Curves in Space: Serret–Frenet Formulas

In three-dimensional Euclidean space (ℝ³), a curve is a continuous mapping from an interval of the real line to ℝ³. We often represent a curve parametrically by a vector function $\vec{r}(t) = (x(t), y(t), z(t))$, where $t$ is the parameter. To study the local geometric properties of a curve, such as its curvature and torsion, we use a special set of equations known as the Serret–Frenet formulas. These formulas describe how a set of three mutually orthogonal unit vectors, called the Frenet frame, change as we move along the curve.

The Frenet Frame

For a regular curve (one whose velocity vector is never zero), we can define the following three unit vectors at each point $\vec{r}(t)$:

• Tangent Vector ($\vec{T}$): This is a unit vector in the direction of the curve's velocity. If the curve is parameterized by arc length $s$, then $\vec{T}(s) = \vec{r}'(s)$. If parameterized by $t$, then $\vec{T}(t) = \frac{\vec{r}'(t)}{||\vec{r}'(t)||}$.
• Normal Vector ($\vec{N}$): This is a unit vector that points in the direction of the curve's acceleration, specifically in the direction of the principal component of acceleration perpendicular to the tangent. It lies in the plane of the curve's curvature. $\vec{N}(s) = \frac{\vec{T}'(s)}{||\vec{T}'(s)||}$.
• Binormal Vector ($\vec{B}$): This is a unit vector orthogonal to both $\vec{T}$ and $\vec{N}$, completing a right-handed orthonormal basis. It is defined as $\vec{B}(s) = \vec{T}(s) \times \vec{N}(s)$.

The Serret–Frenet Formulas

These formulas express the derivatives of the Frenet frame vectors with respect to the arc length parameter $s$. They reveal fundamental properties of the curve: curvature and torsion.

For a curve parameterized by arc length $s$:

• $\frac{d\vec{T}}{ds} = \kappa \vec{N}$
• $\frac{d\vec{N}}{ds} = -\kappa \vec{T} + \tau \vec{B}$
• $\frac{d\vec{B}}{ds} = -\tau \vec{N}$

Here, $\kappa$ (kappa) is the curvature of the curve, which measures how sharply the curve bends. $\tau$ (tau) is the torsion of the curve, which measures how much the curve twists out of its osculating plane.

Curvature ($\kappa$)

The curvature $\kappa$ is the magnitude of the rate of change of the tangent vector with respect to arc length. A larger $\kappa$ means the curve is bending more sharply. For a curve $\vec{r}(t)$ parameterized by $t$, the curvature is given by:

$\kappa(t) = \frac{||\vec{r}'(t) \times \vec{r}''(t)||}{||\vec{r}'(t)||^3}$

If the curve is parameterized by arc length $s$, then $\kappa(s) = ||\vec{r}''(s)||$.

Torsion ($\tau$)

The torsion $\tau$ measures the rate at which the binormal vector changes direction. A non-zero torsion means the curve is twisting. For a curve $\vec{r}(t)$ parameterized by $t$, the torsion is given by:

$\tau(t) = \frac{(\vec{r}'(t) \times \vec{r}''(t)) \cdot \vec{r}'''(t)}{||\vec{r}'(t) \times \vec{r}''(t)||^2}$

If the curve is parameterized by arc length $s$, then $\tau(s) = \vec{r}'''(s) \cdot \vec{B}(s)$.

Example: Helix

Consider the helix parameterized by $\vec{r}(t) = (a \cos(t), a \sin(t), bt)$, where $a > 0$.

$\vec{r}'(t) = (-a \sin(t), a \cos(t), b)$ $||\vec{r}'(t)|| = \sqrt{a^2 \sin^2(t) + a^2 \cos^2(t) + b^2} = \sqrt{a^2 + b^2}$

Let $c = \sqrt{a^2 + b^2}$. Parameterizing by arc length $s = ct$: $\vec{T}(s) = \frac{1}{c}(-a \sin(s/c), a \cos(s/c), b)$

$\vec{T}'(s) = \frac{1}{c^2}(-a \cos(s/c), -a \sin(s/c), 0)$ $||\vec{T}'(s)|| = \frac{a}{c^2} \sqrt{\cos^2(s/c) + \sin^2(s/c)} = \frac{a}{c^2}$

Curvature $\kappa = ||\vec{T}'(s)|| = \frac{a}{c^2} = \frac{a}{a^2+b^2}$.

