>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.
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.
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.
The math might be easier, but the physics is much harder. How do you 3D-print a metal object? It's an easier bet to say that computing power will continue to increase than it is to say some magical high temperature sintering process will be made for home use.
Lost wax casting from a 3D-printed mold would seem like a more straightforward process than CNC machine a brick of solid metal, honestly. People have done this; I've seen blog posts.
Sintered metals are never going to have the same properties that cast, forged, and/or welded metals have. Such parts may be very strong compared to, say, plastics, but there will always be a huge class of parts (especially for machinery) which just cannot reasonably be made that way.
But that's fine, additive manufacturing doesn't have to be the end-all be-all of manufacturing. People should concentrate more on leveraging the strengths of 3D printing as much as possible rather than trying to make it be some jake of all trades manufacturing tool that it can never be.
There's quite a few processes that allow the production of metal parts. The big advantage comes when you produce titanium parts using Selective Laser Sintering. Titanium is very hard to machine using traditional processes, and the waste produced means it's very expensive. SLS deals with both of those issues. The principal downsides right now are inconsistencies in material properties and surface finish. It will be interesting to see if we can solve those problems.
Personally, I'm super excited about the possibility of ultrasonic consolidation hybrid printers:
http://reprap.org/wiki/Ultrasonic_consolidation_Hybrid_print...
As a hybrid approach, it wouldn't be quite as flexible as full 3D printing, but still, it's pretty cool!
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.