Calculate center point of spherical octant

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If I had a sphere and an octant like this, how would I find point $P$?

I can already calculate the other portions of the sphere using the parametric form equations, but I am unsure how I would combine to arrive at the $(x, y, z)$ of $P$ in the image.

For clarification in the image:

$O$ is the center of the sphere; it is not necessarily the origin of the space

Points $A, B, C, D, E, F$ are already known

Point $P$ is a point on the surface of the octant

I would've though that $P$ would've been the center of the arc from $B$ to $F$ in a similar fashion that $C$ is from $E$ to $F$.

A close approximation of this is to take the inital center point $(x, y, z)$ and make $P = (x + epsilon/sqrt3, y + epsilon/sqrt3, z + epsilon/sqrt3$) though I believe this to be incorrect.

geometry spheres spherical-geometry

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edited Jul 19 at 21:31

asked Jul 19 at 20:24

pstatix

1396

add a comment | 

up vote
0
down vote

favorite

If I had a sphere and an octant like this, how would I find point $P$?

I can already calculate the other portions of the sphere using the parametric form equations, but I am unsure how I would combine to arrive at the $(x, y, z)$ of $P$ in the image.

For clarification in the image:

$O$ is the center of the sphere; it is not necessarily the origin of the space

Points $A, B, C, D, E, F$ are already known

Point $P$ is a point on the surface of the octant

I would've though that $P$ would've been the center of the arc from $B$ to $F$ in a similar fashion that $C$ is from $E$ to $F$.

A close approximation of this is to take the inital center point $(x, y, z)$ and make $P = (x + epsilon/sqrt3, y + epsilon/sqrt3, z + epsilon/sqrt3$) though I believe this to be incorrect.

geometry spheres spherical-geometry

share|cite|improve this question

edited Jul 19 at 21:31

asked Jul 19 at 20:24

pstatix

1396

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

If I had a sphere and an octant like this, how would I find point $P$?

I can already calculate the other portions of the sphere using the parametric form equations, but I am unsure how I would combine to arrive at the $(x, y, z)$ of $P$ in the image.

For clarification in the image:

$O$ is the center of the sphere; it is not necessarily the origin of the space

Points $A, B, C, D, E, F$ are already known

Point $P$ is a point on the surface of the octant

I would've though that $P$ would've been the center of the arc from $B$ to $F$ in a similar fashion that $C$ is from $E$ to $F$.

A close approximation of this is to take the inital center point $(x, y, z)$ and make $P = (x + epsilon/sqrt3, y + epsilon/sqrt3, z + epsilon/sqrt3$) though I believe this to be incorrect.

geometry spheres spherical-geometry

share|cite|improve this question

edited Jul 19 at 21:31

asked Jul 19 at 20:24

pstatix

1396

If I had a sphere and an octant like this, how would I find point $P$?

I can already calculate the other portions of the sphere using the parametric form equations, but I am unsure how I would combine to arrive at the $(x, y, z)$ of $P$ in the image.

For clarification in the image:

$O$ is the center of the sphere; it is not necessarily the origin of the space

Points $A, B, C, D, E, F$ are already known

Point $P$ is a point on the surface of the octant

I would've though that $P$ would've been the center of the arc from $B$ to $F$ in a similar fashion that $C$ is from $E$ to $F$.

A close approximation of this is to take the inital center point $(x, y, z)$ and make $P = (x + epsilon/sqrt3, y + epsilon/sqrt3, z + epsilon/sqrt3$) though I believe this to be incorrect.

geometry spheres spherical-geometry

share|cite|improve this question

edited Jul 19 at 21:31

asked Jul 19 at 20:24

pstatix

1396

share|cite|improve this question

share|cite|improve this question

edited Jul 19 at 21:31

edited Jul 19 at 21:31

edited Jul 19 at 21:31

asked Jul 19 at 20:24

pstatix

1396

asked Jul 19 at 20:24

pstatix

1396

asked Jul 19 at 20:24

pstatix

1396

1396

add a comment | 

add a comment | 

1 Answer
1

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up vote
0
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As far as I can tell, you're asking for a point on the sphere.

