I am trying to do the following:

  1. Interprete RGB color values as a 3-D vektor
  2. Transform the RGB color into polar coordinates L, teta and phi
  3. Create colors with constant L

The math behind this is the standard transformation into spherical coordinates:

b = L * Sin(teta) * Cos(phi)
r = L * Sin(teta) * Sin(phi) 
g = L * Cos(teta)

With L beeing the norm of the RGB vektor I want to create.

L=(r^2+b^2+g^2)^0.5 'between 0 and 441.67 = sqrt(3)*255

So far so good. Onto the first issue: r,g and b need to be between 0 and 255

This means that this holds true (I calculated that myself, it's possible that it has errors in it):

If L>255 Then
   arccos(255/L) < teta < pi/2 - arccos(255/L)
   arccos(255/L) < phi < pi/2 - arccos(255/L)
End If

Here my calculations: enter image description here

The idea is that teta goes from (pi/4 - delta) to (pi/4 + delta) where Which basically means that teta and phi are restricted in what they can be according to the ratio of 255/L. This restriction gets upto a point were L=441.67 the maximum length. Here phi and teta can only be pi/4.

I think upto this points my conclusions are valid, but there is a problem I cannot figure out.

If phi = teta = pi/4 and L = 441.67 the RGB value should be (255,255,255) and the forumlas above yield: RGB= (221,255,221)

Which is what the forumlas for r,g and b above yield if phi = teta = pi/4 and L = 441.67 are inserted.

So how would one generate RGB vektors with a constant norm that are still valid RGB values? (0 < r,g,b < 255)

  • $\begingroup$ How did you derive $\arccos(255/L) < \theta < \pi/2 - \arccos(255/L)$? Does your formulas yield (221, 255, 221)? $\endgroup$ – Apass.Jack Feb 27 at 14:16
  • $\begingroup$ @Apass.Jack I added the derivation of the angle description, d in this case specifically is 255. The formulas for x,y and,z in polar coordinates yield (221,255,221) for teta=phi=pi/4 $\endgroup$ – Lucas Raphael Pianegonda Feb 27 at 14:45
  • $\begingroup$ The start of the question makes it sound like you want to convert a RGB color to polar coordinates. If that's not what you want, perhaps it would help to edit that to be clearer. $\endgroup$ – D.W. Feb 27 at 15:26
  • $\begingroup$ Why do you say that "If phi = theta = pi/4 and L = 441.67 the RGB value should be (255,255,255)"? If $L = 255\sqrt 3$ and $L \cos\theta = 255$ then $\theta = \cos^{-1} \frac1{\sqrt3}$. $\endgroup$ – Peter Taylor Feb 28 at 12:01
  • $\begingroup$ @PeterTaylor This was an assumption that I thought should be true, since (255,255,255) is the room diagonal, I thought it should have 45° angles for teta and phi. For phi that is correct but as it turns out teta has a 54,7° angle = cos^-1(1/sqrt(3)). $\endgroup$ – Lucas Raphael Pianegonda Feb 28 at 12:54

So how would one generate RGB vectors with a constant norm that are still valid RGB values? ($0 \le r,g,b \le 255$)

There is a simpler algorithm. Let $L$ be the given norm.

  • Verify that $0\le L\le 255\sqrt3$.
  • Let r be a random number between $\sqrt{L^2-\min(2\cdot255^2, L^2)}$ and $\min(255, L)$.
  • Let $M=\sqrt{L^2-r^2}$. Let g be a random number between $\sqrt{M^2-\min(255^2, M^2)}$ and $\min(255,M)$.
  • Let $b =\sqrt{L^2-r^2-g^2}$.

Some minor adjustment is needed if $r,g,b$ should be integers, in which case only approximate solutions can be found in general.

Here is the right computation if we want to find the range of $\theta$ and $\phi$.

