Regular curves and arc length

Try

The path γ(t)=(2t,0), 0≤t≤1, and the path α(u)=(u,0), 0≤u≤2, trace the same segment at different speeds. Both have length two. Differential geometry must separate a trace in space from the clock used to traverse it.

A C¹ parametrized curve is regular when its velocity never vanishes. Regularity is a condition on the parametrization, not merely on its image. The trace of (t³,0) is a straight line, but its velocity vanishes at zero, so that particular parametrization is not regular there.

Predict

Predict which quantity should survive a smooth change of parameter: velocity, speed, or integrated length. Velocity and speed change; length does not. Predict whether reversing direction changes length. The absolute value in the speed formula should prevent that.

A unit-speed parameter uses distance traveled as its clock. It does not require the curve to be straight, embedded or closed. It requires a positive speed so that the old and new parameters can be locally inverted.

Experiment

Compare three descriptions of the same line segment.

Parametrization and interval	Speed	Length	Regular throughout?
(2t,0), t in [0,1]	2	2	Yes
(u,0), u in [0,2]	1	2	Yes
(2v³,0), v in [0,1]	6v²	2	Not at zero

Integrate the third speed instead of assuming that a zero instantaneous velocity makes the entire length zero. Then identify exactly where its inverse parameter fails to be differentiable. This is a useful distinction between an integral quantity and a local rank condition.

Understand

Fix a regular curve γ and a base parameter t₀. Define s(t)=∫ from t₀ to t ||γ′(r)||dr. The fundamental theorem of calculus gives s′(t)=||γ′(t)||>0. Hence s is strictly increasing and has a local differentiable inverse by the inverse function theorem. For the reparametrized curve, dγ/ds=γ′/||γ′||, so its speed is one.

Under a C¹ reparametrization t=φ(u) with derivative nowhere zero, the chain rule gives new speed ||γ′(φ(u))|| |φ′(u)|. Change of variables proves the length integral is unchanged. The absolute derivative handles orientation reversal correctly.

Endpoint distance is bounded by curve length. If the displacement q−p is nonzero, dot ∫γ′ with its unit direction. The resulting displacement norm is at most ∫||γ′||. If the endpoints coincide, the inequality follows from nonnegativity. Equality constrains the tangent direction almost everywhere; it is stronger than merely having a short-looking trace.

Apply

Repair “Every smooth-looking trace has a unique arc-length parameter.” Arc length is unique only up to an additive constant and a choice of direction on a connected regular parametrized segment. A closed curve also needs a starting point, and a trace traversed several times carries multiplicity.

For optional depth, the length of a merely continuous curve is defined through polygonal approximations. The integral formula agrees with that definition for C¹ curves; a complete proof uses uniform approximation of the derivative on compact intervals. This course uses the differentiable setting while identifying where regularity enters later frame constructions.

Continue

Keep trace, parametrization, orientation and speed separate in every calculation. The MIT 18.950 syllabus motivates a rigorous curvature-centered progression; these are original examples and exercises. Continue to plane curvature and turning to measure how the unit tangent changes.