Track Fitting Degrees of Freedom

A single track in 3D space has 5 degrees of freedom:

• 2 parameters for direction (azimuthal and polar angles)
• 2 parameters for position (intersection with a reference plane)
• 1 parameter for curvature (related to momentum)

When fitting two tracks to a common 3D vertex:

• Initial degrees of freedom: 2 × 5 = 10
• Constraints from common vertex: 3 (x, y, z coordinates must be identical)
• Remaining degrees of freedom: 10 - 3 = 7

These 7 degrees of freedom represent:

• 3 parameters for vertex position
• 4 parameters for track directions (2 per track)

For N tracks constrained to meet at a common 3D vertex:

• Initial degrees of freedom: N × 5 = 5N
• Constraints: 3N (each track must pass through same point)
• Subtract 3 free parameters (which is the position of the vertex)
• Final degrees of freedom: 5N - 3N - 3 = 2N - 3

This formula gives:

• N = 1: 5 degrees of freedom
• N = 2: 7 degrees of freedom
• N = 3: 9 degrees of freedom
• etc.

Physics