  We introduce two new notions called the Daugavet constant and
$\Delta$-constant of a point, which measure quantitatively how far the point is
from being Daugavet point and $\Delta$-point and allow us to study Daugavet and
$\Delta$-points in Banach spaces from a quantitative viewpoint. We show that
these notions can be viewed as a localized version of certain global
estimations of Daugavet and diametral local diameter two properties such as
Daugavet indices of thickness. As an intriguing example, we present the
existence of a Banach space $X$ in which all points on the unit sphere have
positive Daugavet constants despite the Daugavet indices of thickness of $X$
being zero. Moreover, using the Daugavet and $\Delta$-constants of points in
the unit sphere, we describe the existence of almost Daugavet and
$\Delta$-points as well as the set of denting points of the unit ball. We also
present exact values of the Daugavet and $\Delta$-constant on several classical
Banach spaces, as well as Lipschitz-free spaces. In particular, it is shown
that there is a Lipschitz-free space with a $\Delta$-point which is the
furthest away from being a Daugavet point. Finally, we provide some related
stability results concerning the Daugavet and $\Delta$-constant.
