Motion examples

1. Translation \[\vec{x}' = \vec{x}+\vec{t}\]
2. 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

1. 3D Object is a set of points in \(\mathbb{R}^3\)
2. If the object moves, the constituent points move
3. 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 \(t\).

1. Preserve distance between points
   • \(||\vec{X}(t)-\vec{Y}(t)||\) is constant
2. 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 \[g_*(u)\times g_*(v) = g_*(u\times v), \forall u,v\in\mathbb{R}^3\]

Change of Basis

Bases