$\vec{N}(s) = \frac{\vec{T}'(s)}{||\vec{T}'(s)||} = \frac{1}{a/c^2} \frac{1}{c^2}(-a \cos(s/c), -a \sin(s/c), 0) = (-\cos(s/c), -\sin(s/c), 0)$.

$\vec{B}(s) = \vec{T}(s) \times \vec{N}(s) = \frac{1}{c} \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -a \sin(s/c) & a \cos(s/c) & b \\ -\cos(s/c) & -\sin(s/c) & 0 \end{vmatrix}$ $\vec{B}(s) = \frac{1}{c} (b \sin(s/c), -b \cos(s/c), a \sin^2(s/c) + a \cos^2(s/c)) = \frac{1}{c} (b \sin(s/c), -b \cos(s/c), a)$.

$\vec{B}'(s) = \frac{1}{c^2} (b \cos(s/c), b \sin(s/c), 0)$.

Torsion $\tau = -\vec{B}'(s) \cdot \vec{N}(s)$. $\tau = - \left( \frac{1}{c^2} (b \cos(s/c), b \sin(s/c), 0) \right) \cdot (-\cos(s/c), -\sin(s/c), 0)$ $\tau = \frac{1}{c^2} (b \cos^2(s/c) + b \sin^2(s/c)) = \frac{b}{c^2} = \frac{b}{a^2+b^2}$.

For a helix, $\kappa = \frac{a}{a^2+b^2}$ and $\tau = \frac{b}{a^2+b^2}$. If $b=0$, the helix becomes a circle, and $\tau=0$, meaning it has no torsion. If $a=0$, it becomes a straight line, and $\kappa=0$.

Locus of Centers of Curvature

The center of curvature is a point that lies on the normal line to a curve at a given point. It represents the center of the circle that best approximates the curve at that point, known as the osculating circle. The locus of these centers of curvature as we move along the curve forms another curve, which is called the evolute of the original curve.

Defining the Center of Curvature

For a curve $\vec{r}(s)$ parameterized by arc length $s$, the principal normal vector $\vec{N}(s)$ points towards the center of curvature. The radius of curvature is the reciprocal of the curvature, $R(s) = 1/\kappa(s)$.

The center of curvature, denoted by $\vec{C}(s)$, is located at a distance $R(s)$ from the point $\vec{r}(s)$ along the principal normal vector $\vec{N}(s)$. Therefore, the position vector of the center of curvature is:

$\vec{C}(s) = \vec{r}(s) + R(s) \vec{N}(s) = \vec{r}(s) + \frac{1}{\kappa(s)} \vec{N}(s)$

This formula holds when $\kappa(s) \neq 0$. If $\kappa(s) = 0$, the curve is locally straight, and there is no unique center of curvature in this sense.

The Evolute

The evolute is the curve traced by the centers of curvature. If $\vec{r}(s)$ is the original curve, its evolute is given by $\vec{E}(s) = \vec{C}(s) = \vec{r}(s) + \frac{1}{\kappa(s)} \vec{N}(s)$.

A key property of the evolute is that the tangent lines to the evolute are normal to the original curve. Furthermore, the distance from a point on the evolute to the corresponding point on the original curve is equal to the radius of curvature. This means that the original curve is the involute of its evolute.

Relationship to the Frenet Frame

The position of the center of curvature is directly determined by the curve's position ($\vec{r}(s)$), its curvature ($\kappa(s)$), and its principal normal vector ($\vec{N}(s)$). The binormal vector ($\vec{B}(s)$) and torsion ($\tau(s)$) are not directly involved in defining the center of curvature itself, but they are essential for defining the normal vector $\vec{N}(s)$ and understanding the curve's behavior in 3D space.

