Table 4.

Alterations of the $W_{lm}^j$ in Table 2 required for projection isocenters translated off the shim set isocenter by distance (x₀, y₀, z₀)

l	m	W	Woff center
1	0	$W_z^j$	$W_z^j+z_0$
1	1	$W_x^j$	$W_x^j+x_0$
1	−1	$W_y^j$	$W_y^j+y_0$
2	0	$W_{z^2}^j$	$W_{z^2}^j+2z_0{W}_z^j-x_0{W}_x^j-y_0{W}_y^j-\frac{1}{2}(x_0^2+y_0^2)$
2	1	$W_{xz}^j$	$W_{xz}^j+x_0{W}_z^j+z_0{W}_x^j+z_0{x}_0$
2	−1	$W_{yz}^j$	$W_{yz}^j+y_0{W}_z^j+z_0{W}_y^j+z_0{y}_0$
2	2	$W_{x^2-y^2}^j$	$W_{x^2-y^2}^j+2x_0{W}_x^j-2y_0{W}_y^j+(x_0^2-y_0^2)$
2	−2	$W_{xy}^j$	$W_{xy}^j+2(y_0{W}_x^j+x_0{W}_y^j)+2x_0{y}_0$
3	0	$W_{z^3}^j$	$W_{z^3}^j-3(x_0{W}_{xz}^j+y_0{W}_{yz}^j)+3z_0{W}_{z^2}^j+3W_z^j(z_0^2-\frac{1}{2}(x_0^2+y_0^2\left)\right)-3z_0(x_0{W}_x^j+y_0{W}_y^j)-\frac{3}{2}{z}_0(x_0^2+y_0^2)+z_0^3$
3	1	$W_{z^{2}x}^j$	$W_{z^{2}x}^j+x_0(W_{z^2}^j-\frac{1}{4}{W}_{x^2-y^2}^j)-\frac{1}{4}{y}_0{W}_{xy}^j-\frac{1}{2}{x}_0{y}_0{W}_y^j+(z_0^2-\frac{1}{2}{y}_0^2-\frac{3}{4}{x}_0^2)W_x^j+2z_0(W_{xz}^j+x_0{W}_z^j)-\frac{1}{4}{x}_0(x_0^2+y_0^2-4z_0^2)$
3	−1	$W_{z^{2}y}^j$	$W_{z^{2}y}^j+y_0(W_{z^2}^j-\frac{1}{4}{W}_{x^2-y^2}^j)-\frac{1}{4}{x}_0{W}_{xy}^j-\frac{1}{2}{x}_0{y}_0{W}_x^j+(z_0^2-\frac{1}{2}{y}_0^2-\frac{3}{4}{y}_0^2)W_y^j+2z_0(W_{xz}^j+y_0{W}_z^j)-\frac{1}{4}{y}_0(y_0^2+x_0^2-4z_0^2)$
3	2	$W_{z(x^2-y^2)}^j$	$W_{z(x^2-y^2)}^j+2(x_0{W}_{xz}^j+(x_0^2-y_0^2)W_z^j-y_0{W}_{yz}^j+z_0(x_0{W}_x^j-y_0{W}_y^j\left)\right)+z_0{W}_{x^2-y^2}^j+z_0(x_0^2-y_0^2)$
3	3	$W_{y^3}^j$	$W_{y^3}^j+3(-2x_0{W}_{xy}^j+(y_0^2-x_0^2)W_y^j-y_0{W}_{x^2-y^2}^j+2x_0{y}_0{W}_x^j)+y_0^3-3x_0^2{y}_0$
3	−3	$W_{x^3}^j$	$W_{x^3}^j+3(2y_0{W}_{xy}^j-(y_0^2-x_0^2)W_x^j-x_0{W}_{x^2-y^2}^j-2x_0{y}_0{W}_y^j)-x_0^3+3x_0^2{y}_0$