# Clarification of notation used in differential games

I'm working through Rufus Isaacs's work on differential games and I need clarification on the notation used. Some context: The Value of the game is to be the minmax of the payoff which symbolically is

$$V(\mathbf{x})=\min\limits_{\phi(\mathbf{x})}\max\limits_{\psi(\mathbf{x})}\hspace{.2cm}(\text{payoff}).$$

The Lemma on Circular Vectorgrams is where I need help deciphering:

LEMMA 2.8.1. Let $$u$$, $$v$$ be any two numbers such that $$\rho=\sqrt{u^2+v^2}>0.$$ Then $$V(\mathbf{x})=\min\limits_{\phi(\mathbf{x})}\max\limits_{\psi(\mathbf{x})}\hspace{.2cm}(u\cos\phi+v\sin\phi).$$ is furnished by $$\bar{\phi}$$, where $$\cos\bar{\phi}=+[-]\frac{u}{\rho},\hspace{.4cm}\sin\bar{\phi}=+[-]\frac{v}{\rho}.$$ and the max[min] itself is $$+[-]\rho.$$

What does the author mean when there are these plus and minus signs by themselves?

• It's from his book "Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization" – Peetrius Mar 23 at 22:16

LEMMA 2.8.1. Let $$u$$, $$v$$ be any two numbers such that $$\rho=\sqrt{u^2+v^2}>0.$$ Then $$\max\limits_{\phi}\,[\min\limits_{\phi}]\hspace{.2cm}(u\cos\phi+v\sin\phi).$$ is furnished by $$\bar{\phi}$$, where $$\cos\bar{\phi}=+[-]\frac{u}{\rho},\hspace{.4cm}\sin\bar{\phi}=+[-]\frac{v}{\rho}.$$ and the max[min] itself is $$+[-]\rho.$$
LEMMA 2.8.1. Let $$u$$, $$v$$ be any two numbers such that $$\rho=\sqrt{u^2+v^2}>0.$$ Then $$\max\limits_{\phi}\hspace{.2cm}(u\cos\phi+v\sin\phi).$$ is furnished by $$\bar{\phi}$$, where $$\cos\bar{\phi}=+\frac{\vert u\vert}{\rho},\hspace{.4cm}\sin\bar{\phi}=+\frac{\vert v\vert}{\rho}.$$ and the max itself is $$+\rho\ ,$$ while $$\min\limits_{\phi}\hspace{.2cm}(u\cos\phi+v\sin\phi).$$ is furnished by $$\tilde{\phi}$$, where $$\cos\tilde{\phi}=-\frac{\vert u\vert}{\rho},\hspace{.4cm}\sin\tilde{\phi}=-\frac{\vert v\vert}{\rho}.$$ and the min itself is $$-\rho\ .$$