## Revision 526408e4d4d3e277eb6f7644f579439923d38a50 (click the page title to view the current version)

# Change of Basis

# Motion examples

- Translation \[\vec{x}' = \vec{x}+\vec{t}\]
- Rotation \[\vec{x}' = \vec{x}\cdot R\]
- But note that \(R\) is not an arbitrary matrix.
- We’ll return to the restrictions

# Definition: Rigid Body Motion

- 3D Object is a set of points in \(\mathbb{R}^3\)
- If the object moves, the constituent points move
- The points have to move so that they preserve the shape of the object

## Constraints

Let \(\vec{X}(t)\) and \(\vec{Y}(t)\) be the coordinates of points \(\vec{x}\) and \(\vec{x}\) at time \(t\).

- Preserve distance between points
- \(||\vec{X}(t)-\vec{Y}(t)||\) is constant

- Preserve orientation
- i.e. avoid mirroring
- we have to preserve cross-products
- If the right hand rule turns into a left hand rule, we have had mirroring.

Let \(u=\vec{X}-\vec{Y}\) be a vector, and \(g_*(u)=g(\vec{X})-g(\vec{Y})\) the corresponding vector after motion.

Preserving the cross-product means \[g_*(u)\times g_*(v) = g_*(u\times v), \forall u,v\in\mathbb{R}^3\]