# Frenet's formulas

The Frenet formulas ( Frenet formulas ), named after the French mathematician Jean Frédéric Frenet , are the central equations in the theory of space curves , an important part of differential geometry . They are also called derivative equations or Frenet-Serret formulas , the latter after Joseph Serret , who gave the formulas in full. In this article, the Frenetian formulas are first presented in the three-dimensional visual space, followed by the generalization to higher dimensions. ${\ displaystyle \ mathbb {R} ^ {3}}$

## The three-dimensional case

### Overview

The formulas use an orthonormal basis (unit vectors that are perpendicular to each other in pairs) from three vectors (tangent vector , principal normal vector and binormal vector ), which describe the local behavior of the curve , and express the derivatives of these vectors according to the arc length as linear combinations of the three vectors mentioned. The scalar variables curvature and torsion , which are characteristic of the curve, occur. ${\ displaystyle {\ vec {t}}}$${\ displaystyle {\ vec {n}}}$${\ displaystyle {\ vec {b}}}$${\ displaystyle {\ vec {r}} \, (s)}$ ${\ displaystyle s}$ ${\ displaystyle \ kappa}$ ${\ displaystyle \ tau}$

${\ displaystyle {\ frac {\ mathrm {d} {\ vec {r}}} {\ mathrm {d} s}} = {\ vec {t}} \ qquad {\ frac {\ mathrm {d} {\ vec {t}}} {\ mathrm {d} s}} = \ kappa {\ vec {n}} \ qquad {\ vec {t}} \ times {\ vec {n}} = {\ vec {b} }}$
${\ displaystyle {\ frac {\ mathrm {d} {\ vec {b}}} {\ mathrm {d} s}} = - \ tau {\ vec {n}} \ qquad {\ frac {\ mathrm {d } {\ vec {n}}} {\ mathrm {d} s}} = \ tau {\ vec {b}} - \ kappa {\ vec {t}}}$

### Concept formation

The vector connects two points on the path and has the length . For goes against the arc length of the track section between and : ${\ displaystyle \ Delta {\ vec {r}} = {\ vec {r}} {} _ {2} - {\ vec {r}} {} _ {1}}$${\ displaystyle | \ Delta {\ vec {r}} \, | = {\ sqrt {\ Delta {\ vec {r}} \ cdot \ Delta {\ vec {r}}}}}$${\ displaystyle \ Delta {\ vec {r}} \ rightarrow 0}$${\ displaystyle | \ Delta {\ vec {r}} \, |}$${\ displaystyle {\ vec {r}} _ {1}}$${\ displaystyle {\ vec {r}} _ {2}}$

${\ displaystyle \ mathrm {d} s = | \ mathrm {d} {\ vec {r}} \, | = {\ sqrt {\ mathrm {d} {\ vec {r}} \ cdot \ mathrm {d} {\ vec {r}}}}}$

The arc length of the path is from the starting point to the point${\ displaystyle {\ vec {r}} _ {0}}$${\ displaystyle {\ vec {r}}}$

${\ displaystyle s = \ int _ {{\ vec {r}} _ {0}} ^ {\ vec {r}} \ mathrm {d} s = \ int _ {{\ vec {r}} _ {0 }} ^ {\ vec {r}} {\ sqrt {\ mathrm {d} {\ vec {r}} \ cdot \ mathrm {d} {\ vec {r}}}}}$

Given is a space curve parameterized by the arc length : ${\ displaystyle s}$

${\ displaystyle {\ vec {r}} = {\ vec {r}} (s)}$.

For a point on the curve is obtained by deriving by the unit tangent vector , the case of a change of the arc length, indicating the local direction of the curve, so the change in position: ${\ displaystyle {\ vec {r}} (s)}$${\ displaystyle s}$

${\ displaystyle {\ vec {t}} (s) = {\ frac {\ mathrm {d} {\ vec {r}} (s)} {\ mathrm {d} s}} = {\ vec {r} } \, '(s)}$.

