0

Superquadrics are a family of geometric shapes defined by

$$|\frac{x}{A}|^r + |\frac{y}{B}|^s + |\frac{z}{C}|^t =1$$

I have two superquadratics, namely SQ1 and SQ2, with parameters $\{A_1,B_1,C_1,r_1,s_1,t_1\}$ and $\{A_2,B_2,C_2,r_2,s_2,t_2\}$, respectively. The volumes of these superquadratics are the equal, $V_{SQ1}=V_{SQ2}$. I am looking for a way to map SQ1 to SQ2. For example, if they were two ellipsoids, I could use affine transformation to map one ellipsoid to the other one.

Hossein
  • 17

1 Answers1

1

One way is to transform to spherical coordinates. Take any point on one superquadratic, find its angles, then find the point on the other superquadratic that is at the same angles. This does not depend on the volumes being equal.

Another is to set $\frac {x_1}{A_1}=\frac {x_2}{A_2}$ and similarly in $y$, then compute $z_2$ from $z_1$. You can favor any axis you want with this approach.

There are other mappings. You need to describe what makes a mapping be the right one.

Ross Millikan
  • 383,099
  • @ Ross Millikan Thank you for your help. I am looking for any topology-preserving mapping to take points inside the SQ1 and map them to the inside of SQ2. – Hossein Jun 25 '20 at 03:38
  • Both of these satisfy that. The boundaries are preserved, as are the interiors. – Ross Millikan Jun 25 '20 at 03:46
  • It seems that the second approach does not depend on $r,: s,: t$. Am I right? Also, as you said, regarding the first approach, it does not matter if they have the same volume or not. However, their volume should be exactly the same in order to map the interior of one (SQ1) to the interior of the other one (SQ2). Can you please make it clear? I think I did not get your point clearly. Thank you. – Hossein Jun 25 '20 at 04:04
  • 1
    In the second, they will come in when you solve for the third variable. Whether the volume match is important depends on what you want the mapping to preserve. Both approaches are bijections-they map one point to one point. I don't believe either one preserves local volumes-they stretch some areas and shrink others. There are volume preserving mappings as proven in the answer to this question of mine. – Ross Millikan Jun 25 '20 at 04:09