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. 

Please help. 

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/