The Frenet–Serret formulas

Theorem

The Frenet–Serret formulas

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Statement

Let \( \gamma : I \to \mathbb{R}^3 \) be a curve of class \( C^3 \) on an open interval \( I \subseteq \mathbb{R} \), parametrized by arc length \( s \) (so \( |\gamma'(s)| = 1 \) for all \( s \in I \)), and suppose the curvature \( \kappa(s) := |\gamma''(s)| \) satisfies \( \kappa(s) \gt 0 \) for every \( s \in I \). Then there is a uniquely determined orthonormal frame \( \{T(s), N(s), B(s)\} \) along \( \gamma \) — the tangent, principal normal, and binormal — and a scalar function \( \tau(s) \), the torsion, such that \[ T'(s) = \kappa(s)\,N(s), \qquad N'(s) = -\kappa(s)\,T(s) + \tau(s)\,B(s), \qquad B'(s) = -\tau(s)\,N(s). \] Moreover \( \kappa \gt 0 \) and \( \tau \) are, together, a complete set of invariants: two such curves with the same \( \kappa(s), \tau(s) \) on \( I \) differ only by a rigid motion of \( \mathbb{R}^3 \) (the Fundamental Theorem of space curves).

Why it matters

The Frenet–Serret equations are the local structure theorem for curves in \( \mathbb{R}^3 \): they replace the extrinsic, coordinate-dependent description \( \gamma(s) = (x(s), y(s), z(s)) \) with two intrinsic scalar functions, curvature and torsion, that say exactly how much the curve bends and how much it twists out of its osculating plane. This is the first instance, in the simplest possible setting, of the general strategy of differential geometry: attach a moving frame to a geometric object and read off its shape from the derivatives of the frame (the structure equations), rather than from the derivatives of a parametrization.

The formulas also make the classification of space curves an ODE problem: given any continuous \( \kappa(s) \gt 0 \) and \( \tau(s) \), solving the linear system above (and then integrating \( T \) to recover \( \gamma \)) produces a curve with exactly that curvature and torsion, unique up to rigid motion. Curvature and torsion are thus a complete "fingerprint" for the shape of a curve, analogous to the role eigenvalues play for symmetric matrices.

Hypotheses

\( \gamma \) is of class \( C^3 \).Two derivatives are needed to define \( T, N \) and a third to differentiate \( N \) and extract \( \tau \); with only \( C^2 \) regularity, \( \tau \) need not be defined, e.g. \( \gamma \) built by smoothly gluing planar arcs at an angle can have \( \gamma'' \) continuous but \( \gamma''' \) undefined at the join. \( \gamma \) is parametrized by arc length, \( |\gamma'(s)| \equiv 1 \).Without unit speed the frame vectors are still definable, but \( T'(s) = \kappa N \) fails: for a general parameter \( t \), \( \gamma''(t) \) has a tangential component from the changing speed, so \( T'(t) = \kappa\, |\gamma'(t)|\, N(t) \) picks up the extra speed factor and the clean equations require rescaling by \( |\gamma'(t)| \) throughout (the "non-arclength" Frenet formulas involve \( ds/dt \) explicitly). \( \kappa(s) \gt 0 \) for every \( s \in I \).At a point where \( \kappa = 0 \) (e.g. an inflection of a plane curve, or \( \gamma(s) = (s^3, 0, 0) \) reparametrized to unit speed near \( s = 0 \)), \( \gamma''(s) = 0 \) there, so \( N(s) = \gamma''(s)/\kappa(s) \) is a \( 0/0 \) expression: the principal normal — and hence \( B \) and \( \tau \) — is undefined or can jump discontinuously through such a point.

Proof

Since \( \gamma \) is parametrized by arc length, \( \langle \gamma'(s), \gamma'(s)\rangle = 1 \) for all \( s \). Set \( T(s) := \gamma'(s) \).

