Negative radius in Spherical Coordinates

[TaoRich]

Hello Folks,

 

http://en.wikipedia....ordinate_system

Unique coordinates

Any spherical coordinate triplet (r, θ, φ) specifies a single point of three-dimensional space. On the other hand, every point has infinitely many equivalent spherical coordinates. One can add or subtract any number of full turns to either angular measure without changing the angles themselves, and therefore without changing the point. It is also convenient, in many contexts, to allow negative radial distances, with the convention that (−r, θ, φ) is equivalent to (r, θ+180°, φ) for any r, θ, and φ. Moreover, (r, −θ, φ) is equivalent to(r, θ, φ+180°).

 

[1] Can someone please explain that bold text in a bit more detail ?

• When is it convenient (under what contexts) ?

 

http://www.math10.co...c-geometry.html

EQUATION OF SPHERE IN SPHERICAL COORDINATES

r² + r₀² - 2r₀rsinθsinθ₀cos(φ - φ₀) = R²

 

where the sphere has center (r₀,θ₀,φ₀) in spherical coordinates and radius R.

 

If the center is at the origin the equation is: r = R.

[2] If we define a sphere in spherical coordinates, with the centre at the origin, and a positive radius R

• If we then change the sign of the radius to -R what are the implications

Where does this "new sphere" exist in relation to the old sphere ?

(a) in mathematical terms

(b) in practical reality

 

Cheers

Rich

[ydoaPs]

Just like a number line. -r is r units away from zero just like r is, but it is rotated 180 degrees.

[TaoRich]

Just like a number line. -r is r units away from zero just like r is, but it is rotated 180 degrees.

 

In your example, in Spherical Co-ordinates the centre is the zero-point of the number line.

Positive values of R will cover all points on the sphere.

 

Where are the negative values of R located ?

 

Rotating through 180 degrees makes no sense in the context of a 360 plane.

 

Considering that there are two 360 degree planes in the spherical co-ordinate system, it makes even less sense.

 

???

Edited March 14, 2011 by TaoRich

[khaled]

As I know in geometry, positive coordinates heads to the deep-right-top and negative heads out-left-bottom

.. and positive rotation is counter-clockwise & negative rotation is clockwise,

[Xittenn]

I don't think it is so much that there are defined negative radii from the centrally located zero point but that the negation of r in a spherical coordinate literally means r in the opposite direction. It is simply more convenient than doing math on a rotation that has been incremented by 180 degrees.

 

When is this appropriate? In almost any application where spherical coordinates itself is appropriate?

[imatfaal]

Rich - I think you are over complicating matters. To get a negative r flip the elevation over; it's a convention.

 

You could think of it in these terms; take your 'standard' spherical coordinate - you can envisage simplistically turning to your azimuthal angle, setting your elevation, and proceeding R units along that direction. For -R you proceed R units in the opposite direction - which is the same as flipping over your elevation or adding 180deg ( or [imath]\tau/2[/imath] radians)

 

Trying to tie this in with the equation of a sphere in spherical coordinates is a little pointless - every minus r can be equally well represented by a positive and vice versa. I don't know which contexts require this - but if you don't why the worry?

Edited March 14, 2011 by imatfaal

[TaoRich]

Let me try to express my question in a different way:

 

Consider a sphere alpha which expands in radial symmetry about an origin from radius r = 0 to radius r = R

 

So, at any given point in time, the edge of the sphere alpha can be determined by taking a vector radius r(t) and "swinging it around in all directions".

 

Then consider a sphere beta which expands in radial symmetry about the same origin from radius r = 0 to radius r = -R

 

Again, at any given point in time, the edge of the sphere beta can be determined by taking the vector radius r(t) and "swinging it around in all directions", except now the radius r(t) will be 0 or negative.

 

In relation to our original sphere alpha, where does sphere beta exist ?

• Our sphere alpha covers all positive value for r.
• Where do negative values for r lie.

Or expressed another way:

• Positive values of r are "outwards from our point of origin".
• How do we describe the "space" or "sphere" where r goes "inwards from our point of origin".

In my mind, sphere beta is the inverse of sphere alpha (and not some rotated view of alpha).

It is sphere alpha extruded in the opposite direction.

 

Is there no mathematical term (along the lines of orthogonal or something like that) which expresses the relationship between sphere/space alpha and sphere/space beta ?

[imatfaal]

Let me try to express my question in a different way:

 

Consider a sphere alpha which expands in radial symmetry about an origin from radius r = 0 to radius r = R

 

So, at any given point in time, the edge of the sphere alpha can be determined by taking a vector radius r(t) and "swinging it around in all directions".

 

Then consider a sphere beta which expands in radial symmetry about the same origin from radius r = 0 to radius r = -R

 

Again, at any given point in time, the edge of the sphere beta can be determined by taking the vector radius r(t) and "swinging it around in all directions", except now the radius r(t) will be 0 or negative.

