9. Appendix

9.1. Virtual slits

The slit length sl, the slit width sw and the orientation so of the virtual slit, optimized to maintain the spectral resolution in slitless spectroscopy, is computed from the major axis size a_img, the minor axis size b_img and the major axis angle theta_img for each object (the three quantities are given in the SExtractor catalogue columns A_IMAGE, B_IMAGE and THETA_IMAGE).

Without loss of generality we assume that the dispersion direction is parallel to the x-axis, which means the angle between the dispersion direction and the major axis angle (defined with respect to the x-axis) is theta_img. It is then:

\[\begin{split}\begin{aligned} A_{11} & = & (\cos({theta\_img}) / a\_img)^2 + (\sin{(theta\_img)} / b\_img)^2\\ A12 & = & \cos{(theta\_img)} * \sin{(theta\_img)} * (1.0/a\_img^2 - 1.0/b\_img^2)\\ A_{22} & = & (\sin({theta\_img}) / a\_img)^2 + (\cos{(theta\_img)} / b\_img)^2\\ \alpha & = & \arctan{(A12 / A11)}\\\nonumber \\ sl & = & \sqrt{A11} * a\_img * b\_img / \cos{(\alpha)} \\ sw & = & 1.0 / \sqrt{A11} \\ so & = & \alpha + 90.0\end{aligned}\end{split}\]

For further details see [FREUDLING]