Definition of arc-length parametrization; \( T \) is by construction a unit vector field along \( \gamma \). A

Differentiating \( \langle T, T\rangle = 1 \) gives \( \langle T'(s), T(s)\rangle = 0 \), so \( T'(s) \perp T(s) \).

Product rule for the inner product: \( \frac{d}{ds}\langle T,T\rangle = 2\langle T', T\rangle \), and this equals \( \frac{d}{ds}(1) = 0 \). A

Define \( \kappa(s) := |T'(s)| = |\gamma''(s)| \geq 0 \). By hypothesis \( \kappa(s) \gt 0 \), so we may set \( N(s) := T'(s)/\kappa(s) \), a unit vector. By Step 2, \( N(s) \perp T(s) \), and by construction \[ T'(s) = \kappa(s)\,N(s). \]

This is the definition of curvature and principal normal for a unit-speed curve; division is legal exactly because \( \kappa \neq 0 \) — this is where the hypothesis \( \kappa \gt 0 \) is used. This is the first Frenet–Serret equation. B

Define \( B(s) := T(s) \times N(s) \). Then \( \{T(s), N(s), B(s)\} \) is a positively oriented orthonormal basis of \( \mathbb{R}^3 \) at each \( s \).

\( T \perp N \), both unit vectors, and the cross product of two orthonormal vectors is a unit vector orthogonal to both, completing a right-handed orthonormal triad. A

Differentiate \( B = T \times N \): \[ B' = T' \times N + T \times N' = (\kappa N) \times N + T \times N' = T \times N'. \]

Product (Leibniz) rule for the cross product, \( (u\times v)' = u' \times v + u \times v' \), valid since \( T, N \) are \( C^1 \) (as \( \gamma \in C^3 \)); and \( N \times N = 0 \). Uses Step 3 to substitute \( T' = \kappa N \). C

Since \( \{T,N,B\} \) is an orthonormal basis for each \( s \), write \( N'(s) = aT + bN + cB \) for scalar functions \( a,b,c \). Differentiating \( \langle N,N\rangle=1 \) gives \( \langle N',N\rangle = 0 \), so \( b = 0 \).

Expansion of a vector in an orthonormal basis (components are inner products with the basis vectors); same normalization argument as Step 2 applied to \( N \). A

Differentiating \( \langle T,N\rangle = 0 \) gives \( \langle T',N\rangle + \langle T,N'\rangle = 0 \). By Step 3, \( \langle T', N\rangle = \kappa\langle N,N\rangle = \kappa \). Hence \( \langle T,N'\rangle = -\kappa \), i.e. \( a = -\kappa \).

Product rule applied to the orthogonality relation \( T \perp N \), combined with Step 3 and the definition \( a = \langle N',T\rangle \) from the basis expansion in Step 6. B

Define \( \tau(s) := \langle N'(s), B(s)\rangle \), so \( c = \tau \). Combining Steps 6–8, \[ N'(s) = -\kappa(s)\,T(s) + \tau(s)\,B(s), \] the second Frenet–Serret equation.

Definition of torsion as the \( B \)-component of \( N' \); assembly of the coefficients found in Steps 6 and 7. B

From Step 5, \( B' = T \times N' \). Substitute \( N' = -\kappa T + \tau B \) from Step 8: \[ B' = T \times(-\kappa T + \tau B) = -\kappa\,(T\times T) + \tau\,(T \times B) = \tau\,(T\times B). \] Since \( \{T,N,B\} \) is right-handed orthonormal, \( T \times B = -N \), so \[ B'(s) = -\tau(s)\,N(s), \] the third Frenet–Serret equation.

