**Table 1:** Description of the five element sets used in the computations of the LOV, and their respective native units and rescaling parameters.
Cartesian	x	y	z	v_x	v_y	v_z
Units	AU	AU	AU	AU/d	AU/d	AU/d
Scaling	r	r	r	v	v	v
Cometary	e	q	t_p	$\Omega$	$\omega$	i
Units	-	AU	d	rad	rad	rad
Scaling	1	q	Z	$2 \pi$	$2 \pi$	$\pi$
Keplerian	e	a	M	$\Omega$	$\omega$	i
Units	-	AU	rad	rad	rad	rad
Scaling	1	a	$2 \pi$	$2 \pi$	$2 \pi$	$\pi$
Equinoctial	a	h	k	p	q	$\lambda$
Units	AU	-	-	-	-	rad
Scaling	a	1	1	1	1	$2 \pi$
Attributable	$\alpha$	$\delta$	$\dot \alpha$	$\dot \delta$	r	$\dot r$
Units	rad	rad	rad/d	rad/d	AU	AU/d
Scaling	$2 \pi$	$\pi$	$n_\oplus$	$n_\oplus$	1	$n_\oplus$

NOTE: Here r and v are the heliocentric distance and velocity, respectively, d is one day. The angular rate $n_\oplus = 0.01720209895$ rad/day is approximately the mean motion of the Earth and is numerically equivalent to the Gaussian gravitational constant, $k = 0.01720209895~ ({\rm AU}^3/{\rm d}^2)^{-1/2}$ . The quantity $Z=2\pi q^{3/2}n_\oplus^{-1}(1-e)^{-1/2}$ is a characteristic time for a large eccentricity orbit.

Source LaTeX | All tables | In the text

	1 Introduction
	2 The line of variations
	3 Application 1: Impact monitoring
	4 Application 2: Orbit determination
	5 Application 3: Orbit identification
	6 Application 4: Qualitative analysis
	7 Conclusions
	Appendix A: Comparison of LOV definitions
	References