Example: Parabola in 2D (as a space curve)

While the concept is typically applied to space curves, let's consider a planar curve, the parabola $y = x^2$, parameterized by $\vec{r}(t) = (t, t^2)$.

$\vec{r}'(t) = (1, 2t)$ $\vec{r}''(t) = (0, 2)$

Curvature $\kappa(t) = \frac{||\vec{r}'(t) \times \vec{r}''(t)||}{||\vec{r}'(t)||^3}$. For a planar curve, we can embed it in 3D as $\vec{r}(t) = (t, t^2, 0)$. $\vec{r}'(t) = (1, 2t, 0)$ $\vec{r}''(t) = (0, 2, 0)$ $\vec{r}'(t) \times \vec{r}''(t) = (0, 0, 2)$ $||\vec{r}'(t) \times \vec{r}''(t)|| = 2$ $||\vec{r}'(t)|| = \sqrt{1^2 + (2t)^2} = \sqrt{1+4t^2}$ $\kappa(t) = \frac{2}{(1+4t^2)^{3/2}}$

The radius of curvature $R(t) = \frac{(1+4t^2)^{3/2}}{2}$.

To find $\vec{N}(t)$, we first find $\vec{T}(t) = \frac{(1, 2t, 0)}{\sqrt{1+4t^2}}$. $\vec{T}'(t) = \frac{d}{dt} \left( (1+4t^2)^{-1/2} (1, 2t, 0) \right)$ $= -\frac{1}{2}(1+4t^2)^{-3/2}(8t)(1, 2t, 0) + (1+4t^2)^{-1/2}(0, 2, 0)$ $= \frac{-4t(1, 2t, 0) + 2(1+4t^2)(0, 2, 0)}{(1+4t^2)^{3/2}}$ $= \frac{(-4t, -8t^2, 0) + (0, 4+8t^2, 0)}{(1+4t^2)^{3/2}} = \frac{(-4t, 4, 0)}{(1+4t^2)^{3/2}}$ $||\vec{T}'(t)|| = \frac{\sqrt{16t^2 + 16}}{(1+4t^2)^{3/2}} = \frac{4\sqrt{t^2+1}}{(1+4t^2)^{3/2}}$. This is not $\kappa$. The formula for $\vec{N}$ requires parameterization by arc length.

Alternatively, for a planar curve $y=f(x)$, $\kappa(x) = \frac{|f''(x)|}{(1+(f'(x))^2)^{3/2}}$. For $y=x^2$, $f'(x)=2x$, $f''(x)=2$. $\kappa(x) = \frac{2}{(1+(2x)^2)^{3/2}} = \frac{2}{(1+4x^2)^{3/2}}$, which matches our previous result.

The principal normal vector $\vec{N}$ points towards the concave side of the curve. For $y=x^2$, this is the positive y-direction. $\vec{N}(t) = (0, 1, 0)$ (scaled appropriately).

The center of curvature $\vec{C}(t) = \vec{r}(t) + R(t) \vec{N}(t)$. $\vec{C}(t) = (t, t^2, 0) + \frac{(1+4t^2)^{3/2}}{2} (0, 1, 0)$ $\vec{C}(t) = \left( t, t^2 + \frac{(1+4t^2)^{3/2}}{2}, 0 \right)$

The locus of centers of curvature (evolute) for the parabola $y=x^2$ is the curve traced by these points.

Spherical Curvature

Spherical curvature is a concept related to how a curve's tangent vector behaves relative to a sphere. It is particularly relevant when we consider the projection of a curve onto a sphere or the behavior of the tangent vector field on a sphere. A simpler way to think about it is the curvature of the spherical image of the curve's tangent vector.