Bilinearity of the cross product, \( T \times T = 0 \), and the standard identity \( T\times B = -N \) for a right-handed orthonormal frame with \( B = T\times N \) (equivalently \( N = B \times T \), \( T = N\times B \)). C

Uniqueness: any orthonormal frame \( \{\tilde T,\tilde N,\tilde B\} \) along \( \gamma \) satisfying \( \tilde T = \gamma' \) and analogous structure equations with \( \tilde\kappa \gt 0 \) must coincide with \( T,N,B,\kappa,\tau \) constructed above, because \( T=\gamma' \) is forced by arc-length parametrization, \( \kappa = |T'| \) and \( N = T'/\kappa \) are then forced by \( \kappa \gt 0 \), and \( B = T\times N \) is forced by orthonormality plus a fixed (say, right-handed) orientation convention; \( \tau \) is then forced as the coefficient in Step 8. This shows the frame and the two invariants are not a choice but a construction canonically attached to \( \gamma \).

Direct consequence of Steps 1, 3, 4, 8: each object was pinned down uniquely once \( \gamma \), the arc-length parametrization, \( \kappa\gt0 \), and an orientation convention are fixed. C

Result

\[ \begin{pmatrix} T' \\ N' \\ B' \end{pmatrix} = \begin{pmatrix} 0 & \kappa & 0 \\ -\kappa & 0 & \tau \\ 0 & -\tau & 0 \end{pmatrix} \begin{pmatrix} T \\ N \\ B \end{pmatrix} \]

Reading. The tangent turns towards the normal at rate \( \kappa \) (how sharply the curve bends within its osculating plane); the normal turns back against the tangent at the same rate \( \kappa \) but also leaks into the binormal direction at rate \( \tau \) (the plane of bending is itself twisting); the binormal turns against the normal at rate \( \tau \) (how fast the osculating plane rotates about the tangent). The coefficient matrix is skew-symmetric, which is exactly the algebraic expression of the frame staying orthonormal as \( s \) varies.

Scope. Applies to any \( C^3 \) unit-speed curve in \( \mathbb{R}^3 \) with strictly positive curvature at every point of its domain; it is a purely local, pointwise statement (no global assumptions such as closedness or embeddedness are needed). It does not directly apply at points of zero curvature, to curves of lower regularity, or to curves in dimension \( \neq 3 \) (there are analogous but structurally different generalized Frenet formulas for curves in \( \mathbb{R}^n \) and for surfaces).

Corollaries & converses

Planarity criterion (free corollary). \( \tau(s) \equiv 0 \) on \( I \) if and only if \( \gamma(I) \) lies in a single plane; proof is immediate from \( B' = -\tau N = 0 \), so \( B \) is constant, and then \( \frac{d}{ds}\langle \gamma - \gamma(s_0), B\rangle = \langle T, B\rangle = 0 \) forces \( \gamma \) to stay in the plane through \( \gamma(s_0) \) orthogonal to \( B \).
Helix criterion (free corollary). \( \kappa/\tau \) is a nonzero constant if and only if \( \gamma \) is a generalized helix (its tangent makes a constant angle with a fixed axis) — Lancret's theorem; this uses the Frenet equations but requires a short additional argument beyond them.
Converse / Fundamental Theorem. Given any continuous \( \kappa(s) \gt 0 \) and continuous \( \tau(s) \) on \( I \), the linear ODE system in the Result box has a solution \( \{T,N,B\} \) for any orthonormal initial frame at \( s_0 \) (Picard–Lindelöf, since the system is linear with continuous coefficients, hence globally Lipschitz on compacta), and integrating \( T \) from an initial point reconstructs a curve \( \gamma \) with exactly this \( \kappa,\tau \); it is unique up to a rigid motion of \( \mathbb{R}^3 \) fixing the initial data. So the converse holds in the strong sense of full existence-and-uniqueness, not merely a logical converse of the displayed equations.
The construction does not converse to \( \kappa \geq 0 \) curves in general: if \( \kappa \) is allowed to vanish, \( N \) and \( B \) may fail to extend continuously, so the "inverse problem" of prescribing \( \kappa, \tau \) with \( \kappa \) touching zero is genuinely more delicate and outside the scope of this theorem.

