some maths notes

[Isaac Dupree]

Dimensions of things
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In R^3, the set of parabolas that intersect a point V has 5 dimensions.

In R^3, the set of parabolas that intersect a parabola P has 6 dimensions.

In R^3, the set of all parabolas has 7 dimensions.

Let a parabolic trajectory in R^3 be a function P(t) = at^2 + bt + c,
with a, b and c in R^3

The set of parabolic trajectories in R^3 has 9 dimensions.

Given a parabolic trajectory P, the set of parabolic trajectories P'
such that there exists a t such that P(t) = P'(t) has 7 dimensions.

For a point V in R^3 and a time t, the set of parabolic trajectories P
such that P(t) = V has 6 dimensions. For example, if t=0, it is the
6-dimensional set for which c=V, and a and b have arbitrary values.


Spacetime search tree
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(turns out we might not need this)

Let points in the spacetime search tree have 9 dimensions:
- 3 position dimensions (units e.g. m)
- 3 velocity dimensions (units e.g. m/s)
- 3 acceleration dimensions (units e.g. m/s^2)

Let an /object/ in the spacetime search tree be either:
- a point (in the 9D space)
- a line segment (in the 9D space)
- a triangle (in the 9D space)


Let a /region/ be a connected open set bounded by a finite number of
triangles.