By symmetry $x = y = z$. And given that $x^2 + y^2 + z^2 = e = 3 x^2$, we immediately find that $x = y = z = sqrte/3$.

share|cite|improve this answer

edited Jul 19 at 21:31

answered Jul 19 at 20:33

David G. Stork

7,6632929

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So the position would be $p = epsilon/sqrt3$; $(x + p, y + p, z + p)$?
  – pstatix
  Jul 19 at 20:49
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yes. (Though I was assuming that $p = (0,0,0)$.)
  – David G. Stork
  Jul 19 at 20:58
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Sure, fair assumption; need to implement so that initial $(x, y, z)$ is arbitrary with some radius $epsilon$ such that I can compute the center of the 8 octants.
  – pstatix
  Jul 19 at 20:59
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  This actually gives me the centroid of the octant, not the point on the surface
  – pstatix
  Jul 19 at 21:18
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Please re-state your question clearly and unambiguously. What is "center point"? Is point $p$ on the surface of the sphere? (yes or no) Is the radius of the sphere $1$ or $e$? When you say "octant", everyone understands the octant is defined by the origin $(0,0,0$ and three axes.
  – David G. Stork
  Jul 19 at 21:24
  
  

  
  
  
  

 | 
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1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
0
down vote

As far as I can tell, you're asking for a point on the sphere.

By symmetry $x = y = z$. And given that $x^2 + y^2 + z^2 = e = 3 x^2$, we immediately find that $x = y = z = sqrte/3$.

share|cite|improve this answer

edited Jul 19 at 21:31

answered Jul 19 at 20:33

David G. Stork

7,6632929

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So the position would be $p = epsilon/sqrt3$; $(x + p, y + p, z + p)$?
  – pstatix
  Jul 19 at 20:49
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yes. (Though I was assuming that $p = (0,0,0)$.)
  – David G. Stork
  Jul 19 at 20:58
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Sure, fair assumption; need to implement so that initial $(x, y, z)$ is arbitrary with some radius $epsilon$ such that I can compute the center of the 8 octants.
  – pstatix
  Jul 19 at 20:59
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  This actually gives me the centroid of the octant, not the point on the surface
  – pstatix
  Jul 19 at 21:18
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Please re-state your question clearly and unambiguously. What is "center point"? Is point $p$ on the surface of the sphere? (yes or no) Is the radius of the sphere $1$ or $e$? When you say "octant", everyone understands the octant is defined by the origin $(0,0,0$ and three axes.
  – David G. Stork
  Jul 19 at 21:24
  
  

  
  
  
  

 | 
show 2 more comments

up vote
0
down vote

As far as I can tell, you're asking for a point on the sphere.

By symmetry $x = y = z$. And given that $x^2 + y^2 + z^2 = e = 3 x^2$, we immediately find that $x = y = z = sqrte/3$.

share|cite|improve this answer

edited Jul 19 at 21:31

answered Jul 19 at 20:33

David G. Stork

7,6632929

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So the position would be $p = epsilon/sqrt3$; $(x + p, y + p, z + p)$?
  – pstatix
  Jul 19 at 20:49
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yes. (Though I was assuming that $p = (0,0,0)$.)
  – David G. Stork
  Jul 19 at 20:58
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Sure, fair assumption; need to implement so that initial $(x, y, z)$ is arbitrary with some radius $epsilon$ such that I can compute the center of the 8 octants.
  – pstatix
  Jul 19 at 20:59
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  This actually gives me the centroid of the octant, not the point on the surface
  – pstatix
  Jul 19 at 21:18
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Please re-state your question clearly and unambiguously. What is "center point"? Is point $p$ on the surface of the sphere? (yes or no) Is the radius of the sphere $1$ or $e$? When you say "octant", everyone understands the octant is defined by the origin $(0,0,0$ and three axes.
  – David G. Stork
  Jul 19 at 21:24
  
  

  
  
  
  

 | 
show 2 more comments

up vote
0
down vote

up vote
0
down vote

As far as I can tell, you're asking for a point on the sphere.

By symmetry $x = y = z$. And given that $x^2 + y^2 + z^2 = e = 3 x^2$, we immediately find that $x = y = z = sqrte/3$.

share|cite|improve this answer

edited Jul 19 at 21:31

answered Jul 19 at 20:33

David G. Stork

7,6632929

As far as I can tell, you're asking for a point on the sphere.