Fails without

Drop \( \kappa \gt 0 \): take \( \gamma(s) = (s, 0, 0) \), a straight line. Then \( \gamma''(s) = 0 \), so \( \kappa \equiv 0 \) and \( N(s) = \gamma''(s)/\kappa(s) \) is \( 0/0 \) — undefined everywhere, not just at isolated points. No binormal or torsion can be defined, and the "frame" collapses; the theorem's conclusion simply has no content.
Drop unit-speed parametrization: take \( \gamma(t) = (t^2, t^3, 0) \) parametrized by \( t \) (not arc length). Computing \( \frac{d}{dt}\big(\gamma'(t)/|\gamma'(t)|\big) \) directly does not equal \( \kappa(t) N(t) \); it equals \( \kappa(t)\,|\gamma'(t)|\, N(t) \), an extra factor of speed. Using the stated equations verbatim on a non-arclength parametrization gives numerically wrong curvature.
Drop \( C^3 \) regularity: let \( \gamma(s) = (s, f(s), 0) \) with \( f(s) = s^2\sin(1/s) \) for \( s\neq 0 \), \( f(0)=0 \), suitably smoothed to be \( C^2 \) but not \( C^3 \) at \( s=0 \). Then \( \kappa(s) = |\gamma''(s)| \) exists and is continuous, but \( \gamma'''(s) \) — needed to differentiate \( N \) and define \( \tau \) — oscillates without a limit as \( s \to 0 \); torsion is not defined at that point even though the frame \( T, N \) is.

Common errors

Applying \( T' = \kappa N \) directly to a curve parametrized by a general parameter \( t \) without first reparametrizing by arc length or inserting the \( |\gamma'(t)| \) correction factor.
Computing \( N \) as "the second derivative direction" \( \gamma''/|\gamma''| \) for a non-arclength curve, which silently assumes unit speed and gives the wrong vector when speed varies.
Forgetting the minus signs: writing \( N' = \kappa T + \tau B \) or \( B' = \tau N \) instead of \( N' = -\kappa T + \tau B \), \( B' = -\tau N \); the antisymmetry of the coefficient matrix is easy to lose from memory.
Concluding \( \tau = 0 \) forces the curve to be a circle or straight line — it only forces planarity; the planar curve can have varying curvature (e.g. an ellipse).
Treating curvature and torsion as coordinate-dependent quantities and expecting them to change under rotation/translation of \( \gamma \); by the Fundamental Theorem they are rigid-motion invariants, precisely the point of the construction.
Sign confusion in the torsion convention: some texts define \( B' = \tau N \) (opposite sign), which flips the sign of \( \tau \) relative to the convention used here; mixing sources without checking convention corrupts computations such as the helix formula.

Discussion

The Frenet–Serret equations were developed independently by Jean Frédéric Frenet (1847, in his thesis) and Joseph Alfred Serret (1851), building on Cauchy's earlier work on the osculating plane and Euler's study of curvature; they were among the first examples of what would later be recognized as a general phenomenon — a "moving frame" method, systematized by Élie Cartan around the turn of the 20th century into the theory of Cartan connections and structure equations, which now underlies the modern treatment of curvature for arbitrary manifolds and principal bundles.

Conceptually the skew-symmetric matrix \( \begin{pmatrix}0&\kappa&0\\-\kappa&0&\tau\\0&-\tau&0\end{pmatrix} \) is an element of the Lie algebra \( \mathfrak{so}(3) \) of the rotation group; the Frenet equations say the moving frame \( \{T,N,B\}\in SO(3) \) evolves by left multiplication by this Lie-algebra-valued function of \( s \). This is the same structural idea as the angular velocity vector describing a rotating rigid body in mechanics — curvature and torsion literally play the role of the two independent components of an "angular velocity" for the frame as it travels along the curve.