Because of the amount of the derivative is equal to 1; thus it is a unit vector. The tangent unit vector generally changes its direction along the path, but not its length (it always remains a unit vector) or . From this one can conclude that the derivative of the tangent unit vector is perpendicular to it: ${\ displaystyle \ mathrm {d} s = | \ mathrm {d} {\ vec {r}} \, |}$${\ displaystyle | {\ vec {t}} \, | = 1}$${\ displaystyle {\ vec {t}} \ cdot {\ vec {t}} = 1}$

${\ displaystyle {\ vec {t}} \ cdot {\ frac {\ mathrm {d} {\ vec {t}}} {\ mathrm {d} s}} = {\ frac {1} {2}} { \ frac {\ mathrm {d} \ left ({\ vec {t}} \ cdot {\ vec {t}} \, \ right)} {\ mathrm {d} s}} = {\ frac {1} { 2}} {\ frac {\ mathrm {d} 1} {\ mathrm {d} s}} = 0 \ quad \ Longrightarrow \ quad {\ vec {t}} \ perp {\ frac {\ mathrm {d} { \ vec {t}}} {\ mathrm {d} s}}}$

The trajectory can be expanded into a Taylor series : ${\ displaystyle s}$

${\ displaystyle {{\ vec {r}} (s + \ Delta s) = {\ vec {r}} (s) + {\ frac {\ mathrm {d} {\ vec {r}}} {\ mathrm { d} s}} (s) \ Delta s + {\ frac {1} {2}} {\ frac {\ mathrm {d} {} ^ {2} {\ vec {r}}} {\ mathrm {d} s ^ {2}}} (s) \ Delta s ^ {2} + {\ mathcal {O}} (\ Delta s ^ {3}) = {\ vec {r}} (s) + {\ vec { t}} (s) \ Delta s + {\ frac {1} {2}} {\ vec {t}} \, '(s) \ Delta s ^ {2} + {\ mathcal {O}} (\ Delta s ^ {3})}}$
Path curve (red) with tangent unit vectors and osculating circle with radius For illustration purposes, the size is exaggerated.${\ displaystyle {\ boldsymbol {T}}}$${\ displaystyle \ rho.}$${\ displaystyle \ mathrm {d} s}$

The second-order approximation curve in is a parabola that lies in the osculating plane spanned by and . ${\ displaystyle \ Delta s}$${\ displaystyle {\ vec {t}}}$${\ displaystyle {\ vec {t}} \, '}$

In order to calculate the amount of, one considers the osculating circle , which at the observed point on the path clings to its approximation parabola, i. H. the circle that goes through the given curve point has the same direction there as the curve and also coincides with the curve in the second derivative. Let the angle between tangent vectors of neighboring curve points ( and ) be . This applies ${\ displaystyle {\ vec {t}} \, '}$${\ displaystyle {\ vec {t}} \, (s)}$${\ displaystyle {\ vec {t}} \, (s + \ mathrm {d} s)}$${\ displaystyle \ mathrm {d} \ varphi}$

${\ displaystyle \ mathrm {d} \ varphi = | \ mathrm {d} {\ vec {t}} \, | / | {\ vec {t}} \, | = | \ mathrm {d} {\ vec { t}} \, |}$

Since the tangent unit vector is perpendicular to the radius vector of the osculating circle, the angle between neighboring radius vectors ( ) is identical to the angle between the tangent vectors of neighboring curve points ( ). From this it follows as an osculating radius (= radius of curvature): ${\ displaystyle \ mathrm {d} \ varphi = \ mathrm {d} s / \ varrho}$${\ displaystyle \ mathrm {d} \ varphi = | \ mathrm {d} {\ vec {t}} \, |}$${\ displaystyle \ varrho}$

${\ displaystyle {\ frac {\ mathrm {d} \ varphi} {\ mathrm {d} s}} = {\ frac {1} {\ varrho}} \ quad \ Longrightarrow \ quad \ left | {\ frac {\ mathrm {d} {\ vec {t}}} {\ mathrm {d} s}} \ right | = {\ frac {1} {\ varrho}}}$

The reciprocal radius of curvature is called curvature and indicates the strength of the change in direction over the arc length, i.e. the amount of : ${\ displaystyle \ varrho (s) \,}$ ${\ displaystyle \ kappa (s) \,}$${\ displaystyle {\ vec {t}} \, '}$

${\ displaystyle \ kappa (s) = {\ frac {1} {\ varrho (s)}} = | {\ vec {t}} \, '(s) | = | {\ vec {r}} \, '' (s) |}$.

