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Gravitational lensing constrains a galaxy cluster's projected mass independently of its light output or dynamical equilibrium. The lensing convergence is projected surface mass density divided by ${\Sigma }_{\mathrm{crit}}=c^2{D}_s/(4\pi G{D}_d{D}_{ds})$, using the appropriate angular diameter distances. Strong gravitational lensing constrains the central mass through multiple-image geometry, and weak gravitational lensing constrains larger-scale projected mass through coherent image distortions. Converting these observations into a three-dimensional mass needs geometrical modelling and the relevant lensing degeneracies.

Deflection changes one light ray's direction; focusing concerns relative directions of neighboring light rays and their bundle's cross section. To see the difference for a point-mass gravitational lens, send parallel incident light rays past the lens. A ray at impact parameter $b$ meets the optical axis approximately $f(b)=b/\hat{\alpha }=c^2{b}^2/(4GM)$ beyond the lens. Doubling $b$ quadruples this crossing distance. Thus the lens does not have a common focal length for all rays, although it can produce lensing magnification and an Einstein ring for a specified source-observer geometry.

The point-mass gravitational lens also illustrates why focusing must not be identified just with positive lensing convergence. Away from the central mass, differentiate the reduced deflection $\mathbf{\alpha }={\theta }_E^2\mathbf{\theta }/r^2$: its two-dimensional divergence is zero, so $\kappa =0$ there. Nevertheless its radial and tangential derivatives differ, giving lensing shear of magnitude ${\theta }_E^2/r^2$ and $\det A=1-{\theta }_E^4/r^4$. A vacuum ray bundle can be distorted and magnified. In the same ideal transparent geometrical optics model, surface brightness is preserved; the larger apparent solid angle supplies the increased received light.

The local and mean lensing convergence obey $2\kappa =2\bar{\kappa }+r{\bar{\kappa }}^{\prime }$. Hence the radial derivative of the lens mapping is $1-{\alpha }^{\prime }(r)=1-2\kappa +\bar{\kappa }$. Its zeros are radial gravitational-lensing critical curves, where radial stretching diverges and images merge on crossing the mapped gravitational-lensing caustic.

The projected density $\Sigma (R)={\sigma }^2/(2GR)$ gives cylindrical mass $\pi {\sigma }^{2}R/G$, so the reduced deflection magnitude is the constant displayed angular Einstein radius. The signed thin gravitational lens equation is $\beta =\theta -{\theta }_E\mathrm{sgn}\theta$. A source with $0<\beta <{\theta }_E$ has two images; one with $\beta >{\theta }_E$ has one. The lensing convergence and lensing shear magnitude are both ${\theta }_E/(2|\theta |)$.

This circular lensing potential has reduced deflection $\alpha (r)={\theta }_{E}r/\sqrt{r^2+{\theta }_c^2}$, mean lensing convergence $\bar{\kappa }={\theta }_E/\sqrt{r^2+{\theta }_c^2}$ and local convergence $\kappa ={\theta }_E({\theta }_c^2+r^2/2)/(r^2+{\theta }_c^2{)}^{3/2}$. Its central convergence is finite for ${\theta }_c>0$. As the core shrinks to zero, the deflection tends to that of a singular isothermal sphere lens. This softening of the projected lensing potential must not be confused with replacing a three-dimensional density by a different cored profile.