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Given an ellipsoid equation like $$ a x^2 + by^2 + cz^2 + 2 h xy + 2g xz + 2f yz + px + qy + rz + d = 0 $$ , where $x$, $y$ and $z$ are the coordinates in a space, what's the meaning of the constant term $d$?

My intuition tells me the term $d$ is related, in some way, to the volume of the ellipsoid, but I don't see it clear.

Plus, what exactly is the meaning of the other coeficcients?

LFanals
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    The fact that the ellipsoid is rotated in space greatly complicates reading information directly from the coefficients. For instance, $a$, $b$, $c$ alone do not typically define the axis lengths of a rotated ellipsoid, nor do $p$, $q$, $r$ define the center; so $d$ alone has no "meaning". In order to provide appropriate guidance here, it would help if you said something about your skill level in this area. – Blue Mar 30 '20 at 17:47
  • As you may have noticed, I am not an expert on this topic, or anything close to it. I am finishing a degree in EE. I've searched on Internet but I can't solve my question. How can I give meaning to every term of the ellipsoid equation I've written? – LFanals Mar 30 '20 at 17:57
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    To determine what an equation represents, one can manipulate it into "standard form" (without rotations or shifts). This process is already tedious with two-dimensional conics; a site search for "conic general standard form" gives posts showing what's involved. See also this answer, which gives formulas for various elements of a conic in terms of coefficients; note that, for a rotated conic, no individual coefficient has "meaning" in and of itself. Quadric surfaces are "worse". – Blue Mar 30 '20 at 18:13

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You know very well that an ellipsoid in euclidean space has three essential parameters, namely the lengths $\bar a$, $\bar b$, $\bar c$ of its semiaxes.

On the other hand, your displayed equation of an ellipsoid contains $10$ real parameters. Note that you can multiply your equation with an arbitrary real number $\ne0$ without changing the ellipsoid. This already tells you that the value of $d$ in your equation has no meaning whatsoever.

If the given equation contains ${\bf 3}$ linear terms $px+qy+rz$ you can get rid of them by a translation $x=x_0+x'$, $y=y_0+y'$, $z=z_0+z'$ with $(x_0,y_0,z_0)$ suitably chosen, and $(x',y',z')$ as new coordinates. The point $(x_0,y_0,z_0)$ is the center of your ellipsoid in the original $(x,y,z)$ coordinate system. Omitting the $'$ the ellipsoid now has the equation $$a_1x^2+b_1y^2+c_1z^2+2h_1xy+2g_1xz+2f_1yz=1\ ,\tag{1}$$ where the choice $1$ on the RHS means throwing out ${\bf 1}$ more superfluous parameter. On the LHS of $(1)$ we have a quadratic form $q(x,y,z)$. We all know that this form can be diagonalized, i.e., there exists a new orthonormal coordinate system $(\bar x,\bar y,\bar z)$ such that in these new coordinates $q$ appears as $$\bar q(\bar x,\bar y,\bar z)={\bar x^2\over \bar a^2}+{\bar y^2\over \bar b^2}+{\bar z^2\over \bar c^2}\ ,$$ so that $(1)$ is transformed into $${\bar x^2\over \bar a^2}+{\bar y^2\over \bar b^2}+{\bar z^2\over \bar c^2}=1\ .$$ This choice of a new coordinate system involves ${\bf 3}$ real parameters: $2$ for the direction of the $\bar z$-axis, and $1$ for a rotation around this axis in order to fix the $\bar x$-axis. In all we have, by choosing the "optimal" coordinate system, eliminated ${\bf 3}+{\bf 1}+{\bf 3}={\bf 7}$ "superfluous" constants in the given equation, and we are left with $\bar a$, $\bar b$, $\bar c$.

Christian Blatter
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