2.3
Coordinate Systems

- 2.3.1. Conventions
- 2.3.2. Object Space
- 2.3.3. World Space
- 2.3.4. Eye Space
- 2.3.5. Clip Space
- 2.3.6. Normalized-Device Space
- 2.3.7. Screen Space

2.3.1
Conventions
1
This section attempts to describe the mathematical conventions that the r2 package uses with respect to coordinate systems. The r2 package generally does not deviate from standard OpenGL conventions, and this section does not attempt to give a rigorous formal definition of these existing conventions. It does however attempt to establish the naming conventions that the package uses to refer to the standard coordinate spaces [10].
2
The r2 package uses the jtensors package for all mathematical operations on the CPU, and therefore shares its conventions with regards to coordinate system handedness. Important parts are repeated here, but the documentation for the jtensors package should be inspected for details.
3
Any of the matrix functions that deal with rotations assume a right-handed coordinate system. This matches the system conventionally used by OpenGL (and most mathematics literature) . A right-handed coordinate system assumes that if the viewer is standing at the origin and looking towards negative infinity on the Z axis, then the X axis runs horizontally (left towards negative infinity and right towards positive infinity) , and the Y axis runs vertically (down towards negative infinity and up towards positive infinity). The following image demonstrates this axis configuration:
2.3.1.4 Right Handed Coordinate System
5
The jtensors package adheres to the convention that a positive rotation around an axis represents a counter-clockwise rotation when viewing the system along the negative direction of the axis in question.
2.3.1.6 Rotations
7
The package uses the following matrices to define rotations around each axis:
2.3.1.8 Rotation of r radians around the X axis
2.3.1.9 Rotation of r radians around the Y axis
2.3.1.10 Rotation of r radians around the Z axis
11
Which results in the following matrix for rotating r radians around the axis given by (x, y, z), assuming s = sin(r) and c = cos(r):
2.3.1.12 Rotation of r radians around an arbitrary axis
2.3.2
Object Space
1
Object space is the local coordinate system used to describe the positions of vertices in meshes. For example, a unit cube with the origin placed at the center of the cube would have eight vertices with positions expressed as object-space coordinates:
2.3.2.2 Unit cube vertices

    cube = {
      (-0.5, -0.5, -0.5),
      ( 0.5, -0.5, -0.5),
      ( 0.5, -0.5,  0.5),
      (-0.5, -0.5,  0.5),

      (-0.5,  0.5, -0.5),
      ( 0.5,  0.5, -0.5),
      ( 0.5,  0.5,  0.5),
      (-0.5,  0.5,  0.5)
    }

