Math folks:
You need a minimum of two points to define a line and three to define a plane. Is the pattern simply N points to define a space of N-1 dimensions? Why?

A friend on another network pointed out something which makes this seem obvious in retrospect: adding dimensions is just adding undefined variables to a point equation, linearly. The math/notation makes obvious what I was trying to visualize and failing.

So a point in 2D space is p(x, y), in 3D it’s p(x, y, z), and you add a previously undefined variable for each new dimension. That makes the fact that they’re linear super clear.

@a that's quite obvious, but I'm still thinking about the 4D counterpart of a plane, or a "plane" defined with 4 points?

@sirjofri 4 points defines a 3D space (like ours). But visualizing that requires being able to visualize 3D spaces (not really “spaces”; fields?) with a 4D space. I’ve practiced enough that I can do that (not easily), but next I’d need to visualize a 4D space within a 5D space, and there’s no way I was getting to that.

@a wdym 4 points describes a 3d space? A plane with 4 points in 3d space can not be visualized in only one way (that's why we use triangles in CGI). Or do you refer to the axes? (origin, x/y/z)?

And what should "visualizing 3d space in a 4d space" mean? I only know the projection of a 4d object into a 3d space (N to N-1).

@sirjofri Think about a single point (N=1). There are an infinite number of 1-dimensional spaces (that is, lines) which intersect that point. To define a single 1d space, you need a minimum of 2 points: then there is a single line running through them.

Given those two points (N=2), you have an infinite number of 2-dimensional spaces (planes) running through them; easy to visualize as planes rotating on a line. To define a single 2-dimensional space, you need at least one additional point.

@sirjofri So the pattern is always that to define a single N-dimensional space, you need at least N+1 points.

(Again, I think there’s a more precise/correct word than “space” here, which might be “field”, but you get the idea.)

@a ah, yeah, the term "space" here is very misleading... But I guess "field" is also similar. Maybe something like "system" (from coordinate system) or just "object"?

In 3d CGI, we know the term "object space" which doesn't refer to the object itself, but the coordinate system around that object (from the perspective of that object). So in the end, "space" might as well be just right

@a btw following your explanation it also refers to the amount of "axes", basically. To build a 3d coordinate system you have three axes. To build these three axes you need (at a minimum) 4 points, etc. It also fits the explanation 100% and is even easier to explain, imo "I appreciate SDF but it's a general-purpose server and the name doesn't make it obvious that it's about art." - Eugen Rochko