The Spherical Image of a Curve

For a curve $\vec{r}(s)$ parameterized by arc length $s$, its tangent vector $\vec{T}(s)$ is a unit vector field along the curve. We can visualize this vector field by translating each tangent vector so its starting point is at the origin. The endpoints of these translated vectors trace out a curve on the unit sphere centered at the origin. This curve on the unit sphere is called the spherical image of the curve's tangent vector.

Definition of Spherical Curvature

The spherical curvature of a curve $\vec{r}(s)$ is defined as the curvature of the spherical image of its tangent vector $\vec{T}(s)$. Let the spherical image curve be denoted by $\vec{T}(s)$ itself (as a position vector on the unit sphere). Then, the spherical curvature, often denoted by $\kappa_s$, is the magnitude of the derivative of $\vec{T}(s)$ with respect to arc length $s$.

$\vec{T}(s)$ is a unit vector, so $||\vec{T}(s)|| = 1$. The derivative $\vec{T}'(s)$ is a vector tangent to the curve traced by $\vec{T}(s)$ on the unit sphere. The curvature of this spherical image curve is $\kappa_s(s) = ||\vec{T}'(s)||$.

From the Serret–Frenet formulas, we know that $\vec{T}'(s) = \kappa(s) \vec{N}(s)$. Therefore, the spherical curvature is:

$\kappa_s(s) = ||\kappa(s) \vec{N}(s)|| = |\kappa(s)| ||\vec{N}(s)||$

Since $\vec{N}(s)$ is a unit vector, $||\vec{N}(s)|| = 1$. So, $\kappa_s(s) = |\kappa(s)|$.

This means that the spherical curvature of a curve is simply the absolute value of its regular curvature.

Interpretation and Significance

The spherical curvature, $\kappa_s = |\kappa|$, tells us how much the direction of the tangent vector changes per unit arc length. A higher spherical curvature implies that the tangent vector is changing its direction more rapidly, meaning the curve is bending more sharply.

While the regular curvature $\kappa$ is directly related to the osculating circle and the osculating plane, the spherical curvature focuses solely on the change in the tangent direction, irrespective of the normal or binormal components. For a curve parameterized by $t$, the spherical curvature can be calculated using the formula for curvature applied to the unit tangent vector $\vec{T}(t)$:

$\kappa_s(t) = \frac{||\vec{T}'(t) \times \vec{T}''(t)||}{||\vec{T}'(t)||^3}$

However, it's often easier to compute the regular curvature $\kappa(t)$ first and then use $\kappa_s(t) = |\kappa(t)|$.

Example: Circle

Consider a circle of radius $R$ in the xy-plane, parameterized by $\vec{r}(t) = (R \cos(t), R \sin(t), 0)$.

$\vec{r}'(t) = (-R \sin(t), R \cos(t), 0)$ $||\vec{r}'(t)|| = R$

$\vec{T}(t) = (-\sin(t), \cos(t), 0)$

$\vec{T}'(t) = (-\cos(t), -\sin(t), 0)$ $||\vec{T}'(t)|| = \sqrt{\cos^2(t) + \sin^2(t)} = 1$.

The curvature $\kappa(t) = ||\vec{T}'(t)|| = 1$ if parameterized by arc length $s = Rt$. If parameterized by $t$, we use the formula: $\vec{r}''(t) = (-R \cos(t), -R \sin(t), 0)$ $\vec{r}'(t) \times \vec{r}''(t) = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -R \sin t & R \cos t & 0 \\ -R \cos t & -R \sin t & 0 \end{vmatrix} = (0, 0, R^2 \sin^2 t + R^2 \cos^2 t) = (0, 0, R^2)$ $||\vec{r}'(t) \times \vec{r}''(t)|| = R^2$ $||\vec{r}'(t)||^3 = R^3$ $\kappa(t) = \frac{R^2}{R^3} = \frac{1}{R}$.

The spherical curvature $\kappa_s = |\kappa| = |\frac{1}{R}| = \frac{1}{R}$ (since $R>0$).

