Vectors N and B with visuals, Multivariable Calculus

The unit normal N(t) and binormal B(t) vectors for a smoothly parametrized space curve r(t) with visual examples and in examples with T, N, B, κ and the osculating circle. In this lecture, we explore the Frenet Frame, a set of three mutually orthogonal unit vectors—𝑇̂ (tangent), 𝑁̂ (normal), and 𝐵̂ (binormal)—defined along a smoothly parametrized curve 𝐫(𝑡). We begin by discussing the properties of these vectors and their significance in understanding the geometry of the curve. In particular, we look at the definitions and geometric interpretations of 𝑁̂ and 𝐵̂ , emphasizing their roles in describing the curve's orientation in space. The lecture concludes with visual demonstrations of the Frenet Frame for various curves, illustrating the dynamic relationship between the vectors and the curve. (Unit 2 Lecture 10)

Key Points
The Frenet Frame consists of three orthogonal unit vectors that describe the geometry of a curve.
𝑇̂ indicates the forward direction along the curve, 𝑁̂ points towards the curve’s bend, and 𝐵̂ is orthogonal to both.
The normal vector 𝑁̂ does not always exist, particularly at points where the curve does not bend. (Because 𝑁̂ is needed to define 𝐵̂ , (̂ 𝐵) also does not exist at these points.)
Understanding the Frenet Frame enhances our grasp of the curve's spatial orientation and movement.

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