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Is there a name for this class of 'surfaces'? Have they been studied?

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4

$begingroup$

I found a page in one of my notebooks describing a particular class of 'surfaces' which are given by the implicit equations:

$$frac{1}{x}+frac{1}{y}+frac{1}{z}=frac{1}{x+y+z}$$

Or, more generally

$$sum_i left(a_ix_i+b_iright)^{-1}=left(sum_ia_ix_i+b_iright)^{-1}quad:quadmathbf{x}inmathbb{R}^n$$

I only have notes on $mathbf{x}inmathbb{R}^2,mathbb{R}^3$ and I'm not sure what the context was at the time I wrote it, but it looks interesting enough and I would like to know more.

Do these objects have a name? Have they been studied before?

---

Edit:

After working on it for a bit, it occurred to me that in three dimensions this is the same as

$$(u_1+u_2)(u_1+u_3)(u_2+u_3)=0quad:quad u_i=a_ix_i+b_i$$

Which is a particular case of

$$({u_1}^n+{u_2}^n)({u_1}^n+{u_3}^n)({u_2}^n+{u_3}^n)=0$$

For whatever reason, this last bit seems incredibly familiar. I'm not sure why, but it brings to mind something about quaternions, vector norms, and general relativity. Where have I seen this before?

algebraic-geometry reference-request terminology complex-geometry

share|cite|improve this question

edited Mar 21 at 21:57

R. Burton

asked Mar 19 at 16:39

R. BurtonR. Burton

732110

$endgroup$

add a comment | 

4

$begingroup$

I found a page in one of my notebooks describing a particular class of 'surfaces' which are given by the implicit equations:

$$frac{1}{x}+frac{1}{y}+frac{1}{z}=frac{1}{x+y+z}$$

Or, more generally

$$sum_i left(a_ix_i+b_iright)^{-1}=left(sum_ia_ix_i+b_iright)^{-1}quad:quadmathbf{x}inmathbb{R}^n$$

I only have notes on $mathbf{x}inmathbb{R}^2,mathbb{R}^3$ and I'm not sure what the context was at the time I wrote it, but it looks interesting enough and I would like to know more.

Do these objects have a name? Have they been studied before?

---

Edit:

After working on it for a bit, it occurred to me that in three dimensions this is the same as

$$(u_1+u_2)(u_1+u_3)(u_2+u_3)=0quad:quad u_i=a_ix_i+b_i$$

Which is a particular case of

$$({u_1}^n+{u_2}^n)({u_1}^n+{u_3}^n)({u_2}^n+{u_3}^n)=0$$

For whatever reason, this last bit seems incredibly familiar. I'm not sure why, but it brings to mind something about quaternions, vector norms, and general relativity. Where have I seen this before?

algebraic-geometry reference-request terminology complex-geometry

share|cite|improve this question

edited Mar 21 at 21:57

R. Burton

asked Mar 19 at 16:39

R. BurtonR. Burton

732110

$endgroup$

add a comment | 

$begingroup$

I found a page in one of my notebooks describing a particular class of 'surfaces' which are given by the implicit equations:

$$frac{1}{x}+frac{1}{y}+frac{1}{z}=frac{1}{x+y+z}$$

Or, more generally

$$sum_i left(a_ix_i+b_iright)^{-1}=left(sum_ia_ix_i+b_iright)^{-1}quad:quadmathbf{x}inmathbb{R}^n$$

I only have notes on $mathbf{x}inmathbb{R}^2,mathbb{R}^3$ and I'm not sure what the context was at the time I wrote it, but it looks interesting enough and I would like to know more.

Do these objects have a name? Have they been studied before?

---

Edit:

After working on it for a bit, it occurred to me that in three dimensions this is the same as

$$(u_1+u_2)(u_1+u_3)(u_2+u_3)=0quad:quad u_i=a_ix_i+b_i$$

Which is a particular case of

$$({u_1}^n+{u_2}^n)({u_1}^n+{u_3}^n)({u_2}^n+{u_3}^n)=0$$

For whatever reason, this last bit seems incredibly familiar. I'm not sure why, but it brings to mind something about quaternions, vector norms, and general relativity. Where have I seen this before?

algebraic-geometry reference-request terminology complex-geometry

share|cite|improve this question

edited Mar 21 at 21:57

R. Burton

asked Mar 19 at 16:39

R. BurtonR. Burton

732110

$endgroup$

share|cite|improve this question

edited Mar 21 at 21:57

R. Burton

asked Mar 19 at 16:39

R. BurtonR. Burton

732110

share|cite|improve this question

edited Mar 21 at 21:57

R. Burton

asked Mar 19 at 16:39

R. BurtonR. Burton

732110

asked Mar 19 at 16:39

R. BurtonR. Burton

732110

asked Mar 19 at 16:39

R. BurtonR. Burton

732110