Blender 3D: Noob to Pro/3D Geometry
Contents |
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 geometry, the branch of mathematics concerned with spatial relationships, specifically analytical geometry, which expresses these relationships in terms of algebraic formulas. You may have studied geometry in school, in which case, some of the terminology will seem familiar.
[edit] 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. (If there really is a fly buzzing around your room, feel free to substitute that.) The fly is moving in three-dimensional space. In mathematical terms, what that means is, at any moment, its position within the room can be uniquely expressed in terms of three numbers.
Now, there are an infinite number of ways—coordinate systems—in which we could come up with a convention for defining and measuring these numbers—the coordinates. And each convention will give different values for the numbers, 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, of course), 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 is really north, feel free to use that to label the walls of your room; otherwise, choose any wall you like as north.)
Consider the south-west corner of your room, the point at floor level. 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 meters) eastwards from the west wall, the second number will be the distance northwards 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 (the vertical or height), Y (the horizontal or length) and Z (the breadth), 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 conversely that every possible combination of (x,y,z) values, with 
, 
and 
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. In the second picture, the value for x defines a plane parallel to the west wall of the room; in the third picture, the value for y defines a plane parallel to the south wall; and in the fourth picture, the value for z defines a plane parallel to the floor. Put the planes together in the fifth picture, and they intersect at a unique point.
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! Well, people of old were observant, no wonder!
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, switching between these dimensions are simple and easy to do though in Blender.
[edit] Negative Coordinates
Can coordinate values be negative? Depending on the situation, yes. Here we are only considering points within our room. But supposing, 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 where we like, just so long as we all agreed on where it is.) Now, remember that our X-coordinate is the distance east from the origin; How do we express the fact that a point lies west of the origin? Simple: We give it a negative X-coordinate. Similarly, points north of the origin have a positive Y-coordinate, while those south of it get a negative one; and those points above the origin get a positive Z-coordinate, while those below get a negative one.
[edit] Handedness Of Coordinate Systems
It is conventional for most cartesian coordinate systems to be right-handed. To see what this means, hold the thumb, index finger and middle finger of your right hand perpendicular to each other:
Now you can orient your hand so that the thumb points along the X-axis in the positive direction (direction of increasing coordinate numbers), the index finger along the positive Y-axis, and the 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 direction of positive X, positive Y and positive Z as in Figure 1 below, the order X, Y, Z would go clockwise.
[edit] Axes Of Rotation
Consider a spinning ball. Every point on it is moving, except the ones right in the middle: these form a line of stillness around which the rest of the ball spins. This line is called the axis of rotation.
More precisely, the axis of rotation of a rotating object over a period of time 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 positon relative to that object over time.
Conventionally, the direction of the axis of rotation is such that, if you place your eye looking 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:
To remember this convention, hold your right hand in a thumbs-up gesture: 
If the rotation follows the direction of the curled-up fingers, then the direction of the axis of rotation is considered to be the same as the thumb is pointing.
Therefore, when describing the direction of a rotating object, do not say it rotates left to right, right to left, clockwise or counterclockwise. All of the above is relative to the observer. Instead of saying this, find the direction of the axis of rotation and draw an arrow to represent it.
[edit] Additional Resources
- the Geometry wikibook
Analytic geometry at Wikipedia.- Cartesian coordinate system at Wikipedia.
- Right-hand rule at Wikipedia.
- Rotation at Wikipedia.
