hat{X}}_{B}, {\hat{Y}}_{B}, {\hat{Z}}_{B}$ .

_{B}^{A} R = [\begin{matrix} ^{A} {\hat{X}}_{B} & ^{A} {\hat{Y}}_{B} & ^{A} {\hat{Z}}_{B} \end{matrix}]

Each column vector is found by projecting the {B} unit vectors onto the {A} axes. This means the components of the rotation matrix are the dot products of the unit vectors of the two frames

_{B}^{A} R = [\begin{matrix} {\hat{X}}_{B} \cdot {\hat{X}}_{A} & {\hat{Y}}_{B} \cdot {\hat{X}}_{A} & {\hat{Z}}_{B} \cdot {\hat{X}}_{A} \\ {\hat{X}}_{B} \cdot {\hat{Y}}_{A} & {\hat{Y}}_{B} \cdot {\hat{Y}}_{A} & {\hat{Z}}_{B} \cdot {\hat{Y}}_{A} \\ {\hat{X}}_{B} \cdot {\hat{Z}}_{A} & {\hat{Y}}_{B} \cdot {\hat{Z}}_{A} & {\hat{Z}}_{B} \cdot {\hat{Z}}_{A} \end{matrix}]

(Craig, 2014, Eq. 2.3)

Properties of a Rotation Matrix

The matrix is orthogonal. Its columns (and rows) are mutually orthogonal unit vectors.
The inverse of a rotation matrix is equal to its transpose. $(_{B}^{A} R)^{- 1} = (_{B}^{A} R)^{T} =_{A}^{B} R$
The determinant is always +1.

now go read chapter 2 JJ Craig