Normalization of returns the main normal unit vector (curvature vector). Since the tangent unit vector is tangential to the oscillation circle and the main normal unit vector is perpendicular to it, it indicates the direction to the oscillation circle center. It is the direction that is changing. ${\ displaystyle {\ vec {t}} \, '(s)}$ ${\ displaystyle {\ vec {n}} (s)}$${\ displaystyle {\ vec {n}} (s)}$${\ displaystyle {\ vec {t}} (s)}$

${\ displaystyle {\ vec {n}} (s) = {\ frac {{\ vec {t}} \, '(s)} {| {\ vec {t}} \,' (s) |}} = {\ frac {{\ vec {r}} \, '' (s)} {| {\ vec {r}} \, '' (s) |}} = \ varrho (s) \, {\ vec {t}} \, '(s) = \ varrho (s) \, {\ vec {r}} \,' '(s)}$.

The normal vector of the osculating plane is determined with the help of the vector product of the tangent unit vector and the main normal unit vector and is called the binormal unit vector :

${\ displaystyle {\ vec {b}} (s) = {\ vec {t}} (s) \ times {\ vec {n}} (s)}$
Tangent unit vector  T , main normal unit vector  N and binormal unit vector  B form the accompanying tripod of a space curve.
The oscillation plane is also shown; it is spanned by the main normal and tangent unit vector.

Tangent, principal normal and binormal unit vectors form an orthonormal basis of , i. that is, these vectors all have the magnitude 1 and are mutually perpendicular in pairs. This orthonormal basis is also called the accompanying tripod of the curve. The Frenet formulas express the derivatives of the mentioned basis vectors as linear combinations of these basis vectors: ${\ displaystyle \ mathbb {R} ^ {3}}$

${\ displaystyle {\ vec {t}} \, '(s) = \ kappa (s) \, {\ vec {n}} (s)}$
${\ displaystyle {\ vec {n}} \, '(s) = - \ kappa (s) \, {\ vec {t}} (s) + \ tau (s) \, {\ vec {b}} (s)}$
${\ displaystyle {\ vec {b}} \, '(s) = - \ tau (s) \, {\ vec {n}} (s)}$

or in a memorable matrix notation

${\ displaystyle {\ begin {pmatrix} {\ vec {t}} \\ {\ vec {n}} \\ {\ vec {b}} \ end {pmatrix}} '\, = \, {\ begin { pmatrix} 0 & \ kappa & 0 \\ - \ kappa & 0 & \ tau \\ 0 & - \ tau & 0 \ end {pmatrix}} {\ begin {pmatrix} {\ vec {t}} \\ {\ vec {n}} \ \ {\ vec {b}} \ end {pmatrix}}}$.

Here stand for the curvature and for the twist (torsion) of the curve in the curve point under consideration. ${\ displaystyle \ kappa (s) \,}$${\ displaystyle \ tau (s) \,}$

Animation of the accompanying tripod, as well as the curvature and torsion function. The rotation of the binormal vector can be clearly seen at the peak values ​​of the torsion function.

Using the accompanying tripod, the curvature and torsion can each be illustrated as a change in direction of a certain tangent unit vector. There are some (partly animated) graphic illustrations for this .