The spherical image of the tangent vector $\vec{T}(t) = (-\sin(t), \cos(t), 0)$ is itself a circle of radius 1 on the unit sphere. The curvature of this unit circle is $1/1 = 1$. This seems inconsistent. Let's re-evaluate.

If we parameterize the circle by arc length $s$, then $\vec{r}(s) = (R \cos(s/R), R \sin(s/R), 0)$. $\vec{T}(s) = \vec{r}'(s) = (-\sin(s/R), \cos(s/R), 0)$. $\vec{T}'(s) = (-\frac{1}{R} \cos(s/R), -\frac{1}{R} \sin(s/R), 0)$. $\kappa_s(s) = ||\vec{T}'(s)|| = \sqrt{\frac{1}{R^2} \cos^2(s/R) + \frac{1}{R^2} \sin^2(s/R)} = \sqrt{\frac{1}{R^2}} = \frac{1}{R}$.

The curvature of the circle itself is $\kappa = 1/R$. So, the spherical curvature is equal to the regular curvature for a circle.

The spherical image of $\vec{T}(s)$ is the curve $(-\sin(s/R), \cos(s/R), 0)$. This is a unit circle traced on the sphere. The curvature of a unit circle is 1.

There seems to be a subtle point here. The definition $\kappa_s(s) = ||\vec{T}'(s)||$ directly gives the curvature of the spherical image curve. And we found $\vec{T}'(s) = \kappa(s) \vec{N}(s)$. So $\kappa_s(s) = ||\kappa(s) \vec{N}(s)|| = |\kappa(s)| ||\vec{N}(s)|| = |\kappa(s)|$.

For the circle $\kappa = 1/R$. So $\kappa_s = 1/R$. The spherical image is a unit circle, its curvature is indeed 1. Why is $1/R$ not equal to 1?

The mistake lies in assuming the spherical image is parameterized by arc length $s$ of the original curve. The spherical image curve is $\vec{u}(\theta) = (-\sin\theta, \cos\theta, 0)$ if we use the parameter $t$ (or $\theta$). The curvature of this unit circle $\vec{u}(\theta)$ is $\kappa_{unit} = \frac{||\vec{u}'(\theta) \times \vec{u}''(\theta)||}{||\vec{u}'(\theta)||^3}$. $\vec{u}'(\theta) = (-\cos\theta, -\sin\theta, 0)$, $||\vec{u}'(\theta)|| = 1$. $\vec{u}''(\theta) = (\sin\theta, -\cos\theta, 0)$. $\vec{u}'(\theta) \times \vec{u}''(\theta) = (0, 0, \cos^2\theta + \sin^2\theta) = (0, 0, 1)$. $||\vec{u}'(\theta) \times \vec{u}''(\theta)|| = 1$. $\kappa_{unit} = \frac{1}{1^3} = 1$.

The definition $\kappa_s = ||\vec{T}'(s)||$ is correct. For the circle $\vec{T}(s) = (-\sin(s/R), \cos(s/R), 0)$. $\vec{T}'(s) = (-\frac{1}{R} \cos(s/R), -\frac{1}{R} \sin(s/R), 0)$. $\kappa_s(s) = ||\vec{T}'(s)|| = \frac{1}{R}$.

The issue is that $\kappa_s(s)$ is the curvature of the spherical image *as parameterized by $s$*. The curvature of a curve is independent of its parameterization. The spherical image curve is a unit circle. Its curvature is 1.

The correct interpretation is that the spherical curvature is the curvature of the spherical image curve *when parameterized by arc length*. The arc length of the spherical image curve $\vec{T}(s)$ is $s_{spherical} = \int_0^s ||\vec{T}'(\sigma)|| d\sigma = \int_0^s \kappa_s(\sigma) d\sigma$.

For the circle, $\kappa_s(s) = 1/R$. The arc length parameter for the spherical image is $s_{spherical} = \int_0^s (1/R) d\sigma = s/R$. The spherical image curve parameterized by its own arc length $s'$ is $\vec{T}(s'(s)) = (-\sin(s'/R), \cos(s'/R), 0)$. The curvature of this unit circle with respect to its arc length $s'$ is 1.

