Multiple Coordinate Spaces

A coordinate space is defined by an origin and a set of axes. Every coordinate space must be defined relative to some other coordinate space. The full pipeline in a real-time 3D engine is:

The Key Spaces

• World Space — the global reference frame. All objects ultimately exist here.
• Object Space (Local Space / Model Space) — the coordinate system of a single object. The object's pivot is the origin. Vertices in a mesh file are in object space.
• Upright Space — a hybrid: the origin is aligned with the object, but the axes are aligned with world space. Used when separating position from orientation.
• Camera Space (View Space) — world space reoriented so the camera is at the origin looking down −Z (in OpenGL convention).
• Clip Space — post-projection. The GPU frustum-clips here.
• Screen Space — final 2D pixel coordinates after the perspective divide and viewport transform.

A coordinate space is specified by its origin and axes. The origin defines position; the axes define orientation. Both must be described relative to some parent space.

The axes of a coordinate space are called basis vectors. Any vector in a space can be expressed as a linear combination of the basis vectors:

Where p, q, r are the basis vectors and vx, vy, vz are the coordinates of v in that basis.

Converting Between Spaces

To convert a point from object space to world space, you need:

• o — the object's origin in world space coordinates
• p, q, r — the object's local axes expressed in world space
• b — the point's coordinates in object space

In matrix form, p, q, r become the rows (or columns, depending on convention) of a transformation matrix. This is exactly what the model matrix does in a vertex shader.

The span of a set of basis vectors is the set of all vectors that can be expressed as linear combinations of those basis vectors — in other words, every point that can be reached by moving along those axes.

The rank of a basis is the number of dimensions in its span:

• One non-zero vector → span is a line (rank 1)
• Two non-parallel vectors → span is a plane (rank 2)
• Three non-coplanar vectors → span is all of 3D space (rank 3)

Full rank means the number of dimensions in the span equals the number of basis vectors. For a 3D game coordinate space, you need full rank (rank 3) — otherwise some directions cannot be represented.

If a basis vector lies within the span of the other basis vectors, the vectors are linearly dependent. Consequences:

• The basis does not have full rank
• Vector coordinates in that basis are not uniquely determined

If no basis vector can be expressed as a linear combination of the others, they are linearly independent. A proper 3D coordinate space requires three linearly independent axes.

Orthogonal Basis

All basis vectors are mutually perpendicular — the dot product of any two is zero:

Orthonormal Basis

Orthogonal and all basis vectors are unit length. The standard world-space axes {i, j, k} = {(1,0,0), (0,1,0), (0,0,1)} form the canonical orthonormal basis.

Finding Coordinates in an Orthonormal Basis

When the basis is orthonormal, finding a vector's coordinates is just three dot products:

This is why orthonormal bases are so computationally convenient — no matrix inversion needed to go from world space coordinates to basis coordinates.

1. Linear Independence Check

Are the vectors (1, 0, 0), (0, 1, 0), and (1, 1, 0) linearly independent?

2. Coordinate Space Conversion

An object's local axes in world space are p = (1,0,0), q = (0,0,-1), r = (0,1,0), and its origin is at o = (5, 0, 3). A vertex in object space is at b = (2, 1, 0). What is its world space position?

3. Orthonormal Basis Coordinates

Given orthonormal basis vectors p = (1,0,0), q = (0,1,0), r = (0,0,1) and world vector u = (3, -1, 2), find the coordinates of u in this basis.

Interview Question

Explain the difference between world space and object space. Give a practical example of when you'd work in each.

Interview Question

What is an orthonormal basis and why is it important for transformation matrices used in games?