The change in direction of the binormal unit vector corresponds to the torsion : ${\ displaystyle \ displaystyle \ tau (s)}$

• The greater the torsion, the faster the binormal unit vector changes depending on its direction. If the torsion is 0 everywhere, the space curve is a flat curve , i.e. that is, there is a common plane on which all points of the curve lie.${\ displaystyle {\ vec {b}} (s)}$${\ displaystyle \ displaystyle s}$

The change in direction of the tangent unit vector corresponds to the curvature : ${\ displaystyle \ displaystyle \ kappa (s)}$

• The stronger the curvature , the faster the tangent unit vector changes depending on its direction.${\ displaystyle \ displaystyle \ kappa (s)}$${\ displaystyle {\ vec {t}} (s)}$${\ displaystyle \ displaystyle s}$

Points of the space curve with the curvature 0, in which there is no osculating circle, in which the derivative of the tangent unit vector is the zero vector , are called turning points and must be treated separately. There the terms normal vector and binormal vector lose their meaning. If all points have the curvature 0, the space curve is a straight line .

The Frenet formulas can also be formulated with the Darboux vector .

### Frenet's formulas as a function of other parameters

The above formulas are defined as a function of the arc length s. Often, however, the space curves are dependent on other parameters, e.g. B. given by time. To express the relationships by the new parameter t, one uses the following relation:

${\ displaystyle {\ frac {\ mathrm {d} s} {\ mathrm {d} t}} = \ left | {\ frac {\ mathrm {d} {\ vec {r}}} {\ mathrm {d} t}} \ right | = \ left | {\ dot {\ vec {r}}} \ right |}$

thus one can rewrite the derivatives from to : ${\ displaystyle \ textstyle {\ frac {\ mathrm {d}} {\ mathrm {d} s}}}$${\ displaystyle \ textstyle {\ frac {\ mathrm {d}} {\ mathrm {d} t}}}$

${\ displaystyle {\ frac {\ mathrm {d}} {\ mathrm {d} s}} = {\ frac {\ mathrm {d} t} {\ mathrm {d} s}} {\ frac {\ mathrm { d}} {\ mathrm {d} t}} = {\ frac {1} {\ tfrac {\ mathrm {d} s} {\ mathrm {d} t}}} {\ frac {\ mathrm {d}} {\ mathrm {d} t}} = {\ frac {1} {| {\ dot {\ vec {r}}} \, |}} {\ frac {\ mathrm {d}} {\ mathrm {d} t}}}$

Consequently, the Frenet formulas of a space curve that is parameterized with respect to (the derivatives according to are marked with a point) are: ${\ displaystyle {\ vec {r}} (t)}$${\ displaystyle t}$${\ displaystyle t}$

${\ displaystyle {\ frac {{\ dot {\ vec {r}}} (t)} {| {\ dot {\ vec {r}}} (t) |}} = {\ vec {t}}}$
${\ displaystyle {\ frac {{\ dot {\ vec {t}}} (t)} {| {\ dot {\ vec {r}}} (t) |}} = \ kappa {\ vec {n} }}$
${\ displaystyle {\ vec {t}} \ times {\ vec {n}} = {\ vec {b}}}$
${\ displaystyle {\ frac {{\ dot {\ vec {b}}} (t)} {| {\ dot {\ vec {r}}} (t) |}} = - \ tau {\ vec {n }}}$
${\ displaystyle {\ frac {{\ dot {\ vec {n}}} (t)} {| {\ dot {\ vec {r}}} (t) |}} = \ tau {\ vec {b} } - \ kappa {\ vec {t}}}$

A curve that can be differentiated three times according to t has the following characteristic vectors and scalars at every parameter position with : ${\ displaystyle {\ vec {r}} (t)}$${\ displaystyle {\ dot {\ vec {r}}} (t) \ times {\ ddot {\ vec {r}}} (t) \ neq 0}$

