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### Three-dimensional Plane Diagrams

```
Date: 03/10/99 at 17:05:12
From: Lindsay
Subject: Three-dimensional Plane Diagrams

I am having trouble learning how to draw planes, and the special ways
they are supposed to be drawn. For example, draw two parallel planes
with another plane intersecting them, or draw two parallel planes with
an intersecting line.

```

```
Date: 03/11/99 at 12:48:48
From: Doctor Peterson
Subject: Re: Three-dimensional Plane Diagrams

Actually it is not too hard, as long as we do not try to get too fancy.
Here is a horizontal plane:

+----------------------------+
/                            /
/                            /
/                            /
/                            /
/                            /
/                            /
/                            /
+----------------------------+

It looks horizontal because there's a horizontal line across the front.

Here is a vertical plane turned so we see the right side, and another
turned to show the left side:

+               +
/|               |\
/ |               | \
/  |               |  \
/   |               |   \
/    |               |    \
+     |               |     +
|     |               |     |
|     |               |     |
|     |               |     |
|     |               |     |
|     |               |     |
|     +               +     |
|    /                 \    |
|   /                   \   |
|  /                     \  |
| /                       \ |
|/                         \|
+                           +

These look vertical because there's a vertical line along the front
(or side). I can make them intersect just by putting them together at
an edge:

+
/|\
/ | \
/  |  \
/   |   \
/    |    \
+     |     +
|     |     |
|     |     |
|     |     |
|     |     |
|     |     |
|     +     |
|    / \    |
|   /   \   |
|  /     \  |
| /       \ |
|/         \|
+           +

Two parallel planes will look identical; here are two vertical planes
intersected by a horizontal plane:

+             +
/|            /|
/ |           / |
/  |          /  |
/   |         /   |
+--/----+--------/----+--------+
/  +    /|       +    /|       /
/   |   / |       |   / |      /
/    |  /  |       |  /  |     /
/     | /   |       | /   |    /
/      |/    |       |/    |   /
+-------+-------------+--------+
|     +       |     +
|    /        |    /
|   /         |   /
|  /          |  /
| /           | /
|/            |/
+             +

I made the edges of the planes intersect so it is easy to see how they
are related. (Of course, planes do not really have edges - this really
pictures only part of the planes, so you can see something.) I also
drew lines where the planes intersect, which join the points where the
edges intersect. I chose to make the planes transparent; you can make
them opaque by erasing the parts that are hidden behind planes:

+             +
/|            /|
/ |           / |
/  |          /  |
/   |         /   |
+--/    +--------/    +--------+
/  +    /        +    /        /
/   |   /         |   /        /
/    |  /          |  /        /
/     | /           | /        /
/      |/            |/        /
+-------+-------------+--------+
|     +       |     +
|    /        |    /
|   /         |   /
|  /          |  /
| /           | /
|/            |/
+             +

That is starting to get pretty impressive, isn't it! You could make the
intersecting plane tilted so it is not perpendicular to the others, but
I am not going to push my luck.

For my last performance, I will give you your parallel planes with an
intersecting line. To make it clear that the line is not parallel to
the planes, I will send it in a different direction than the edges of
the planes; to show where it intersects, I will make points, and make
the planes opaque:

\   +             +
\ /|            /|
/ |           / |
/ +|          /  |
/   \         /   |
/    |\       /    |
+     | \     +     |
|     |  \    |     |
|     |   \   |     |
|     |    \  |     |
|     |     \ |     |
|     |      \|     |
|     +       |     +
|    /        |    /
|   /         | + /
|  /          |  \
| /           | / \
|/            |/   \
+             +     \

It really does not make much difference where you put those points of
intersection, as long as they are on the line and within the plane.

I hope that helps. If I have left out something, please write back.
This is fun!

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry
Middle School Geometry
Middle School Higher-Dimensional Geometry

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