What is a first and second order cone?

Question

I am aware of the definition of a cone set $C$. $x_1, x_2 \in C$ and $\theta_1, \theta_2 \ge 0$

$$\theta_1x_1+\theta_2x_2 \in C$$

However, what is a first-order and second-order cone?

Specifically, relating to second order cone programming, why do we call the constraint

$$ \lVert Ax+b \rVert_2 \le c^Tx+d$$

a second order constraint and why do we call it a cone constraint?

Answer 1

It's called the second order cone because it's defined by a quadratic equation rather than a linear one. This naming is consistent with a quadratic polynomials being of degree two while a linear polynomial is of degree one.