Appendix C: The slant orthographic projection

The slant orthographic or generalized SIN projection derives from the basic interferometer equation (e.g. Thompson et al. 1986). The phase term in the Fourier exponent is

$$\wp = \vec{(e - e_0) \cdot B} ,$$ (C.1)

where $\vec{e_0}$ and $\vec{e}$ are unit vectors pointing towards the field center and a point in the field, $\vec{B}$ is the baseline vector, and we measure the phase $\wp$ in rotations so that we don't need to carry factors of $2\pi$. We can write

$$\wp = p_u u + p_v v + p_w w ,$$ (C.2)

where (u,v,w) are components of the baseline vector in a right-handed coordinate system with the w-axis pointing from the geocenter towards the source and the u-axis lying in the equatorial plane, and

$$\begin{array}{rccl} p_u & = & & \cos \theta \sin \phi , \\ ... ...eta \cos \phi , \\ p_w & = & & \sin \theta - 1 , \end{array}$$ (C.3)

are the coordinates of $\vec{(e - e_0)}$, where $(\phi,\theta)$ are the longitude and latitude of $\vec{e}$ in the spherical coordinate system with the pole towards $\vec{e_0}$ and origin of longitude towards negative v, as required by Fig. 3. Now, for a planar array we may write

n_u u + n_v v + n_w w = 0 (C.4)

where (n_u,n_v,n_w) are the direction cosines of the normal to the plane. Using this to eliminate w from Eq. (C.2) we have

$$\wp = \left[p_u - \frac{n_u}{n_w} p_w\right] u + \left[p_v - \frac{n_v}{n_w} p_w\right] v\cdot$$ (C.5)

Being the Fourier conjugate variables, the quantities in brackets become the Cartesian coordinates, in radians, in the plane of the synthesized map. Writing

$$\begin{array}{rcl} \xi & = & n_u / n_w , \\ \eta & = & n_v / n_w , \end{array}$$ (C.6)

Eqs. (61) and (62) are then readily derived from Eqs. (C.3) and (C.5). For 12-hour synthesis by an east-west interferometer the baselines all lie in the Earth's equatorial plane whence $(\xi,\eta) = (0, \cot\delta_0)$, where $\delta_0$ is the declination of the field center. For a "snapshot'' observation by an array such as the VLA, Cornwell & Perley (1992) give $(\xi,\eta) = (-\tan Z\sin\chi, \tan Z\cos\chi)$, where Z is the zenith angle and $\chi$ is the parallactic angle of the field center at the time of the observation.

In synthesizing a map a phase shift may be applied to the visibility data in order to translate the field center. If the shift applied is

$$\Delta\wp = q_u u + q_v v + q_w w$$ (C.7)

where (q_u,q_v,q_w) is constant then Eq. (C.2) becomes

$$\wp = (p_u - q_u) u + (p_v - q_v) v + (p_w - q_w) w ,$$ (C.8)

whence Eq. (C.5) becomes

$$\begin{array}{rcl} \wp & = & [(p_u - q_u) - \xi (p_w - q_w)] u ~ \\ & & +[(p_v - q_v) - \eta(p_w - q_w)] v . \end{array}$$ (C.9)

Equations (61) and (62) become

$$\hphantom{-}\frac{180\hbox{$^\circ$ }}{\pi}\left[~ \cos\theta \si... ...ta - 1)~ \right] - \frac{180\hbox{$^\circ$ }}{\pi}\left[q_u - \xi q_w \right] ,$$

$$- \frac{180\hbox{$^\circ$ }}{\pi}\left[~ \cos\theta \cos\phi - \e... ...a - 1)~ \right] - \frac{180\hbox{$^\circ$ }}{\pi}\left[q_v - \eta q_w \right] ,$$ (C.10)

from which on comparison with Eqs. (61) and (62) we see that the field center is shifted by

$$(\Delta x, \Delta y) = \frac{180\hbox{$^\circ$ }}{\pi}(q_u - \xi q_w, ~ q_v - \eta q_w) .$$ (C.11)

The shift is applied to the coordinate reference pixel.