Reconstruct a curve from bending data

Try

Suppose a plane path should turn at signed rate k(s)=s per unit distance. Set its initial tangent angle to zero. Then θ(s)=s²/2, and integrating (cos(s²/2),sin(s²/2)) produces a unit-speed curve with exactly the prescribed turning law.

The resulting coordinates need not have elementary antiderivatives. Existence of a geometric curve does not require a convenient closed formula. The mechanism is integration in the plane and a linear moving-frame ODE in space.

Predict

Predict what initial data are needed in addition to curvature. A starting point and tangent fix the plane construction. In space, an oriented orthonormal frame fixes the initial bending plane as well.

Predict whether periodic curvature and torsion force the curve to close. They do not: the frame may return with a rotation, and even a returning tangent can have a nonzero displacement integral over a period.

Experiment

Inspect three plane curvature prescriptions.

Signed curvature k(s)	Tangent angle from θ(0)=0	Geometric behavior
0	0	Straight line
c≠0	cs	Circle of radius 1/
s	s²/2	Increasing turning rate for positive s

Integrate the constant-curvature case explicitly and identify its center. For the variable case, keep the integral representation and verify its derivatives directly. A numerical approximation may display the path, but the derivative identities establish the prescribed curvature.

Understand

Given continuous k(s), define θ(s)=θ₀+∫k and γ(s)=p₀+∫(cos θ,sin θ). Then γ′ has norm one and its tangent-angle derivative is k. This proves existence. Two such curves with matched initial point and tangent have identical angle and position integrals, proving uniqueness.

For space curves, prescribe continuous κ>0 and τ and solve the Frenet frame ODE. If F has columns T,N,B, write F′=FA with skew-symmetric coefficient matrix A determined by κ and τ. Standard linear ODE existence and uniqueness supplies F on the interval. Its Gram matrix satisfies (FᵀF)′=Aᵀ(FᵀF)+(FᵀF)A. The identity matrix solves this equation with the orthonormal initial data; uniqueness forces FᵀF=I. Continuity preserves the positive orientation.

Now integrate γ′=T. The frame equations give T′=κN, so the prescribed κ and τ are indeed its Frenet invariants. If two constructions have the same functions, match initial frames by a rotation and initial points by a translation. ODE uniqueness and tangent integration then make the curves coincide. This proves uniqueness up to orientation-preserving rigid motion.

Apply

The ODE theorem is an explicit analysis dependency; the geometric part of the proof explains why its solution stays an orthonormal frame and realizes the requested invariants. Positive κ is essential for identifying the chosen N with the principal normal.

Repair “The fundamental theorem of curves classifies embedded closed curves.” It is a local or interval reconstruction theorem. Closing requires both a frame compatibility condition and zero net displacement. Avoiding self-intersections is another global requirement. None follows merely from prescribing smooth periodic coefficients.

For optional depth, changing torsion sign under a reflection explains why orientation-preserving and arbitrary rigid motions should be distinguished in uniqueness statements.

Continue

Use reconstruction as a proof of sufficiency: verify the ODE, preserved constraints and recovered geometric invariants. Continue to closed curves and global bounds for restrictions that cannot be read from a small interval alone.