3
In other rendering systems, object space is sometimes referred to as local space, or model space.
4
In the r2 package, object space is represented by the R2SpaceObjectType.
2.3.3
World Space
1
In order to position objects in a scene, they must be assigned a transform that can be applied to each of their object space vertices to yield absolute positions in so-called world space.
2
As an example, if the unit cube described above was assigned a transform that moved its origin to (3, 5, 1), then its object space vertex (-0.5, 0.5, 0.5) would end up at (3 + -0.5, 5 + 0.5, 1 + 0.5) = (2.5, 5.5, 1.5) in world space.
3
In the r2 package, a transform applied to an object produces a 4x4 model matrix. Multiplying the model matrix with the positions of the object space vertices yields vertices in world space.
4
Note that, despite the name, world space does not imply that users have to store their actual world representation in this coordinate space. For example, flight simulators often have to transform their planet-scale world representation to an aircraft relative representation for rendering to work around the issues inherent in rendering extremely large scenes. The basic issue is that the relatively low level of floating point precision available on current graphics hardware means that if the coordinates of objects within the flight simulator's world were to be used directly, the values would tend to be drastically larger than those that could be expressed by the available limited-precision floating point types on the GPU. Instead, simulators often transform the locations of objects in their worlds such that the aircraft is placed at the origin (0, 0, 0) and the objects are positioned relative to the aircraft before being passed to the GPU for rendering. As a concrete example, within the simulator's world, the aircraft may be at (1882838.3, 450.0, 5892309.0), and a control tower nearby may be at (1883838.5, 0.0, 5892809.0). These coordinate values would be far too large to pass to the GPU if a reasonable level of precision is required, but if the current aircraft location is subtracted from all positions, the coordinates in aircraft relative space of the aircraft become (0, 0, 0) and the coordinates of the tower become (1883838.5 - 1882838.3, 0.0 - 450.0, 5892809.0 - 5892309.0) = (1000.19, -450.0, 500.0). The aircraft relative space coordinates are certainly small enough to be given to the GPU directly without risking imprecision issues, and therefore the simulator would essentially treat aircraft relative space and r2 world space as equivalent [11].
5
In the r2 package, world space is represented by the R2SpaceWorldType.
2.3.4
Eye Space
1
Eye space represents a coordinate system with the observer implicitly fixed at the origin (0.0, 0.0, 0.0) and looking towards infinity in the negative Z direction.
2
The main purpose of eye space is to simplify the mathematics required to implement various algorithms such as lighting. The problem with implementing these sorts of algorithms in world space is that one must constantly take into account the position of the observer (typically by subtracting the location of the observer from each set of world space coordinates and accounting for any change in orientation of the observer). By fixing the orientation of the observer towards negative Z, and the position of the observer at (0.0, 0.0, 0.0), and by transforming all vertices of all objects into the same system, the mathematics of lighting are greatly simplified. The majority of the rendering algorithms used in the r2 package are implemented in eye space.
3
In the r2 package, the observer produces a 4x4 view matrix. Multiplying the view matrix with any given world space position yields a position in eye space. In practice, the view matrix v and the current object's model matrix m are concatenated (multiplied) to produce a model-view matrix mv = v * m [5], and mv is then passed directly to the renderer's vertex shaders to transform the current object's vertices [6].
4
Additionally, as the r2 package does all lighting in eye space, it's necessary to transform the object space normal vectors given in mesh data to eye space. However, the usual model-view matrix will almost certainly contain some sort of translational component and possibly a scaling component. Normal vectors are not supposed to be translated; they represent directions! A non-uniform scale applied to an object will also deform the normal vectors, making them non-perpendicular to the surface they're associated with:
2.3.4.5 Unit cube vertices
6
With the scaled triangle on the right, the normal vector is now not perpendicular to the surface (in addition to no longer being of unit length). The red vector indicates what the surface normal should be.
7
Therefore it's necessary to derive another 3x3 matrix known as the normal matrix from the model-view matrix that contains just the rotational component of the original matrix. The full derivation of this matrix is given in Mathematics for 3D Game Programming and Computer Graphics, Third Edition [4]. Briefly, the normal matrix is equal to the inverse transpose of the top left 3x3 elements of an arbitrary 4x4 model-view matrix.
8
In other rendering systems, eye space is sometimes referred to as camera space, or view space.
9
In the r2 package, eye space is represented by the R2SpaceEyeType.
2.3.5
Clip Space
1
Clip space is a homogeneous coordinate system in which OpenGL performs clipping of primitives (such as triangles). In OpenGL, clip space is effectively a left-handed coordinate system by default [7]. Intuitively, coordinates in eye space are transformed with a projection (normally either an orthographic or perspective projection) such that all vertices are projected into a homogeneous unit cube placed at the origin - clip space - resulting in four-dimensional (x, y, z, w) positions. Positions that end up outside of the cube are clipped (discarded) by dedicated clipping hardware, usually producing more triangles as a result.
2.3.5.2 Primitive Clipping
3
A projection effectively determines how objects in the three-dimensional scene are projected onto the two-dimensional viewing plane (a computer screen, in most cases) . A perspective projection transforms vertices such that objects that are further away from the viewing plane appear to be smaller than objects that are close to it, while an orthographic projection preserves the perceived sizes of objects regardless of their distance from the viewing plane.
2.3.5.4 Perspective projection
2.3.5.5 Orthographic projection
6
Because eye space is a right-handed coordinate system by convention, but by default clip space is left-handed, the projection matrix used will invert the sign of the z component of any given point.
7
In the r2 package, the observer produces a 4x4 projection matrix. The projection matrix is passed, along with the model-view matrix, to the renderer's vertex shaders. As is normal in OpenGL, the vertex shader produces clip space coordinates which are then used by the hardware rasterizer to produce color fragments onscreen.
8
In the r2 package, clip space is represented by the R2SpaceClipType.
2.3.6
Normalized-Device Space
1
Normalized-device space is, by default, a left-handed [8] coordinate space in which clip space coordinates have been divided by their own w component (discarding the resulting w = 1 component in the process), yielding three dimensional coordinates. The range of values in the resulting coordinates are effectively normalized by the division to fall within the ranges [(-1, -1, -1), (1, 1, 1)] [9]. The coordinate space represents a simplifying intermediate step between having clip space coordinates and getting something projected into a two-dimensional image (screen space) for viewing.
2
The r2 package does not directly use or manipulate values in normalized-device space; it is mentioned here for completeness.
2.3.7
Screen Space
1
Screen space is, by default, a left-handed coordinate system representing the screen (or window) that is displaying the actual results of rendering. If the screen is of width w and height h, and the current depth range of the window is [n, f], then the range of values in screen space coordinates runs from [(0, 0, n), (w, h, f)]. The origin (0, 0, 0) is assumed to be at the bottom-left corner.
2
The depth range is actually a configurable value, but the r2 package keeps the OpenGL default. From the glDepthRange function manual page:
2.3.7.3 glDepthRange

    After clipping and division by w, depth coordinates range from -1 to 1,
    corresponding to the near and far clipping planes. glDepthRange specifies a
    linear mapping of the normalized depth coordinates in this range to window
    depth coordinates. Regardless of the actual depth buffer implementation,
    window coordinate depth values are treated as though they range from 0
    through 1 (like color components). Thus, the values accepted by
    glDepthRange are both clamped to this range before they are accepted.
    The setting of (0,1) maps the near plane to 0 and the far plane to 1.
    With this mapping, the depth buffer range is fully utilized.

4
As OpenGL, by default, specifies a depth range of [0, 1], the positive Z axis points away from the observer and so the coordinate system is left handed.

[4]
See section 4.5, Transforming normal vectors .
[5]
Note that matrix multiplication is not commutative.
[6]
The reason for producing the concatenated matrix on the CPU and then passing it to the shader is efficiency; if a mesh had 1000 vertices, and the shader was passed m and v separately, the shader would repeatedly perform the same mv = v * m multiplication to produce mv for each vertex - yielding the exact same mv each time!
[7]
Because normalized device space is a left-handed system by default, with the viewer looking towards positive Z, and because the transformation from clip space to normalized device space for a given point is the division of the components of that point by the point's own w component.
[8]
The handedness of the coordinate space is dependent on the depth range configured for screen space.
[9]
It is actually the division by w that produces the scaling effect necessary to produce the illusion of perspective in perspective projections.
[10]
Almost all rendering systems use different names to refer to the same concepts, without ever bothering to document their conventions. This harms comprehension and generally wastes everybody's time.
[11]
A classic example of a modern game title that failed to anticipate precision issues is Minecraft.