sections of a sphere

[Eliot]

I have a ball in 3d space (x,y,z)
the ball is divided to 12 equal slices - A,B,C,D,E,F,G,H,I,J,K,L
the center of the ball is (127.5, 127.5,127.5)
the radius is 127.5

I need the equation of each section
So when I get a point(x,y,z) inside the ball
I can return the section which contain the point

from the point of view of x,y it looks like this:

 

[lev888]

I have a ball in 3d space (x,y,z)
the ball is divided to 12 equal slices - A,B,C,D,E,F,G,H,I,J,K,L
the center of the ball is (127.5, 127.5,127.5)
the radius is 127.5

I need the equation of each section
So when I get a point(x,y,z) inside the ball
I can return the section which contain the point

from the point of view of x,y it looks like this:
View attachment 18967

First of all, if the sphere is divided by planes normal to the xy plane, you can consider it a 2d problem. Provided, of course, you are guaranteed that the points are inside the sphere. Or you can check that first by computing the radius and then disregard the z coordinate.
So, given a point, what determines which section the point is in?

 

[Eliot]

First of all, if the sphere is divided by planes normal to the xy plane, you can consider it a 2d problem. Provided, of course, you are guaranteed that the points are inside the sphere. Or you can check that first by computing the radius and then disregard the z coordinate.
So, given a point, what determines which section the point is in?

First of all, if the sphere is divided by planes normal to the xy plane, you can consider it a 2d problem. Provided, of course, you are guaranteed that the points are inside the sphere. Or you can check that first by computing the radius and then disregard the z coordinate.
So, given a point, what determines which section the point is in?

Thank you very match for your quick answer. The problem in disregarding the z is that ,for example, 2 points: point(120,120,7) ,(120,120,120) are in the same section according to xy plane but they are not in the same section in x,y,z space. How can I find the equation of each section? sorry for my ignorance , but I've study this material a long time ago , so even if you'll just show me how a general equation of the shape of a section looks like in 3d space will help me get started ... i think ? . thanks in advance, Eliot

 

[lev888]

point(120,120,7) ,(120,120,120) are in the same section according to xy plane but they are not in the same section in x,y,z space.

Ok, what are the 2 different sections?

 

[Eliot]

for example
if the ball's center is (0,0,0)
and its diameter is 1
then 2 points are equal in x,y but differ in z
p1 is part of B and P2 is part of E

So what i try to find is by given any (x,y,z) with in the ball, to which section it belongs (i.e A | B | C | D ...) As far as I understand , if i have the equation of each section , I'll be able to test which equation works with the specific point if you have other ideas I'll be happy to here. Or the way I can find those equations.

here is a look of this ball example from perspective of x,y,z



Thanks in advanced,
Eliot

 

[lev888]

for example
if the ball's center is (0,0,0)
and its diameter is 1
then 2 points are equal in x,y but differ in z
p1 is part of B and P2 is part of E
View attachment 18997
So what i try to find is by given any (x,y,z) with in the ball, to which section it belongs (i.e A | B | C | D ...) As far as I understand , if i have the equation of each section , I'll be able to test which equation works with the specific point if you have other ideas I'll be happy to here. Or the way I can find those equations.

here is a look of this ball example from perspective of x,y,z

View attachment 18998

Thanks in advanced,
Eliot

In your first post you wrote: "from the point of view of x,y it looks like this: "
In my reply I wrote: "First of all, if the sphere is divided by planes normal to the xy plane, you can consider it a 2d problem "
Now you are showing the sphere as divided by planes normal to the xz plane.
So, which is it?
Regardless of which coordinate plane it is, we can view this problem as 2d in that plane.

 

[Eliot]

Sorry I need in 3d , I just show my intention from x,y perspective and from x,z perspective - just because i find it hard to draw it in 3d.
but what i need is the equation of E section, for Example in the 3d space of (x,y,z)

 

[lev888]

Sorry I need in 3d , I just show my intention from x,y perspective and from x,z perspective - just because i find it hard to draw it in 3d.
but what i need is the equation of E section, for Example in the 3d space of (x,y,z)

Even in general 3d case, the sphere _appears_ to be divided by planes passing through the same axis, therefore, you can use the normal plane of the axis and deal with lines in that plane instead of planes in 3d space. Up to you.
Regarding equations of sections - we can use equations to represent the 2 bounding planes and the surface of the sphere, but I don't know how to represent each section by "an equation".
Consider a simple example: given coordinates of square vertices in xy plane determine whether a given point is inside it. Can you represent the square by "an equation"?

 

[Eliot]

Here is the shape of a section in 3D each point (x,y,z) can also be inside of the shape or on it's surface



Do you still think I can solve it with 2d planes? This is for an application I'm building . So I need to find a way by programming this is why I'm looking for equation or absolute definition for this space. Inside this space there are infinite planes (not that I know how to find them because of the arc) ... Anyway any direction can help

 

[lev888]

Here is the shape of a section in 3D each point (x,y,z) can also be inside of the shape or on it's surface

View attachment 19031

Do you still think I can solve it with 2d planes? This is for an application I'm building . So I need to find a way by programming this is why I'm looking for equation or absolute definition for this space. Inside this space there are infinite planes (not that I know how to find them because of the arc) ... Anyway any direction can help

1. Calculate the distance from the point to the center of the sphere to check whether it's inside.
2. Project the point onto the plane normal to the axis.
3. Now you have a point in circle divided into 12 sectors on a plane. Each sector is defined by a range of the central angle. To find which sector the point is in what measure do you need to calculate?

 

[Eliot]

lets see if I got it:
suppose the point is p(x,y,z),
the center is a(a1,a2,a3),
r is the radios
S is a plane horizontal to xz plane and contain the point a (half of the ball)



Is this the correct solution?

Thanks in advance ,
Eliot

 

[lev888]

lets see if I got it:
suppose the point is p(x,y,z),
the center is a(a1,a2,a3),
r is the radios
S is a plane horizontal to xz plane and contain the point a (half of the ball)

View attachment 19051

Is this the correct solution?

Thanks in advance ,
Eliot

#3 is not very clear (sin calculation). A diagram would help. Anyway, just look up how to convert cartesian to polar coordinates and make sure the resulting angle range is 0 to 360.