By symmetry $x = y = z$. And given that $x^2 + y^2 + z^2 = e = 3 x^2$, we immediately find that $x = y = z = sqrte/3$.

share|cite|improve this answer

edited Jul 19 at 21:31

answered Jul 19 at 20:33

David G. Stork

7,6632929

share|cite|improve this answer

share|cite|improve this answer

edited Jul 19 at 21:31

edited Jul 19 at 21:31

edited Jul 19 at 21:31

answered Jul 19 at 20:33

David G. Stork

7,6632929

answered Jul 19 at 20:33

David G. Stork

7,6632929

answered Jul 19 at 20:33

David G. Stork

7,6632929

7,6632929

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So the position would be $p = epsilon/sqrt3$; $(x + p, y + p, z + p)$?
  – pstatix
  Jul 19 at 20:49
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yes. (Though I was assuming that $p = (0,0,0)$.)
  – David G. Stork
  Jul 19 at 20:58
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Sure, fair assumption; need to implement so that initial $(x, y, z)$ is arbitrary with some radius $epsilon$ such that I can compute the center of the 8 octants.
  – pstatix
  Jul 19 at 20:59
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  This actually gives me the centroid of the octant, not the point on the surface
  – pstatix
  Jul 19 at 21:18
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Please re-state your question clearly and unambiguously. What is "center point"? Is point $p$ on the surface of the sphere? (yes or no) Is the radius of the sphere $1$ or $e$? When you say "octant", everyone understands the octant is defined by the origin $(0,0,0$ and three axes.
  – David G. Stork
  Jul 19 at 21:24
  
  

  
  
  
  

 | 
show 2 more comments

• 
  
  
  

  
  

  
  
  
  
  
  

  
  So the position would be $p = epsilon/sqrt3$; $(x + p, y + p, z + p)$?
  – pstatix
  Jul 19 at 20:49
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Yes. (Though I was assuming that $p = (0,0,0)$.)
  – David G. Stork
  Jul 19 at 20:58
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Sure, fair assumption; need to implement so that initial $(x, y, z)$ is arbitrary with some radius $epsilon$ such that I can compute the center of the 8 octants.
  – pstatix
  Jul 19 at 20:59
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  This actually gives me the centroid of the octant, not the point on the surface
  – pstatix
  Jul 19 at 21:18
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Please re-state your question clearly and unambiguously. What is "center point"? Is point $p$ on the surface of the sphere? (yes or no) Is the radius of the sphere $1$ or $e$? When you say "octant", everyone understands the octant is defined by the origin $(0,0,0$ and three axes.
  – David G. Stork
  Jul 19 at 21:24
  
  

  
  
  
  

So the position would be $p = epsilon/sqrt3$; $(x + p, y + p, z + p)$?
– pstatix
Jul 19 at 20:49

So the position would be $p = epsilon/sqrt3$; $(x + p, y + p, z + p)$?
– pstatix
Jul 19 at 20:49

Yes. (Though I was assuming that $p = (0,0,0)$.)
– David G. Stork
Jul 19 at 20:58

Yes. (Though I was assuming that $p = (0,0,0)$.)
– David G. Stork
Jul 19 at 20:58

Sure, fair assumption; need to implement so that initial $(x, y, z)$ is arbitrary with some radius $epsilon$ such that I can compute the center of the 8 octants.
– pstatix
Jul 19 at 20:59

Sure, fair assumption; need to implement so that initial $(x, y, z)$ is arbitrary with some radius $epsilon$ such that I can compute the center of the 8 octants.
– pstatix
Jul 19 at 20:59

This actually gives me the centroid of the octant, not the point on the surface
– pstatix
Jul 19 at 21:18

This actually gives me the centroid of the octant, not the point on the surface
– pstatix
Jul 19 at 21:18

Please re-state your question clearly and unambiguously. What is "center point"? Is point $p$ on the surface of the sphere? (yes or no) Is the radius of the sphere $1$ or $e$? When you say "octant", everyone understands the octant is defined by the origin $(0,0,0$ and three axes.
– David G. Stork
Jul 19 at 21:24

Please re-state your question clearly and unambiguously. What is "center point"? Is point $p$ on the surface of the sphere? (yes or no) Is the radius of the sphere $1$ or $e$? When you say "octant", everyone understands the octant is defined by the origin $(0,0,0$ and three axes.
– David G. Stork
Jul 19 at 21:24

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