I'm reading Forsyth and Ponce's computer vision a modern approach. In Section 2.2.2., the shape of specularities, there is an equation I am not sure about. The authors write
We consider a smooth specular surface and assume that the brightness reflected in the direction V is a function of $V \cdot P$, where $P$ is the specular direction. We expect the specularity to be small and isolated, so we can assume the source direction $S$ and the viewing direction $V$ are constant over its extent. Let us further assume that the specularity can be defined by a threshold on the specular energy, i.e $V \cdot P \geq 1 - \epsilon$ for some constant $\epsilon$, denote by $N$, the unit surface normal and define the half-angle direction as $H=(S+V)/2$. Using the fact that the vectors $S,V,P$ have unit length and a whit of plane geometry, it can easily be shown that the boundary of the specularity is defined by
$$ 1 - \epsilon = V \cdot P = 2 \frac{(H \cdot N)^2}{H \cdot H} -1 $$
I'm not sure why $V \cdot P = 2 \frac{(H \cdot N)^2}{H \cdot H} -1$, not sure where to start really.
$$ V\cdot P=\cos\alpha=2\cos^2{\alpha\over2}-1= 2{(H\cdot N)^2\over H\cdot H}-1 $$