A moving point

Now consider the motion of a point in a 3D world $ {\mathbb{R}}^3$. Imagine that a geometric model, as defined in Section 3.1, is moving over time. This causes each point $ (x,y,z)$ on the model to move, resulting a function of time for each coordinate of each point:

$\displaystyle (x(t),y(t),z(t)) .$ (8.8)

The velocity $ v$ and acceleration $ a$ from Section 8.1.1 must therefore expand to have three coordinates. The velocity $ v$ is replaced by $ (v_x,v_y,v_z)$ to indicate velocity with respect to the $ x$, $ y$, and $ z$ coordinates, respectively. The magnitude of $ v$ is called the speed:

$\displaystyle \sqrt{v_x^2 + v_y^2 + v_z^2}$ (8.9)

Continuing further, the acceleration also expands to include three components: $ (a_x,a_y,a_z)$.

Steven M LaValle 2016-12-31