 Tangent vector ${\ displaystyle {\ vec {t}} (t) = {\ frac {{\ dot {\ vec {r}}} (t)} {\ left | {\ dot {\ vec {r}}} (t ) \ right |}}}$ Binormal vector ${\ displaystyle {\ vec {b}} (t) = {\ frac {{\ dot {\ vec {r}}} (t) \ times {\ ddot {\ vec {r}}} (t)} { \ left | {\ dot {\ vec {r}}} (t) \ times {\ ddot {\ vec {r}}} (t) \ right |}} = {\ frac {{\ dot {\ vec { r}}} (t) \ times {\ dot {\ vec {t}}} (t)} {\ left | {\ dot {\ vec {r}}} (t) \ right | \ left | {\ dot {\ vec {t}}} (t) \ right |}}}$ Principal normal vector ${\ displaystyle {\ vec {n}} (t) = {\ vec {b}} (t) \ times {\ vec {t}} (t) = {\ frac {\ left ({\ dot {\ vec {r}}} (t) \ times {\ ddot {\ vec {r}}} (t) \ right) \ times {\ dot {\ vec {r}}} (t)} {\ left | {\ dot {\ vec {r}}} (t) \ times {\ ddot {\ vec {r}}} (t) \ right | \ left | {\ dot {\ vec {r}}} (t) \ right |}} = {\ frac {{\ dot {\ vec {t}}} (t)} {\ left | {\ dot {\ vec {t}}} (t) \ right |}}}$ curvature ${\ displaystyle \ kappa (t) = {\ frac {\ left | {\ dot {\ vec {r}}} (t) \ times {\ ddot {\ vec {r}}} (t) \ right |} {\ left | {\ dot {\ vec {r}}} (t) \ right | ^ {3}}} = {\ frac {\ left | {\ dot {\ vec {t}}} (t) \ right |} {\ left | {\ dot {\ vec {r}}} (t) \ right |}}}$ torsion ${\ displaystyle \ tau (t) = {\ frac {\ left ({\ dot {\ vec {r}}} (t) \ times {\ ddot {\ vec {r}}} (t) \ right) \ cdot {\ overset {\ ldots} {\ vec {r}}} (t)} {\ left | {\ dot {\ vec {r}}} (t) \ times {\ ddot {\ vec {r}} } (t) \ right | ^ {2}}}}$

## The Frenet's formulas in n dimensions

For the -dimensional case, some technical requirements are first required. A curve parameterized according to arc length and continuously differentiable times is called a Frenet curve if the vectors of the first derivatives are linearly independent at each point . The accompanying Frenet leg consists of vectors that meet the following conditions: ${\ displaystyle n}$${\ displaystyle n}$${\ displaystyle s \ mapsto {\ vec {r}} (s)}$${\ displaystyle {\ vec {r}} \, '(s), {\ vec {r}} \,' '(s), \ ldots, {\ vec {r}} ^ {(n-1)} (s)}$${\ displaystyle n-1}$${\ displaystyle s}$${\ displaystyle n}$${\ displaystyle n}$${\ displaystyle {\ vec {e_ {i}}} = {\ vec {e_ {i}}} (s)}$

1. ${\ displaystyle {\ vec {e_ {1}}}, \ ldots, {\ vec {e_ {n}}}}$are orthonormal and positively oriented .
2. For each, the linear hulls of and match.${\ displaystyle i = 1, \ ldots n-1}$${\ displaystyle {\ vec {e_ {1}}}, \ ldots {\ vec {e_ {i}}}}$${\ displaystyle {\ vec {r}} \, ', \ ldots, {\ vec {r}} ^ {(i)}}$
3. ${\ displaystyle \ langle {\ vec {r}} ^ {(i)}, {\ vec {e_ {i}}} \ rangle> 0}$for everyone .${\ displaystyle i = 1, \ ldots n-1}$

You have to read these conditions point by point, that is, they apply to every parameter point . In the three-dimensional case described above, the vectors , and form an accompanying Frenet tripod. With the help of the Gram-Schmidt orthogonalization method, one can show that Frenet legs for Frenet curves exist and are uniquely determined. In the -dimensional case, differential equations are obtained for the components of the accompanying Frenet- leg: ${\ displaystyle s}$${\ displaystyle {\ vec {e_ {1}}} = {\ vec {t}}}$${\ displaystyle {\ vec {e_ {2}}} = {\ vec {n}}}$${\ displaystyle {\ vec {e_ {3}}} = {\ vec {b}}}$${\ displaystyle n}$${\ displaystyle n}$${\ displaystyle n}$