The Fundamental Theorem of space curves (existence and uniqueness given \( \kappa \gt 0, \tau \)) is a genuinely different kind of statement from the Frenet equations themselves: the equations are a computation (that such \( \kappa,\tau\) exist and satisfy a linear ODE once a curve is given), while the Fundamental Theorem is an inverse existence-and-uniqueness result proved via Picard–Lindelöf for linear systems together with an integration step to recover the curve from its tangent field; conflating "the formulas hold" with "the formulas characterize the curve" is a common source of imprecision even in otherwise careful treatments.

Common misconception: that "torsion measures how far the curve is from being planar" in an absolute, un-normalized sense — in fact \( \tau \) is a rate of rotation of the osculating plane per unit arc length, so a curve can wind tightly (large \( \tau \)) while staying visually close to a plane over a short arc, and comparing torsion values across curves of very different curvature scale is not directly meaningful without accounting for \( \kappa \) as well (as in the Lancret ratio \( \kappa/\tau \)).

Worked examples

Circular helix. Let \( \gamma(t) = (a\cos t, a\sin t, bt) \) with \( a\gt 0 \). Find \( \kappa,\tau \).

\( \gamma'(t) = (-a\sin t, a\cos t, b) \), \( |\gamma'(t)| = \sqrt{a^2+b^2} =: c \), constant. Arc length is \( s = ct \), so reparametrize: \( T(s) = \gamma'(t)/c \).

Speed is constant, so arc length is an affine rescaling of \( t \); this lets us convert derivatives in \( t \) to derivatives in \( s \) via \( \frac{d}{ds} = \frac{1}{c}\frac{d}{dt} \). A

\[ T'(s) = \frac{1}{c}\frac{dT}{dt} = \frac{1}{c^2}\gamma''(t) = \frac{1}{c^2}(-a\cos t, -a\sin t, 0). \] So \( \kappa = |T'(s)| = a/c^2 = \dfrac{a}{a^2+b^2} \), constant, and \( N(s) = (-\cos t,-\sin t,0) \).

Direct computation using Step 3 of the proof, \( \kappa = |T'(s)| \), \( N = T'/\kappa \). B

\( B = T\times N = \dfrac{1}{c}(b\sin t, -b\cos t, a) \). Then \( \dfrac{dB}{dt} = \dfrac{b}{c}(\cos t,\sin t,0) \), so \( B'(s) = \dfrac{1}{c}\dfrac{dB}{dt} = \dfrac{b}{c^2}(\cos t,\sin t,0) = -\dfrac{b}{c^2}N(s) \).

Definition \( B=T\times N \), chain rule for \( d/ds \), and comparison with \( B' = -\tau N \) from the third Frenet equation. B

\[ \kappa = \frac{a}{a^2+b^2}, \qquad \tau = \frac{b}{a^2+b^2} \]

Both constant — consistent with the helix criterion, since \( \kappa/\tau = a/b \) is constant.

Recovering planarity from \( \tau \equiv 0 \). Suppose a unit-speed \( \gamma:I\to\mathbb{R}^3 \) with \( \kappa\gt0 \) has \( \tau(s)\equiv 0 \). Show \( \gamma \) lies in a plane.

By the third Frenet equation, \( B'(s) = -\tau(s)N(s) = 0 \) for all \( s \), so \( B(s) \equiv B_0 \), a fixed unit vector.

Direct substitution of the hypothesis \( \tau\equiv0 \) into the Result box equation \( B' = -\tau N \). A

Fix \( s_0 \in I \) and define \( h(s) := \langle \gamma(s) - \gamma(s_0), B_0\rangle \). Then \( h(s_0)=0 \) and \[ h'(s) = \langle \gamma'(s), B_0\rangle = \langle T(s), B_0\rangle = 0, \] since \( T\perp B \) always (orthonormal frame).

Fundamental theorem of calculus setup plus orthogonality of the Frenet frame established in Step 4 of the proof. A

Since \( h'(s)\equiv 0 \) on the interval \( I \), \( h \) is constant, and \( h(s_0)=0 \) forces \( h(s)\equiv 0 \) on \( I \).

