# 3D Math Primer for Game Programmers (Coordinate Systems)

Left-hand coordinate system.

In this article, I would like to provide a brief math primer for people who would like to get involved in game programming.  This is not an exhaustive explanation of all the math theory that one will have to know in order to be a successful game programmer, but it’s the very minimum amount of information that is necessary to know before you can begin as a game programmer.

This article assumes you have a minimum understanding vectors, and matrices.  I will simply show applications of vectors and matrices and how they apply to game programming.

# Conventions

Throughout this article, I will use a convention when referring to vectors, scalars, and matrices.

1. Scalars are represented by lower-case italic characters ($\mathif{a,b,\theta,\lambda}$).
2. Vectors are represented by lower-case bold characters ($\mathbf{x,y,z}$)
3. Matrices are represented by upper-case bold characters ($\mathbf{R,S,T,M}$)

# Coordinate Systems

Before we can talk about transformations, we must make a formal definition of what our coordinate system is.  Throughout this article, I will be using a left-handed coordinate system.  This is the default coordinate system used by DirectX.  The default coordinate system used by OpenGL is a right-handed coordinate system.

The easiest way to remember what coordinate system you are using is to use your hands.  If you point your thumb, your index finger, and your middle finger orthogonal to each other, then each finger will point in the direction of a positive cardinal axis in your coordinate space.  Using your left hand, your thumb points towards you (the $+Z$ axis), your index finger points up (the $+Y$ axis), and your middle finger points to the right (the $+X$ axis).

Using your right-hand, your thumb and your index finger still point in the same direction, but the middle finger (the $+X$ axis) points in the opposite direction.  If we rotated our hand around the index finger (the $+Y$ axis) in order to get the middle finger to point to the right, our thumb (the $+Z$ axis) would then be pointing away from you.

Another important note to remember is the direction of rotation.  If you point your thumb in the positive direction of the axis of rotation and curl your fingers around that imaginary axis, your fingers will curl in the positive direction of rotation.  If you do this on your left hand your fingers will curl in a clockwise direction when looking down at your thumb.  However if you do this on your right hand, your fingers will curl in a counter-clockwise direction when looking down at your thumb.

This can be seen in the images below:

 Left-hand coordinate system. Right-hand coordinate system.

The following table shows the direction of rotation for positive and negative rotations depending on the handedness of the coordinate system.

 Left-hand coordinate system Right-hand coordinate system Viewed From Positive Rotation Negative Rotation Positive Rotation Negative Rotation The negative end looking toward the positive end Counter-Clockwise Clockwise Clockwise Counter-Clockwise The positive end looking toward the negative end Clockwise Counter-Clockwise Counter-Clockwise Clockwise

# Coordinate Spaces

In a 3D game engine, we usually deal with several different coordinates spaces.  Among the most commonly used spaces are Object space, World space, Inertial space, and Camera space.

## Object Space

In object space, also known as local space, or modeling space, an object’s vertices are expressed relative to the object that they describe.  That is, the way the artist intended for them to be displayed if you didn’t move the object from the origin.

The image below shows an example of an object in object space.  As you can see from the image, the object is placed at it’s relative origin.

Object shown in local space.

## World Space

The world coordinate space is the global coordinate space for which all other object spaces are described.  The image below shows an object described in world space.  Notice how the object’s world transformation places it away from the world-origin by some translation, and rotation.

Object shown in world space.

## Inertial Space

The inertial space is the “halfway” space between object space and world space.  The origin of inertial space has the same origin as object space, and the axes of inertial space are parallel to that of the world space axes.

To transform a point from object space to inertial space requires only a rotation and to transform a point from inertial space to world space requires only a translation.

The image blow shows an example of inertial space.

Object show in inertial space.

## Camera Space

Camera space is the coordinate space that is associated with the observer.  Camera space is considered to be the origin and orientation of what we are looking at.  The camera’s coordinate axis usually assume the positive $\mathbf{X}$ axis points right, the positive $\mathbf{Y}$ axis points up, and in a left-handed coordinate system, the positive $\mathbf{Z}$ axis points forward, or at the scene.  In a right-handed coordinate system, the $\mathbf{Z}$ axis is reversed, so the negative $\mathbf{Z}$ axis points forward, or at the object in our scene.  A more common name for these camera axes are the “Right”, “Up”, and “At” axes.

The camera space transformation together with the projection transformation is useful for answering such questions as:

1. Is an object completely in view of the camera, partially in view, or not in view at all,
2. Is one object closer to the camera than the other,
3. Is an object directly in front of, above, below, to the left, or to the right of the camera.

## Combining Coordinate Spaces

In 3D computer graphics, coordinate spaces are described using a homogeneous coordinate system.  A homogeneous coordinate system allows us to represent all of our affine transformations (translation, rotation, scale, and perspective projection) in a similar way so they can easily be combined into a single representation.

Any number coordinate spaces can be combined using matrix multiplication which results in a single matrix that can be applied to all the vertices of an object.

Even multiple world coordinate spaces can be combined in order to derive a final coordinate space that describes the location of all of our vertices in an object.  This is useful for nested coordinate spaces where the position of an object is expressed relative to a “parent” object.  When the parent object’s world transform is changed, the transform of the child object is also changed implicitly.  Using this method, complex scenes can be constructed from several smaller scenes and placed into the larger scene to create a complete world.

The following video shows this concept of combining coordinate spaces of smaller objects to build a larger complex scene.

# References

3D Math Primer for Graphics and Game Development

Fletcher Dunn and Ian Parberry (2002). 3D Math Primer for Graphics and Game Development. Wordware Publishing.

 $\leftarrow$ Vector Operations $\rightarrow$

## 6 thoughts on “3D Math Primer for Game Programmers (Coordinate Systems)”

1. Hey Jeremiah,

Your blog is an awesome resource keep em coming, Ill be watching this blog!

Gr,

Mark

• Thanks Mark. I appreciate the complement. I haven’t been very active on the site during the summer but I’d like to start writing again. If you have any suggestions for a great tutorial, I’d love to hear them!

Your coordinate system discription disagree with the camera space discription. At camera space discription the left- and right-handed are correct.

thumb = X (and not Z)
indexfinger = Y
middlefinger = Z (and not X)

• Thanks Ju Wu for your comments but my explanation is not wrong, it’s simply another way of looking at it. The point is, the difference between a right-handed coordinate system and the left-handed coordinate system is one of the axis is reversed while the other two axis stay the same.

However I will accept that the most common interpretation of the handedness mnemonic is the way you describe it and I will consider revising it when I have the time to do so.

But it’s great to know somebody is reading!

Thanks for your comments. It definitely helps me to improve my articles.

3. You should really correct your description of left-handed/right-handed coordinate systems, to agree with the rest of the world. You are introducing confusion into the minds of people who are trying to learn. If you were the only person to speak in these terms then it would be different, but as it stands, this is common terminology and your definition is incorrect.

• Shea,

Agreed. The common notation is much better and I will try to get this fixed as soon as possible.
Thanks for pointing this out.