The definition $\kappa_s = ||\vec{T}'(s)||$ where $s$ is the arc length of the original curve, gives the rate of change of the tangent direction with respect to the original curve's arc length.

So, for the circle, $\kappa_s = 1/R$. This is the correct value for spherical curvature. The curvature of the spherical image curve itself (as a geometric object) is 1. The relationship $\kappa_s = |\kappa|$ is the fundamental definition.

Intrinsic Equations of a Curve

Intrinsic equations of a curve are equations that relate the fundamental geometric quantities of the curve, such as arc length, curvature, and torsion, without reference to a coordinate system. They describe the curve's shape intrinsically, meaning the equations remain the same regardless of how the curve is embedded in space.

The Concept of Intrinsic Properties

In differential geometry, certain properties of a curve are considered intrinsic. These are properties that do not depend on the choice of coordinate system or the curve's position and orientation in space. The fundamental intrinsic properties of a space curve are its curvature ($\kappa$) and torsion ($\tau$). The arc length ($s$) is also an intrinsic measure.

The Fundamental Theorem of Curves

The Fundamental Theorem of Space Curves states that a curve in three-dimensional Euclidean space is uniquely determined (up to rigid motion, i.e., translation and rotation) by its curvature $\kappa(s)$ and torsion $\tau(s)$ as functions of the arc length $s$.

This theorem implies that if we know $\kappa(s)$ and $\tau(s)$ for all $s$, we can, in principle, reconstruct the curve. The Serret–Frenet formulas provide the differential equations that govern this relationship:

• $\frac{d\vec{T}}{ds} = \kappa \vec{N}$
• $\frac{d\vec{N}}{ds} = -\kappa \vec{T} + \tau \vec{B}$
• $\frac{d\vec{B}}{ds} = -\tau \vec{N}$

These equations form a system of first-order linear differential equations for the Frenet frame vectors $\vec{T}, \vec{N}, \vec{B}$. Given initial conditions for these vectors (e.g., at $s=0$) and the functions $\kappa(s)$ and $\tau(s)$, the Frenet frame is uniquely determined. The curve itself is then obtained by integrating the tangent vector: $\vec{r}(s) = \vec{r}(0) + \int_0^s \vec{T}(u) du$.

Intrinsic Equations

The intrinsic equations of a curve are the pair of equations:

$k = \kappa(s)$ $\tau = \tau(s)$

where $k$ and $\tau$ are the curvature and torsion, respectively, and $s$ is the arc length parameter. These two functions completely characterize the shape of the curve up to rigid motion.

Examples of Intrinsic Equations

1. Helix: For a circular helix $\vec{r}(t) = (a \cos t, a \sin t, bt)$, we found $\kappa = \frac{a}{a^2+b^2}$ and $\tau = \frac{b}{a^2+b^2}$. If we parameterize by arc length $s$, these values are constant. So, the intrinsic equations are: $\kappa(s) = \frac{a}{a^2+b^2}$ (constant) $\tau(s) = \frac{b}{a^2+b^2}$ (constant) This signifies that a helix is a curve with constant curvature and constant torsion.

2. Circle (in 3D): Consider a circle of radius $R$ in the xy-plane. Its curvature is $\kappa = 1/R$ (constant). Its torsion is $\tau = 0$ (since it lies in a plane). The intrinsic equations are: $\kappa(s) = 1/R$ (constant) $\tau(s) = 0$ A curve with $\tau=0$ is planar.

3. Straight Line: A straight line has zero curvature ($\kappa = 0$) and zero torsion ($\tau = 0$). The intrinsic equations are: $\kappa(s) = 0$ $\tau(s) = 0$

The Role of Arc Length

The arc length parameter $s$ is crucial because the Serret–Frenet formulas are most elegantly expressed in terms of derivatives with respect to $s$. Using arc length ensures that the derivatives represent rates of change intrinsically related to the curve's geometry, independent of the speed at which a parameter traverses the curve.