[MauranKilom]

> A point in the sense you mean is a collection of real numbers equal to the number of dimensions of the space it’s in.

Not necessarily. In 3D graphics, it is common to represent points with homogeneous coordinates, where points in N-dimensional space are represented by N+1 real numbers. Using 4x4 matrices [0] to describe affine transformations of 3D points is very convenient.

(Agreed with your overall point though. Just goes to show how different some fundamental perceptions/definitions can be.)

[0]: https://en.wikipedia.org/wiki/Affine_transformation

[meheleventyone]

The extra real isn’t really part of the definition of the point in space though and isn’t necessary to store to apply a 4x4 matrix. See for example applyMatrix4 in this file: https://github.com/mrdoob/three.js/blob/master/src/math/Vect...

[MauranKilom]

> The extra real isn’t really part of the definition of the point in space though

It actually is. It's generally assumed to be equal to one, but it need not be.

> isn’t necessary to store to apply a 4x4 matrix

...if you assume it is equal to one, yes.

However, actually representing the fourth component is both mathematically sound and occasionally useful. For example, the midpoint of two affine points, such as (1, 2, 3, 1) and (3, 6, 9, 1) is actually just their sum: (4, 8, 12, 2), which represents the same 3D point as (2, 4, 6, 1). The fourth component can also be zero, in which case you describe a point at infinity in the given direction.

But yes, if you only use homogeneous coordinates for chaining 3D transformations, storing the extra component it pointless.