The distances from chamfer are discrete distances using of the masks whose weights are entireties. The distances from chamfer are fast to calculate on a machine. Although they are more precise than the discrete distances d4 and d8, they provide only one approximation of the Euclidean distance.

Definition

A weighting is a couple $(\backslash \; mathbf\; \{X\},\; W)$ where $\backslash \; mathbf\; \{X\}$ is a point and $w$ an entirety. A mask $\backslash \; mathcal\; \{M\}$ is a finished whole of pondérations : $\backslash \; mathcal\; \{M\}\; =\; \backslash \; left\; \backslash \; \{(\backslash \; mathbf\; \{X\}\; \_i,\; w\_i)\; \backslash \; right\; \backslash \}$. A mask of chamfer is a mask having a central symmetry, whose weights are strictly positive and nonnull displacements.

It is possible to find a way between two points $p$ and $q$ by using displacements of a mask of chamfer. That is to say $\backslash \; mathcal\; \{M\}\; =\; \backslash \; left\; \backslash \; \{(\backslash \; mathbf\; \{X\}\; \_i,\; w\_i)\; \backslash \; right\; \backslash \}$ a mask of chamfer, then

$q\; =\; p\; +\; \backslash \; sum\; \backslash \; lambda\_i\; \backslash \; mathbf\; \{X\}\; \_i$,

where $\backslash \; lambda\_i$ is an entirety which corresponds to the number of times that displacement $\backslash \; mathbf\; \{X\}\; \_i$.

Outdistance chamfer

A distance from chamfer $d\_\; \{\backslash \; mathcal\; \{M\}\}$ between two points $p$ and $q$ of a discrete space is the minimal cost $\backslash \; sum\; \backslash \; lambda\_i\; w\_i$ of all the ways finite length $\backslash \; sum\; \backslash \; lambda\_i\; \backslash \; mathbf\; \{X\}\; \_i$ between $p$ and $q$ and using displacements $\backslash \; mathbf\; \{X\}\; \_i$ of the mask $\backslash \; mathcal\; \{M\}$ :

$d\_\; \{\backslash \; mathcal\; \{M\}\}\; =\; \backslash \; min\; \backslash \; left\; \backslash \; \{\backslash \; sum\; \backslash \; lambda\_i\; w\_i\; \backslash \; left|\; Q\; =\; p\; +\; \backslash \; sum\; \backslash \; lambda\_i\; \backslash \; mathbf\; \{X\}\; \_i\; \backslash \; right.\; \backslash right\backslash \}$.