Skip to content

Commit

Permalink
add preformat to all math symbols on 4 elbow blog
Browse files Browse the repository at this point in the history
  • Loading branch information
Fidel Coria committed Aug 11, 2024
1 parent ec5700b commit e88afb4
Showing 1 changed file with 7 additions and 7 deletions.
14 changes: 7 additions & 7 deletions blog/2024-08-10-Four_Elbows/index.md
Original file line number Diff line number Diff line change
Expand Up @@ -53,26 +53,26 @@ Here is a simplified drawing with conventional mathematical notation. Pretend it
fig. 3
```

Let there be a plane at point A perpendicular to ->AA' called P.
Let there be a plane at point A perpendicular to `->AA'` called `P`.

- This plane includes all points in the rays that a 90 degree "elbow" at point A could draw when rotated.
- This plane includes all points in the rays that a 90 degree "elbow" at point `A` could draw when rotated.

Now, let there also be a plane at point B perpendicular to ->BB' called Q.
Now, let there also be a plane at point `B` perpendicular to `->BB'` called `Q`.

These planes are useful because they get us closer to identifying the orientation of the "vertical" pipes in fig. 2. The reason for this is that each plane contains all of the possible orientations of the pipes independently.

So all we need to find is a line in plane P through point A that is parallel to a line in plane Q through point B. Then it is trivial to select a point C and draw a perpendicular line to a point D as illustrated in fig. 3.
So all we need to find is a line in plane `P` through point `A` that is parallel to a line in plane `Q` through point `B`. Then it is trivial to select a point `C` and draw a perpendicular line to a point `D` as illustrated in fig. 3.

Notice that the planes may either intersect or not.

1. In the case that the planes do not intersect

This means the planes are parallel. Which means any line we chose through point A will have a parallel line through point B. So there are infinitely many lines to select and an infinite choice of points C and D. In other words there are infinitely many ways to orient the fitting in fig. 2 by rotating it around the pipes. The length of the "vertical" pipes is the other infinite choice.
This means the planes are parallel. Which means any line we chose through point `A` will have a parallel line through point `B`. So there are infinitely many lines to select and an infinite choice of points `C` and `D`. In other words there are infinitely many ways to orient the fitting in fig. 2 by rotating it around the pipes. The length of the "vertical" pipes is the other infinite choice.

2. In the case that the planes intersect

We can take the line of intersection -call it X- and draw a line parallel to X in plane P through point A. Likewise we can draw a line parallel to X in plane Q through point B.
By transitivity the line in plane P through point A is parallel to the line in plane Q through point B.
We can take the line of intersection -call it `X`- and draw a line parallel to `X` in plane `P` through point `A`. Likewise we can draw a line parallel to `X` in plane `Q` through point `B`.
By transitivity the line in plane `P` through point `A` is parallel to the line in plane `Q` through point `B`.
This time we only have one pair of parallel lines instead of an infinite choice. But there are 2 orientations for the "vertical" pipes as shown in fig. 4 and fig. 5.
The length of the "vertical" pipes is still an infinite choice.

Expand Down

0 comments on commit e88afb4

Please sign in to comment.