# Simple 3d

Every point of a thought plane, edge or point (thought in 3-dimensional space) gets *projected* onto the screen

For the first realisation of simple projections, we resign on using a camera location (read more about that later on, on the page after the next) and choose the frame of projection to be spanned up by the four points (-1,1,0), (1,1,0), (1,-1,0) and (-1,-1,0) - marked orange in the following graphic:

Also, i drew in a Point called P at the location (1,1,1) that we're going to project to the orange plane now.

**The formula for doing that is basic and simple**: The X and Y coordinates
get just divided by the Z coordinate each:

X_{screen} = X_{3d coordinate system} / Z_{3d coordinate system} and

Y_{screen} = Y_{3d coordinate system} / Z_{3d coordinate system}

so that any *points that are ***more in the distance** (higher Z) get transformed to *points that are ***closer to the mid of the screen** (the result of the projection) - the 2d origin (0, 0).

For example, you have the point (1,1,1) Projected to (1,1) like this.

But the point (1,1,100) gets transformed to (1/100, 1/100) = (0.01, 0.01)

The next graphic shows that a little bit more clearer, maybe:

I entered the two points P(1,1,1) and Q(1,1,10) into the projection system.

The yellow line highlights their direct connection in 3d-space. The orange line is the same line projected onto the screen. There (on the screen) you find projected P and Q highlighted with green and purple X-es. On the right,
i showed how a real painter would interpret the situation, adding in two additional points in between P and Q, drawing all points as pins connected to the earth or lets say trees.

### Outlook

Now, if you are mathematically aware, you might have noticed that actually, doing so, the points all become projected to a plane spanned up by the four points (-1,1,1), (1,1,1), (1,-1,1) and (-1,-1,1) instead of (-1,1,0), (1,1,0), (1,-1,0) and (-1,-1, 0) (that means, the frame residing on plane z=1 instead of plane z=0):

And also note that you must be aware of not dividing through zero. Read more about that (and what we do about it) on the next page: Let's continue 3d III