= Image parity in gravitational lensing
{title2=$\operatorname{sgn}\det A$}
<Image parity in gravitational lensing> is the sign of the <determinant> of the source-from-image <Jacobian matrix> $A=\partial\boldsymbol\beta/\partial\boldsymbol\theta$. A positive sign preserves the local handedness of a small source pattern; a negative sign reverses it. Negative signed <lensing magnification> therefore means a reflected orientation, not negative received light. This is a two-dimensional property: the <derivative> of a signed radial equation alone does not determine it.
For a <point-mass gravitational lens>, differentiate $\boldsymbol\beta=(1-\theta_E^2/r^2)\boldsymbol\theta$, where $r=|\boldsymbol\theta|$. The radial and tangential <eigenvalues> of $A$ are
$$
\lambda_r=1+\frac{\theta_E^2}{r^2},\qquad\lambda_t=1-\frac{\theta_E^2}{r^2},\qquad\det A=1-\frac{\theta_E^4}{r^4}.
$$
Consequently the outer member of the <two images of a point-mass lens> has positive <image parity in gravitational lensing>; the inner member has negative <image parity in gravitational lensing>. At $r=\theta_E$ the tangential <eigenvalue> vanishes: this is the <tangential critical curve of an axisymmetric lens>, and its <gravitational-lensing caustic> is the aligned source point. The signed <lensing magnification> is $\mu=1/\det A$, but measured total light uses $|\mu_+|+|\mu_-|$.
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