Be a Frenet curve with an accompanying Frenet leg . Then there are clearly certain functions , where -mal is continuously differentiable and only assumes positive values, so that the following Frenetian formulas apply: ${\ displaystyle s \ mapsto {\ vec {r}} (s)}$${\ displaystyle n}$${\ displaystyle {\ vec {e_ {1}}}, \ ldots, {\ vec {e_ {n}}}}$${\ displaystyle \ kappa _ {1}, \ ldots, \ kappa _ {n-1}}$${\ displaystyle \ kappa _ {i} \,}$ ${\ displaystyle (n-1-i)}$${\ displaystyle i \ leq n-2}$

${\ displaystyle {\ begin {pmatrix} {\ vec {e_ {1}}} \\ {\ vec {e_ {2}}} \\\ vdots \\\ vdots \\ {\ vec {e_ {n-1 }}} \\ {\ vec {e_ {n}}} \ end {pmatrix}} '\, = \, {\ begin {pmatrix} 0 & \ kappa _ {1} & 0 & \ ldots & \ ldots & 0 \\ - \ kappa _ {1} & 0 & \ kappa _ {2} & 0 & \ ldots & \ vdots \\ 0 & - \ kappa _ {2} & 0 & \ ddots & \ ddots & \ vdots \\\ vdots & 0 & \ ddots & \ ddots & \ ddots & 0 \\\ vdots & \ vdots & \ ddots & \ ddots & 0 & \ kappa _ {n-1} \\ 0 & \ ldots & \ ldots & 0 & - \ kappa _ {n-1} & 0 \ end {pmatrix}} { \ begin {pmatrix} {\ vec {e_ {1}}} \\ {\ vec {e_ {2}}} \\\ vdots \\\ vdots \\ {\ vec {e_ {n-1}}} \ \ {\ vec {e_ {n}}} \ end {pmatrix}}}$

${\ displaystyle \ kappa _ {i}}$is called the -th Frenet curvature, the last one is also called the torsion of the curve . The curve is contained in a hyperplane exactly when the torsion disappears. In many applications it can be differentiated as often as desired; this property is then transferred to the Frenet curvatures. ${\ displaystyle i}$${\ displaystyle \ kappa _ {n-1}}$${\ displaystyle s \ mapsto {\ vec {r}} (s)}$

## Law of local curve theory

Conversely, one can construct curves for given Frenet curvatures, more precisely the so-called main theorem of local curve theory applies :

Let there be any number of differentiable real-valued functions defined on an interval , whereby they only assume positive values. A point and a positively oriented orthonormal system are given for a point . Then there is exactly one infinitely often differentiable Frenet curve with ${\ displaystyle I}$${\ displaystyle \ kappa _ {1}, \ ldots, \ kappa _ {n-1}}$${\ displaystyle \ kappa _ {1}, \ ldots, \ kappa _ {n-2}}$${\ displaystyle s_ {0} \ in I}$${\ displaystyle p_ {0} \ in \ mathbb {R} ^ {n}}$${\ displaystyle {\ vec {e_ {1}}} ^ {0}, \ ldots, {\ vec {e_ {n}}} ^ {0}}$${\ displaystyle {\ vec {r}}: I \ rightarrow \ mathbb {R} ^ {n}, \, s \ mapsto {\ vec {r}} (s)}$

• ${\ displaystyle {\ vec {r}} (s_ {0}) = p_ {0}}$,
• ${\ displaystyle {\ vec {e_ {1}}} ^ {0}, \ ldots, {\ vec {e_ {n}}} ^ {0}}$is the accompanying Frenet leg in the parameter point ,${\ displaystyle n}$${\ displaystyle s_ {0}}$
• ${\ displaystyle \ kappa _ {1}, \ ldots, \ kappa _ {n-1}}$are the Frenet curvatures of .${\ displaystyle {\ vec {r}}}$

The first two conditions determine the location and directions at the parameter point , the further course of the curve is then determined by the curvature specifications of the third condition. The proof is based on the Frenet formulas given above and the solution theory of linear differential equation systems is used . ${\ displaystyle s_ {0}}$