  • If $0\le L\le 255$, then $0\le\theta,\phi\le\pi/2$
  • If $255\lt L\le 255\sqrt3$, let $M=\sqrt{\max(0, L^2-2\cdot255^2})$. we have $$\begin{align} &\quad\quad\begin{cases}b = L \sin(\theta) \cos(\phi)\\ r = L \sin(\theta) \sin(\phi)\\ g = L \cos(\theta)\\ 0\le b,r,g\le 255\\ 0\le\theta,\phi\le\pi/2 \end{cases}\\ &\Longleftrightarrow \begin{cases}b = L \sin(\theta) \cos(\phi)\\ r = L \sin(\theta) \sin(\phi)\\ g = L \cos(\theta)\\ \sqrt{\max\left(0,(L\sin(\theta))^2-255^2\right)}\le b,r\le \min(255,L\sin(\theta)) \\ M\le g\le 255\\ 0\le\theta,\phi\le\pi/2,\, \theta\not=0 \end{cases}\\ &\Longleftrightarrow \begin{cases} \sqrt{\max\left(0,1-\left(\dfrac{255}{L\sin(\theta}\right)^2\right)}\le \cos(\phi)\le \min\left(1, \dfrac{255}{L\sin(\theta)}\right)\\ \sqrt{\max\left(0,1-\left(\dfrac{255}{L\sin(\theta}\right)^2\right)}\le \sin(\phi)\le \min\left(1, \dfrac{255}{L\sin(\theta)}\right)\\ \dfrac{M}{L}\le \cos(\theta)\le \dfrac{255}L\\ 0\le\theta,\phi\le\pi/2,\, \theta\not=0 \end{cases}\\ &(\text{the inequalities on}\cos\phi\text{ put on the same restrictions on }\phi\\ &\text { as the inequalities on }\sin\phi)\\ &\Longleftrightarrow \begin{cases} \arccos\left(\min\left(1, \dfrac{255}{L\sin(\theta)}\right)\right)\le \phi\\ \phi\le\arccos\left(\sqrt{\max\left(0,1-\left(\dfrac{255}{L\sin(\theta)}\right)^2\right)}\right)\\ \arccos(\dfrac{255}L)\le\theta\le\arccos(\dfrac{M}{L}) \end{cases} \end{align} $$

The formula above tells a way to generate RGB vectors with a given norm $L$. Select $\theta$ and $\phi$ in the range given by the inequalities. Compute $r,g,b$ accordingly.

In particular, if you take $L=255\sqrt3=441.67$, then $\dfrac{255}L=\dfrac{M}L=\dfrac3{\sqrt3}$. $\theta=\arccos(255/L)=0.9553$. Since $\dfrac{255}{L\sin(\theta)}=0.7071$, we will get $\arccos(0.7071)\le \phi\le\arccos(0.7171)$. So $\phi=\pi/4$ and we will arrive at $b=255$, $g=255$ and $r=255$, which is the only possible combination of RGB vector for that value of $L$, as expected.

There have been several computational mistakes in the question. There has been a few in the previous versions of this answer as well. The comments below this answer might have become irrelevant.

  • $\begingroup$ 1. Instead of Lsin(teta) I used L assuming that L would be normal to teta. $\endgroup$ – Lucas Raphael Pianegonda Feb 28 at 7:28
  • $\begingroup$ The "much simpler algorithm" doesn't really work. You have no system that keeps r,g or b below 255. This only works for L<255, there it works fine but for greater L it doesn't yield valid r,g,b tripletts. $\endgroup$ – Lucas Raphael Pianegonda Feb 28 at 7:58
  • 1
    $\begingroup$ Oh, right. I meant something, but wrote something total different. I am correcting it. Just a moment or two. $\endgroup$ – Apass.Jack Feb 28 at 8:00
  • $\begingroup$ Another potential error: the minimal teta is arccos(255/L) which is between 0 and 0.955 as you already pointed out. But if you'd insert teta =0 which is fine from the teta perspective and insert it into the restrictions for phi, you'll get a divided by 0 because sin(teta) is zero. $\endgroup$ – Lucas Raphael Pianegonda Feb 28 at 8:30
  • $\begingroup$ It looks like the exercise should be "Find at least two places where this answer made a mistake." ... I will update in a moment. $\endgroup$ – Apass.Jack Feb 28 at 8:46

Generate a random three-dimensional vector $v$ with non-negative components (you can generate each of the three components at random). Then, fix up the norm to be $L$ by setting

$$w = L \cdot {v \over \|v\|}.$$

Now $w$ will have norm $L$. Interpret $w$ as a RGB color.

Next, check that all three components of $w$ are in the range 0..255. If yes, you've got your RGB color. If not, throw it away and go back to the beginning and generate a new $v$, and repeat. (This is called rejection sampling.)


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