 

In relation to our original sphere alpha, where does sphere beta exist ?

 

 

The two spheres are the same. You could define a sphere as the locus of all points a distance R away from a certain point - the same sphere could by convention be defined as the locus of all points -R from a certain point. Each point defined by a +R would be diametrically opposite the point defined by -R, but as you are looking at all points in order to create the sphere that doesn't really matter.

 

Our sphere alpha covers all positive value for r.

• Where do negative values for r lie.

Or expressed another way:

• Positive values of r are "outwards from our point of origin".
• How do we describe the "space" or "sphere" where r goes "inwards from our point of origin".

In my mind, sphere beta is the inverse of sphere alpha (and not some rotated view of alpha).

It is sphere alpha extruded in the opposite direction.

 

Is there no mathematical term (along the lines of orthogonal or something like that) which expresses the relationship between sphere/space alpha and sphere/space beta ?

 

It doesn't, in my thoughts, require any concept of a negative / different space and it isn't an inverse. I think you are getting hung up on your definition of what R is (and forgive me if you are not); it is not merely away from the origin, it is away from the origin in a direction specified by the other two components. A negative R is a movement away from the origin in the opposite direction. There is a mathematical expression for -R values - you gave it in your first post. For the sphere - no there is no relationship expression because, if I am right, they are exactly the same. Think of the differences of two lines - one formed by moving +2 in the x-axis from the origin and the other -2 in the x-axis from the point 2,0,0; they are formed differently, but are exactly the same.

[ajb]

You are over thinking this. The distance from the origin to the surface of the sphere is given by [math]|R|[/math], which is positive.

[khaled]

You are over thinking this. The distance from the origin to the surface of the sphere is given by [math]|R|[/math], which is positive.

 

distance is always positive, signed properties are given when there is a direction

 

since we are talking about spherical coordinates .. do the coordinates advance in spherical lines ?

[ajb]

distance is always positive...

 

I think it is this that might be the source of TaoRich's confusion. The -R seems "unnatural" at first, but in reality it is no trouble. It is not suggesting that the radius, the distance from the origin to the surface of the sphere is some how a negative quantity.

[TaoRich]

since we are talking about spherical coordinates .. do the coordinates advance in spherical lines ?

I'm not sure what you mean by "do the coordinates advance in spherical lines ?" Can you elaborate ?

 

distance is always positive, signed properties are given when there is a direction

That relates to what I am on about - it's the direction that is significant.

 

With regard to other comments above about rotation through 180 degrees, or taking the absolute, I'll re-express my query yet again in other terms:

 

[1] We can fold or reflect a one dimensional system about a point of symmetry:

 

By way of example, take the number line argument: we can take a value of n and reflect that value about the zero point to give us -n.

 

[2] We can fold or reflect a two dimensional system about a point of symmetry or an axis of symmetry:

 

By way of example, we can take co-ordinates (x,y):

a) and reflect the x value about the x axis zero point to give us (-x,y)

b) or reflect the y value about the y axis zero point to give us (x,-y)

or

c) reflect the x AND y values about any line or axis we chose within (x,y) Cartesian system

a) and b) are reflections about one dimension

c) is a reflection about two dimensions

[3] We can fold or reflect a three dimensional system about about a point of symmetry or an axis of symmetry:

By way of example, we can take any co-ordinate system for our three dimensions, say (x,y,z) or (r,θ,φ):

 

a) and reflect the x value about the x axis zero point to give us (-x,y,z)

b) or reflect the y value about the y axis zero point to give us (x,-y,z)

c) or reflect the z value about the z axis zero point to give us (x,y,-z)

or

d) reflect the x AND y values about any plane we chose within the (x,y,z) Cartesian Coordinate system or the Spherical Coordinate system(r,θ,φ)

a) b) and c) are reflections about one dimension

d) is a reflection about two dimensions

To visualise what I mean by d) consider holding up a mirror in a room:

• You can angle the mirror to reflect the 3 dimensional world about any plane you chose.You can reflect your left hand to become your right hand (and vice versa)

Back now to my exploration / query:

 

What happens when we reflect 3 dimensions about 3 dimensions ?

 

Not changing the sign of a one or two dimensional vector or radius.

Not rotating the 3 dimensional object through some angle

 

If we view a sphere in our 3 dimension reality ... and then we reflect that sphere about all 3 dimensions ... what would the reflection sphere look like ?... where would it be ?

 

- - -

 

Please note that I am not trying to solve any particular problem, or come to grips with any existing mathematics.

 

I'm exploring ... questing, questioning.

( Yes, I'm talking about the edge or fringe of our physical reality, not a "conventional view of our reality" ... but bear in mind in Mathematics, we can manipulate numbers, descriptions, objects, fields, planes into "non-physical realms" and "unreal dimensions". )

Edited March 16, 2011 by TaoRich

[Xittenn]

If you wish to inflect the field for your purposes, feel free do so, I for one will not inhibit your motions!