I don't think you are misunderstanding so much as conflating two things, the co-ordinate space / construction of the world and construction of objects within that world.
It's correct to say that every 'point' in a voxel based world can be represented by a voxel in that point. It's incorrect to say that every point in a world has voxel data associated with it. Perhaps this is the place where Euclideon and Notch diverge as well.
Using a quantized world representation view (where co-ordinates in the world are quantized to voxel boundaries) then as a voxel moves through the world space it moves from point to point in that space. So a 'cloud' of voxels has an origin and an orientation in worldspace, that allows the translation of a local voxel space (the object) into world voxel space. Since you can't do sub-voxel positioning it behooves you to have an oversampling rate between voxel space and display pixels as well.
A 'feature' here is that it also allows for fine grain collision detection (harder to do in polygon space) but again its pretty expensive computationally. (boundary surface intersection of the local voxel space with the world voxel space).
What I'm not understanding is the object space. If a voxel's coordinates are implied based on its position, then there would be voxel data for every "point" in the object space. Unless we use a sparse representation of some sort, in which case it seems that we're basically using point clouds.
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I don't think you are misunderstanding so much as conflating two things, the co-ordinate space / construction of the world and construction of objects within that world.
It's correct to say that every 'point' in a voxel based world can be represented by a voxel in that point. It's incorrect to say that every point in a world has voxel data associated with it. Perhaps this is the place where Euclideon and Notch diverge as well.
Using a quantized world representation view (where co-ordinates in the world are quantized to voxel boundaries) then as a voxel moves through the world space it moves from point to point in that space. So a 'cloud' of voxels has an origin and an orientation in worldspace, that allows the translation of a local voxel space (the object) into world voxel space. Since you can't do sub-voxel positioning it behooves you to have an oversampling rate between voxel space and display pixels as well.
A 'feature' here is that it also allows for fine grain collision detection (harder to do in polygon space) but again its pretty expensive computationally. (boundary surface intersection of the local voxel space with the world voxel space).
What I'm not understanding is the object space. If a voxel's coordinates are implied based on its position, then there would be voxel data for every "point" in the object space. Unless we use a sparse representation of some sort, in which case it seems that we're basically using point clouds.