TNB Frame

The TNB frame, also known as the Frenet-Serret frame, is a coordinate system associated with a point on a space curve. It consists of three mutually perpendicular vectors:

1. Tangent vector (T)
2. Normal vector (N)
3. Binormal vector (B)

This frame moves along the curve, providing insight into the curve’s geometry in 3D space.

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1. **Tangent Vector $\mathbf{T}$ **

The tangent vector at a point on the curve represents the direction of the curve at that point. It is the unit vector in the direction of the velocity vector.

where $\mathbf{r}(t)$ is the vector-valued function describing the curve, and ${\mathbf{r}}^{\prime }(t)$ is its derivative (velocity vector).

2. **Normal Vector $\mathbf{N}$ **

The normal vector points in the direction of the curve’s instantaneous acceleration, perpendicular to the tangent vector. It indicates how the curve is bending.

• The normal vector is obtained by differentiating the tangent vector and normalizing it.
• It points towards the center of curvature of the curve.

3. Binormal Vector $\mathbf{B}$

The binormal vector is perpendicular to both the tangent and normal vectors. It completes the right-handed coordinate system and indicates the direction in which the curve is twisting out of the plane formed by ( \mathbf{T} ) and ( \mathbf{N} ).

• It is the cross product of the tangent and normal vectors.
• ( \mathbf{B}(t) ) is orthogonal to both ( \mathbf{T}(t) ) and ( \mathbf{N}(t) ).

4. Curvature And Torsion

• Curvature (( \kappa )): Measures how fast the curve is changing direction (bending). It is related to the normal component of the acceleration.

  $\kappa =\left|\frac{d\mathbf{T}}{ds}\right|$

• **Torsion $\tau$ **: Measures how fast the curve is twisting out of the plane defined by $\mathbf{T}$ and $\mathbf{N}$.

5. Frenet-Serret Formulas

The TNB frame obeys a set of differential equations known as the Frenet-Serret formulas, which describe how the TNB vectors change as you move along the curve:

Where:

• ( s ) is the arc length.
• ( \kappa ) is the curvature.
• ( \tau ) is the torsion.

6. Geometric Interpretation

• The tangent vector $$$ \mathbf{T}$$ tells you the direction of the curve.
• The normal vector ( \mathbf{N} ) tells you how the curve bends.
• The binormal vector ( \mathbf{B} ) tells you how the curve twists.

Together, these vectors form an orthonormal basis that fully describes the local geometry of the curve at any point.

7. Example: Helix

For a 3D helix given by ( \mathbf{r}(t) = \langle a\cos(t), a\sin(t), bt \rangle ), the TNB frame can be computed as follows:

• Tangent vector ( \mathbf{T}(t) ): Direction of motion along the helix.
• Normal vector ( \mathbf{N}(t) ): Points toward the center of the helix.
• Binormal vector ( \mathbf{B}(t) ): Points along the axis of the helix, perpendicular to the plane formed by ( \mathbf{T}(t) ) and ( \mathbf{N}(t) ).

8. Applications Of the TNB Frame

• Physics: Describing the motion of particles along curved paths.
• Robotics: Path planning for robotic arms or autonomous vehicles.
• Computer Graphics: Modeling curves and surfaces in 3D space.
• Engineering: Analyzing stresses in curved beams or surfaces.

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References

• Stewart, James. Calculus: Early Transcendentals. (For further exploration of the TNB frame)
• Frenet-Serret Equations - Wolfram MathWorld