1.3 Tangent vectors and normal vectors

Tangent Vectors and Normal Vectors

Tangent and normal vectors give you a way to describe how a curve behaves locally: what direction it's heading and how it's bending at any given point. These vectors form the foundation for the Frenet frame, a moving coordinate system that travels along the curve and captures its geometry at every point.

Tangent and Normal Vectors

Tangent Vectors and Unit Tangent Vectors

Given a smooth curve defined by a position vector $\vec{r}(t)$, the tangent vector at a point is the derivative:

$\vec{r}'(t)$

This vector points in the direction of motion and its magnitude equals the speed along the curve. Note that $\vec{r}'(t)$ is often called the velocity vector, not the unit tangent vector. The distinction matters.

The unit tangent vector strips away speed information and isolates pure direction:

$\vec{T}(t) = \frac{\vec{r}'(t)}{|\vec{r}'(t)|}$

Because $|\vec{T}(t)| = 1$ always, this vector tells you which way the curve is going without any information about how fast.

Normal Vectors and Unit Normal Vectors

The unit normal vector captures the direction in which the curve is turning. You get it by differentiating the unit tangent vector and normalizing:

$\vec{N}(t) = \frac{\vec{T}'(t)}{|\vec{T}'(t)|}$

A few things to keep straight here:

• $\vec{T}'(t)$ itself is perpendicular to $\vec{T}(t)$. This follows from the fact that $\vec{T}(t) \cdot \vec{T}(t) = 1$; differentiating both sides gives $2\,\vec{T}(t) \cdot \vec{T}'(t) = 0$.
• $\vec{N}(t)$ always points toward the center of curvature, the center of the circle that best approximates the curve at that point.
• $\vec{N}(t)$ describes the direction of bending, not the magnitude. The magnitude of bending is captured by curvature $\kappa$.

Computing $\vec{N}(t)$ step by step:

1. Find $\vec{r}'(t)$ and compute $|\vec{r}'(t)|$.
2. Form $\vec{T}(t) = \frac{\vec{r}'(t)}{|\vec{r}'(t)|}$.
3. Differentiate $\vec{T}(t)$ with respect to $t$ to get $\vec{T}'(t)$.
4. Normalize: $\vec{N}(t) = \frac{\vec{T}'(t)}{|\vec{T}'(t)|}$.

Step 3 is usually the messiest part because the quotient rule on $\vec{T}(t)$ can get algebraically heavy. Simplify $\vec{T}(t)$ as much as possible before differentiating.

Frenet Frame

Frenet Frame and Its Components

The Frenet frame (also called the TNB frame) is a right-handed, orthonormal coordinate system that moves along the curve. It consists of three vectors at each point:

• $\vec{T}(t)$ — unit tangent vector (direction of motion)
• $\vec{N}(t)$ — unit normal vector (direction of bending)
• $\vec{B}(t)$ — binormal vector (perpendicular to both)

The binormal vector is defined as:

$\vec{B}(t) = \vec{T}(t) \times \vec{N}(t)$

Since $\vec{T}$ and $\vec{N}$ are orthogonal unit vectors, their cross product $\vec{B}$ is automatically a unit vector perpendicular to both. Together, $\{\vec{T}, \vec{N}, \vec{B}\}$ form a right-handed orthonormal basis that adapts to the curve's geometry as you move along it.

The Frenet frame is the natural setting for defining curvature (how fast the curve bends, measured by changes in $\vec{T}$) and torsion (how fast the curve twists out of a plane, measured by changes in $\vec{B}$).

Osculating Plane and Its Relationship to the Frenet Frame

The osculating plane at a point on the curve is the plane spanned by $\vec{T}(t)$ and $\vec{N}(t)$. Think of it as the plane that the curve is "trying to lie in" at that instant.

• It contains the best-fit circle (the osculating circle) to the curve at that point.
• The binormal vector $\vec{B}(t)$ is the normal to the osculating plane.
• For a planar curve, the osculating plane is the same everywhere: it's just the plane the curve lives in. For a space curve, the osculating plane tilts and rotates as you move along the curve, and the rate of that rotation is exactly what torsion measures.