Notice, how we have snuck up on the idea of Homogeneous Coordinates,
based on simple logical arguments. Keep the following in mind,
coordinates are not geometric,
they are just scales for basis elements. Thus, you should
not be bothered by the fact that our coordinates suddenly have 4 numbers.
We could have had more (no one said we have to have a linearly independent basis set).
Note how this approach to coordinates is completely consistent with our intuitions.
Subtracting two points yields a vector.
Adding a vector to a point produces a point.
If you multiply a vector by a scalar you still get a vector.
And, in most cases, when you scale points you'll get some nonsense 4th
coordinate element that reminds you that the thing you're left with is no longer a point.
Isn't it strange how seemingly bizarre things
make sense sometimes?
Notice, how we have snuck up on the idea of Homogeneous Coordinates,
based on simple logical arguments. Keep the following in mind,
coordinates are not geometric,
they are just scales for basis elements. Thus, you should
not be bothered by the fact that our coordinates suddenly have 4 numbers.
We could have had more (no one said we have to have a linearly independent basis set).
