# Blender 3D: Noob to Pro/3D Geometry

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If you haven\'t previously studied 3D graphics, technical drawing, or
analytic geometry, you are about to learn a new way of visualizing the
world, an ability that\'s fundamental to working with Blender or any 3D
modeling tool.

3D modeling is based on , the
branch of mathematics concerned with spatial relationships, specifically
, which expresses
these relationships in terms of algebraic formulas. If you have studied
geometry, some of the terminology will be familiar.

## Coordinates And Coordinate Systems

Look around the room you're in. The odds are it will have a cuboidal
shape, with four vertical walls at right angles to each other, a flat,
horizontal floor, and a flat, horizontal ceiling.

Now imagine there's a fly buzzing around the room. The fly is moving in
three-dimensional space. In mathematical terms, that means its position
within the room at any given moment, can be expressed in terms of a
unique combination of three numbers.

There are an infinite number of ways ---*coordinate systems*--- in which
we could come up with a convention for defining and measuring these
numbers, i.e. the *coordinates*. Each convention will yield different
values even if the fly is in the same position. Coordinates only make
sense with reference to a specific coordinate system! To narrow down the
possibilities (in a purely arbitrary fashion), let us label the walls of
the room with the points of the compass: in a clockwise direction,
North, East, South and West. (If you know which way really is north,
feel free to use that to label the walls of your room. Otherwise, choose
any wall you like as north.)

Consider the point at floor level in the south-west corner of the room.
We will call this (arbitrary) point the *origin* of our coordinate
system, and the three numbers at this point will be $(0, 0, 0)$. The
first of the three numbers will be the distance (in some suitable units,
let's say metres) eastwards from the west wall, the second number will
be the distance north from the south wall, and the third number will be
the height above the floor.

Each of these directions is called an *axis* (plural: *axes*), and they
are conventionally labelled X, Y and Z, in that order. With a little bit
of thought, you should be able to convince yourself that every point
within the space of your room corresponds to exactly one set of
$(x, y, z)$ values, and that every possible combination of $(x, y, z)$
values, with $0 \le x \le W$, $0 \le y \le L$ and $0 \le z \le H$ (where
$W$ is the east-west dimension of your room, $L$ is its north-south
dimension, and $H$ is the height between ceiling and floor) corresponds
to a point in the room.

The following diagram illustrates how the coordinates are built up,
using the same colour codes that Blender uses to label its axes: red for
X, green for Y and blue for Z (an easy way to remember this if you\'re
familiar with RGB is the order \-- Red X, Green Y, Blue Z). In the
second picture, the *x* value defines a plane parallel to the west wall
of the room. In the third picture, the *y* value defines a plane
parallel to the south wall, and in the fourth picture, the *z* value
defines a plane parallel to the floor. Put the planes together in the
fifth picture, and they intersect at a unique point.

![](BlenderCoordinates.png "BlenderCoordinates.png"){width="800"}

Another simple way to understand what the coordinates of a point say
(x,y,z) means is, if one starts from origin and moves x, y, and z units
of distance parallel to x, y, and z axes respectively, in any sequence,
one will reach that point. Thus, for example, a coordinate of (3,4,5)
means the point which is reached when one moves, starting from origin, 3
units of distance along x-axis, 4 units of distance along y-axis and 5
units of distance along z-axis.

This style of coordinate system, with the numbers corresponding to
distances along perpendicular axes, is called *Cartesian coordinates*,
named after René Descartes, the 17th-century mathematician who first
introduced the concept. Legend has it that he came up with the idea
after watching a fly buzzing around his bedroom!

There are other ways to define coordinate systems, for example by
substituting direction angles in place of one or two of the distance
measurements. These can be useful in certain situations, but usually all
coordinate systems in Blender are Cartesian. However, in Blender,
switching between these coordinate systems is simple and easy to do.

### Negative Coordinates

Can coordinate values be negative? Depending on the situation, yes. Here
we are only considering points within our room. But suppose instead of
placing our origin in the bottom southwest corner, we put it in the
middle of the room, halfway between the floor and ceiling. (After all,
it is an arbitrary point, we can place it wherever we like, as long as
we agree on its location.) If the X-coordinate is the distance *east*
from the origin, how do we define a point *west* of the origin? We
simply give it a negative X-coordinate. Similarly, points north of the
origin have a positive Y-coordinate, those south of it, have negative
Y-coordinates. Points above the origin have a positive Z-coordinate,
those below it, a negative Z-coordinate.

### Handedness Of Coordinate Systems

It is conventional for most Cartesian coordinate systems to be
*right-handed*. To understand this, hold the thumb, index finger and
middle finger of your right hand perpendicular to each other: !**Figure
1:** The three axes form a right-handed
system{width="200"}
Now orient your hand so your thumb points along the X-axis in the
positive direction (direction of increasing coordinate numbers), your
index finger along the positive Y-axis, and your middle finger along the
positive Z-axis. Another way of looking at it is, if you placed your eye
at the origin, and you could see the three arrows pointing in the
directions of positive X, positive Y and positive Z as in Figure 1, the
order X, Y, Z would go counter clockwise.

!**Figure 2:** Another view of right-handed
system
Another way to visualize this is to make a fist with your right hand,
with your curled fingers towards you. Stick out your thumb directly to
the right (X). Now aim your pointer finger straight up (Y). Finally,
make your middle finger point toward yourself (Z). This is the view from
directly above the origin.

## Axes Of Rotation

Consider a spinning sphere. Every point on it is moving, except the ones
along the axis. These form a motionless line around which the rest of
the sphere spins. This line is called the *axis of rotation*.

More precisely, the axis of rotation is a point or a line connecting
points that do not change position while that object rotates, drawn when
the observer assumes he/she does not change position relative to that
object over time.

Conventionally, the *direction* of the axis of rotation is such that if
you look in that direction, the rotation appears clockwise, as
illustrated below, where the yellow arrow shows the rotational movement,
while the purple one shows the rotation axis:

![](RotationAxisDirections.png "RotationAxisDirections.png")

To remember this convention, hold your right hand in a thumbs-up
gesture:

![](Right-hand_grip_rule.svg "Right-hand_grip_rule.svg"){width="100"}

If the rotation follows the direction of your curled fingers, then the
direction of the axis of rotation is considered to be the same as the
direction which the thumb is pointing in.

This gesture is a different form of the right-hand rule and is sometimes
called *the right-hand grip
rule*,
*the corkscrew-rule* or *the right-hand thumb rule*. From now on we will
refer to it as \'the right-hand grip rule\'.

When describing the direction of a rotating object, do not say that it
rotates *left-to-right/clockwise*, or *right-to-left/counterclockwise*.
Each of these on their own are meaningless, because they\'re relative to
the observer. Instead of saying this, find the direction of the axis of
rotation and draw an arrow to represent it. Those who know the
right-hand grip rule will be able to figure out what the direction of
rotation of the object is, by using the rule when interpreting your
drawing.

## Additional Resources

-   the *Geometry* wikibook

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