A geodesic transports its own tangent with an affine parameter

Given a curve x(λ), its tangent contracts the covariant derivative of a vector carried along it. Parallel transport sets that derivative to zero. The components need not remain constant because the coordinate basis changes. With the Levi-Civita connection, inner products of parallel vectors are preserved. Comparing vectors at separated points requires such a rule; subtracting their coordinate arrays without specifying transport is generally not a coordinate-independent comparison. The result can depend on the chosen path when curvature or global holonomy is present.

In the flat polar plane a constant Cartesian vector pointing along x has orthonormal polar components (cosφ,−sinφ). Those numbers rotate as the point moves around a circle even though the Cartesian vector is parallel. This is basis variation rather than a physical torque. On a curved manifold, transporting around a small loop can yield a genuine holonomy. Flatness eliminates such local contractible-loop effects, but nontrivial global identifications can still matter. One should not infer global path independence merely from a local flat patch.

In polar coordinates, the affine geodesic equations are r double-dot minus r times angular-rate squared equals zero, and angular double-dot plus twice radial rate times angular rate divided by r equals zero. A straight line missing the origin has changing r and φ, so neither coordinate is generally linear in affine distance. The connection terms account for that changing basis. A fixed-r circle in a Euclidean plane instead requires acceleration; it is not a geodesic of the plane even though its speed can be constant.

A nonlinear reparameterization of a geodesic preserves its unparameterized path but changes the equation. The left side becomes a function times the tangent. Only transformations between affine parameters of the form λ′=aλ+b, with constant nonzero a, preserve the homogeneous form. A vanishing ordinary second derivative in some arbitrary parameter is therefore neither necessary nor sufficient for the full geometric statement. The metric norm of an affinely transported tangent stays constant, preserving whether it is timelike, null, or spacelike.

For a positive-metric geometry, geodesics are locally length-stationary and minimize length over sufficiently short segments. In Lorentzian geometry, timelike geodesics are proper-time stationary and locally maximizing before the relevant conjugate-point limitations. Null geodesics are derived from the affine tangent or a suitable constrained action, not by dividing by their zero proper time. Multiple geodesics can connect the same events, and a long stationary path need not provide a global optimum. These are geometric conditions separate from the coordinate ODE representation.

The vector lab illustrates the flat polar component rotation. Reset and set both controls to angle zero and magnitude one, predicting (1,0). Reset and set angle −90 degrees and magnitude one, predicting (0,−1). Interpret these as orthonormal polar components of the same fixed Cartesian vector at φ=0 and φ=π/2. They are separately entered component arrays; the lab has no moving coordinate frame, transport connection, or curved manifold. This comparison tests basis geometry without presenting playback as a physical transport trajectory.