Fullscreen map with draggable overlay panels. Use Settings to show/hide panels.
Variables: x, y, r = sqrt(x*x+y*y).
Use JavaScript/Math syntax, e.g. 0.3*Math.exp(-r*r/0.1).
If γt is blank, it is estimated from the radial average of κ.
The complex shear is \[ \gamma(p)=\gamma_1(p)+i\gamma_2(p). \] Natural components are denoted \(\Gamma_0,\Gamma_1,\Gamma_2,\Gamma_3\), and use the triangle centroid as the reference point. Tangential/cross products are denoted \(I_{abc}\), where \(a,b,c\in\{t,\times\}\). Legacy raw products are denoted \(S_0,S_1,S_2,S_3\); they are coordinate dependent.
The shear is treated as a spin-2 complex field \[ \gamma(p)=\gamma_1(p)+i\gamma_2(p). \] For the single axisymmetric halo in this demo, the physical shear is generated as \[ \gamma(p)=-\gamma_{t,h}(R)\exp(2i\phi), \] where \(\phi\) is the polar angle of \(p\) with respect to the halo center.
Let the triangle vertices be \(p_1,p_2,p_3\), and define the centroid \[ c=\frac{p_1+p_2+p_3}{3}. \] For each vertex, \[ \zeta_i=\arg(p_i-c). \] The centroid-reference projected shear is \[ \gamma(p_i;\zeta_i) = -\gamma(p_i)\exp(-2i\zeta_i). \]
The displayed natural components are \[ \begin{aligned} \Gamma_0 &= \gamma(p_1;\zeta_1) \gamma(p_2;\zeta_2) \gamma(p_3;\zeta_3),\\ \Gamma_1 &= \gamma^*(p_1;\zeta_1) \gamma(p_2;\zeta_2) \gamma(p_3;\zeta_3),\\ \Gamma_2 &= \gamma(p_1;\zeta_1) \gamma^*(p_2;\zeta_2) \gamma(p_3;\zeta_3),\\ \Gamma_3 &= \gamma(p_1;\zeta_1) \gamma(p_2;\zeta_2) \gamma^*(p_3;\zeta_3). \end{aligned} \] In a real estimator these would be ensemble or triplet-bin averages.
The projected shear at each vertex is decomposed as \[ \gamma(p_i;\zeta_i) = \gamma_t(p_i;\zeta_i) + i\gamma_\times(p_i;\zeta_i). \] The displayed real-valued components are \[ I_{abc} = \gamma_a(p_1;\zeta_1) \gamma_b(p_2;\zeta_2) \gamma_c(p_3;\zeta_3), \qquad a,b,c\in\{t,\times\}. \]
The eight components are \[ I_{ttt},\ I_{tt\times},\ I_{t\times t},\ I_{\times tt},\ I_{t\times\times},\ I_{\times t\times},\ I_{\times\times t},\ I_{\times\times\times}. \] These are the real tangential/cross basis associated with the same centroid reference used by \(\Gamma_0,\ldots,\Gamma_3\).
The legacy raw products are \[ \begin{aligned} S_0 &= \gamma(p_1)\gamma(p_2)\gamma(p_3),\\ S_1 &= \gamma^*(p_1)\gamma(p_2)\gamma(p_3),\\ S_2 &= \gamma(p_1)\gamma^*(p_2)\gamma(p_3),\\ S_3 &= \gamma(p_1)\gamma(p_2)\gamma^*(p_3). \end{aligned} \] These are useful for debugging, but they depend on the global coordinate frame and are not the preferred rotation-adapted visualization.
Press Record start, then drag vertices or use the controlled scale/angle
sliders. The lower plot records selected components as a function of the deformation step.
The horizontal dashed line marks zero whenever the plotted range crosses zero.