Undergrad Bilateral and unilateral constraints

[hmparticle9]

I am going through this course on collision detection: https://siggraphcontact.github.io/

In this link is a PDF called course notes. Scrolling down to section 1.3, called constraints.

In this section it is said that we can write bilateral constraints as $\phi(\mathbf{x}) = 0$ and unilateral constraints as $\phi(\mathbf{x}) \ge 0$.

I understand that, but then it says that these constraints call also be written as:
$$\mathbf{J} \mathbf{u} = 0, \mathbf{J} \mathbf{u} \ge 0,$$
where
$$\mathbf{J} = \frac{\partial \phi}{\partial \mathbf{q}} \in \mathbb{R}^{m \times n},$$
and $\mathbf{u}$ is the velocity vector.

I am struggling with most of section 1.3, I will add bits as we progress.

 

[wrobel]

strange story indeed