Standard one-variable calculus fact: a differentiable function with zero derivative on an interval is constant (Mean Value Theorem). A

\[ \langle \gamma(s) - \gamma(s_0), B_0\rangle = 0 \ \ \forall s\in I \]

So \( \gamma(I) \) lies entirely in the plane through \( \gamma(s_0) \) with normal vector \( B_0 \) — the constant binormal is exactly the plane's normal.

Problems

Show directly from the Frenet equations that \( \dfrac{d}{ds}(T\cdot N \times B)=0\), confirming the frame stays right-handed orthonormal (i.e. the triple scalar product \( \det[T\,N\,B] \) is constant, equal to its initial value 1).
Solution
Let \( \Delta(s) = \det[T\ N\ B] \) (columns). Differentiating a determinant of a matrix whose columns each depend on \( s \): \( \Delta' = \det[T'\ N\ B] + \det[T\ N'\ B] + \det[T\ N\ B'] \). Substitute the Frenet equations: \( T'=\kappa N \) makes the first determinant \( \kappa\det[N\ N\ B]=0 \) (repeated column). \( N' = -\kappa T+\tau B \) makes the second \( \det[T\ N'\ B] = -\kappa\det[T\ T\ B]+\tau\det[T\ B\ B] = 0+0=0 \) (each term has a repeated column). \( B'=-\tau N \) makes the third \( -\tau\det[T\ N\ N]=0 \). So \( \Delta'(s)\equiv 0 \), hence \( \Delta \) is constant, and since \( \{T,N,B\} \) is orthonormal and right-handed by construction (Step 4 of the proof), \( \Delta(s)\equiv 1 \).
A unit-speed curve has constant curvature \( \kappa = \kappa_0 \gt 0 \) and torsion \( \tau \equiv 0 \). Using the corollary on planarity together with the classical fact that a plane curve with constant curvature is a circular arc, identify the curve.
Solution
By the planarity corollary, \( \tau\equiv0 \) forces \( \gamma \) to lie in a fixed plane. Within that plane, curvature for a plane curve (with the induced signed curvature equal in magnitude to \( \kappa_0 \)) being constant and positive characterizes a circle of radius \( 1/\kappa_0 \) traversed at unit speed; hence \( \gamma \) is (a portion of) a circle of radius \( 1/\kappa_0 \).
Let \( \gamma \) be unit-speed with Frenet frame \( T,N,B \) and invariants \( \kappa,\tau \). Compute \( \gamma'''(s) \) in terms of \( T,N,B,\kappa,\tau \) and their derivatives, and hence express \( \tau \) using \( \gamma',\gamma'',\gamma''' \) (the classical formula \( \tau = \dfrac{(\gamma'\times\gamma'')\cdot\gamma'''}{|\gamma''|^2} \)).
Solution
\( \gamma' = T \). \( \gamma'' = T' = \kappa N \). Differentiate again using the product rule and the second Frenet equation: \( \gamma''' = \kappa' N + \kappa N' = \kappa' N + \kappa(-\kappa T + \tau B) = -\kappa^2 T + \kappa' N + \kappa\tau B \). Now \( \gamma'\times\gamma'' = T\times \kappa N = \kappa B \) (using \( T\times N = B \)). Then \( (\gamma'\times\gamma'')\cdot\gamma''' = \kappa B\cdot(-\kappa^2T+\kappa'N+\kappa\tau B) = \kappa\cdot\kappa\tau\,(B\cdot B) = \kappa^2\tau \), using orthonormality of \( T,N,B \) to kill the \( T,N \) terms. Also \( |\gamma''|^2 = \kappa^2 \). Dividing, \( \tau = \dfrac{(\gamma'\times\gamma'')\cdot\gamma'''}{\kappa^2} = \dfrac{(\gamma'\times\gamma'')\cdot\gamma'''}{|\gamma''|^2} \), as claimed (valid since \( \kappa\gt0 \)).
