3 ms·

>If you have some points round a 2D circle you can visit them in order one after another; but if you have some points on a 3D sphere, there is no order that pla

by pliny 13y ago

>If you have some points round a 2D circle you can visit them in order one after another; but if you have some points on a 3D sphere, there is no order that places them one after another; and Can someone clarify why this is true? If we consider the sphere in spherical coordinates, we choose an arbitrary starting point (as in the case of the 2D circle visit) them in the following order: find the next point for which θ(pointA)-θ(pointB) (A is the point we are now, B is the point we are considering) is minimal and greater than 0, if there are two or more points that are the same number of radians away on the θ-plane (and they are all minimal), find the point, out of those, for which φ(pointA)-φ(pointB) is minimal and greater than 0, this uniquely identifies all points on the surface of the sphere and allows us to order them if we have some arbitrary starting point.

te_platt 13y ago
I don't think he meant it in a strict mathematical sense. Just that it's easy/natural to pick a direction on a line and say point C is after A, and B is in-between; whereas on a surface there isn't an obvious ordering.
- starky 13y ago
  It however isn't difficult to do this for a 3-axis machine. in fact, the way he describes building layer by layer is exactly how you make complex shapes on a 3-axis mill. You essentially trace a line on the surface that you are trying to machine. The problem simply reduces to the same line problem with the points defined in 3 dimensions. Of course, there are notable complexity benefits of 3D printing over 3-axis machining, but many of them can be overcome with other machining methods such as EDM.
  - mdturnerphys 13y ago
    "The problem simply reduces to the same line problem with the points defined in 3 dimensions." Except when you have overhangs.
    - theon144 13y ago
      To be fair, overhangs add a bit of fun to 3D printing too.
    - starky 13y ago
      There are ways to deal with some of the overhang issues with proper tooling though. Regardless, as I said, 3D printing definitely wins in the complexity capability, overhangs are just one of them.
zeckalpha 13y ago
https://en.wikipedia.org/wiki/Hairy_ball_theorem https://en.wikipedia.org/wiki/Hairy_ball_theorem

by pliny 13y ago

te_platt 13y ago

I don't think he meant it in a strict mathematical sense. Just that it's easy/natural to pick a direction on a line and say point C is after A, and B is in-between; whereas on a surface there isn't an obvious ordering.

starky 13y ago
It however isn't difficult to do this for a 3-axis machine. in fact, the way he describes building layer by layer is exactly how you make complex shapes on a 3-axis mill. You essentially trace a line on the surface that you are trying to machine. The problem simply reduces to the same line problem with the points defined in 3 dimensions. Of course, there are notable complexity benefits of 3D printing over 3-axis machining, but many of them can be overcome with other machining methods such as EDM.
- mdturnerphys 13y ago
  "The problem simply reduces to the same line problem with the points defined in 3 dimensions." Except when you have overhangs.
  - theon144 13y ago
    To be fair, overhangs add a bit of fun to 3D printing too.
  - starky 13y ago
    There are ways to deal with some of the overhang issues with proper tooling though. Regardless, as I said, 3D printing definitely wins in the complexity capability, overhangs are just one of them.

zeckalpha 13y ago

https://en.wikipedia.org/wiki/Hairy_ball_theorem https://en.wikipedia.org/wiki/Hairy_ball_theorem