I am stuck and a little embarrassed.

I'm trying to implement this answer. I have a unit vector $w$ pointing in the direction of a radio source on the celestial sphere.

$u$ and $v$ are the two other, mutually orthogonal unit vectors that point east and north from that point, tangent to the sphere.

Without back-converting to R.A. and Dec or using trigonometry, is there a simple way to generate $u$ and $v$ vectorially?

Slide 38 here may or may not be helpful, I think my description is sufficient by itself.

  • $\begingroup$ How did you determine w? $\endgroup$
    – Mike G
    Apr 11, 2019 at 11:42
  • $\begingroup$ @MikeG In my question, $w$ is a given. It can come from any source. If it comes from orbital mechanics, it's the normalized relative position vector $\mathbf{r}/|r|$, or it can come from an initial RA/Dec value that's been spun around the Earth's axis for 24 hours like I did here when I built SgrA_star (which is what would be $w$), or it can be from somewhere else entirely. From any given $w$ I'm asking how to get $u$ and $v$ without going back to trigonometry. Can it be done using just vectors somehow? $\endgroup$
    – uhoh
    Apr 11, 2019 at 11:49

1 Answer 1


If $\mathbf{\hat{n}}$ points to the north celestial pole, then the eastward tangent vector is

$$\mathbf{\hat{u} = \frac{\hat{n} \times \hat{w}}{\|\hat{n} \times \hat{w}\|}}$$

and the northward tangent vector is

$$\mathbf{\hat{v} = \frac{\hat{w} \times \hat{u}}{\|\hat{w} \times \hat{u}\|}}$$

$\mathbf{\|\hat{w} \times \hat{u}\|}$ analytically should be 1 but can be slightly different numerically.

  • $\begingroup$ Bingo! renormalize, facepalm, repeat... $\endgroup$
    – uhoh
    Apr 11, 2019 at 21:00

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