Consider \( \gamma(t) = (t,t^2,t^3) \) for \( t \) near \( 0 \) (not arc-length parametrized). Using the invariant formulas \( \kappa = \dfrac{|\gamma'\times\gamma''|}{|\gamma'|^3} \) and \( \tau = \dfrac{(\gamma'\times\gamma'')\cdot\gamma'''}{|\gamma'\times\gamma''|^2} \) (which agree with the arc-length definitions after reparametrization — state why), compute \( \kappa(0) \) and \( \tau(0) \).
Solution
These formulas are the reparametrization-invariant versions of \( \kappa=|T'(s)|,\ \tau=\) coefficient in \( N' \); they must reduce to the arc-length formulas when \( |\gamma'|\equiv1 \) (direct substitution recovers the Problem 3 formula and the definition \( \kappa=|\gamma''|\) since \(|\gamma'\times\gamma''|=|\gamma''|\) when \(\gamma'\perp\gamma''\) after arclength differentiation), and are invariant under \( C^3 \) orientation-preserving reparametrization by the chain rule, which is why they may legally be applied to any regular \( C^3 \) parametrization, not just unit speed. At \( t=0 \): \( \gamma'(t)=(1,2t,3t^2)\Rightarrow \gamma'(0)=(1,0,0) \). \( \gamma''(t)=(0,2,6t)\Rightarrow\gamma''(0)=(0,2,0) \). \( \gamma'''(t)=(0,0,6)\Rightarrow\gamma'''(0)=(0,0,6) \). Then \( \gamma'(0)\times\gamma''(0) = (1,0,0)\times(0,2,0) = (0\cdot0-0\cdot2,\ 0\cdot0-1\cdot0,\ 1\cdot2-0\cdot0) = (0,0,2) \), so \( |\gamma'\times\gamma''| = 2 \), \( |\gamma'|=1 \), giving \( \kappa(0) = 2/1^3 = 2 \). And \( (\gamma'\times\gamma'')\cdot\gamma''' = (0,0,2)\cdot(0,0,6) = 12 \), so \( \tau(0) = 12/2^2 = 3 \).
Suppose two unit-speed curves \( \gamma_1,\gamma_2:I\to\mathbb{R}^3 \) with \( \kappa\gt0 \) have identical curvature and torsion functions \( \kappa(s),\tau(s) \) on \( I \), and at some \( s_0\in I \) satisfy \( \gamma_1(s_0)=\gamma_2(s_0) \) and \( T_1(s_0)=T_2(s_0),\ N_1(s_0)=N_2(s_0) \) (hence also \( B_1(s_0)=B_2(s_0) \)). Explain, citing the relevant uniqueness theorem, why \( \gamma_1\equiv\gamma_2 \) on \( I \), without assuming any additional rigid motion is needed.
Solution
Both \( (T_1,N_1,B_1) \) and \( (T_2,N_2,B_2) \) solve the same linear ODE system \( F' = A(s)F \) with \( A(s)=\begin{pmatrix}0&\kappa&0\\-\kappa&0&\tau\\0&-\tau&0\end{pmatrix} \) (Result box), viewing \( F=(T,N,B) \) as a matrix-valued function, and by hypothesis they share the same initial condition \( F_1(s_0)=F_2(s_0) \). Since \( A(s) \) is continuous, the Picard–Lindelöf existence-uniqueness theorem for linear ODEs guarantees a unique solution through any initial condition, so \( F_1(s)\equiv F_2(s) \) on all of \( I \); in particular \( T_1(s)\equiv T_2(s) \). Then \( \gamma_1(s) = \gamma_1(s_0)+\int_{s_0}^s T_1(u)\,du = \gamma_2(s_0)+\int_{s_0}^s T_2(u)\,du = \gamma_2(s) \) for all \( s\in I \), since the integrands and the base points agree. No rigid motion is needed because the initial frames were assumed already equal, not merely congruent; this is the "pinned" special case of the Fundamental Theorem's uniqueness clause.