# Lectures on the differential geometry of curves and surfaces by Blaga P. By Blaga P.

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7. 1) we obtain ρ k(t) = r ×r . 2) Remarks. 1. 2) it follows that a parameterized curve r = r(t) is biregular at a point t0 iff k(t0 ) 0. 2. Since for equivalent parameterized curve the naturally parameterized curve to each of them are equivalent between them, the notion of curvature makes sense also for regular curves. Examples. 1. For the straight line r = r0 + ta the curvature vector (and, therefore, also the curvature) is identically zero. 2. For the circle S R1 = {(x, y, z) ∈ R3 |x2 + y2 = R2 , z = 0} we choose the parameterization    x = R cos t     0 < t < 2π.

4) and the previous relation, we get χ(t) = (r , r , r ) . 1. Let ω = χτ + kβ. 6) ν =ω×ν      β = ω × β The vector ω is called the Darboux vector. 1 The geometrical meaning of the torsion The torsion is, in a way, an analogue of the curvature (this is the reason why in the oldfashioned books the torsion is called the second curvature). What we mean is that the torsion can be also interpreted as being the speed of rotation of a straight line, this time the binormal. In other words, we have 54 Chapter 1.

1 The behaviour of the Frenet frame at a parameter change A notion defined for parameterized curves makes sense for regular curves iff it is invariant at a parameter change, in other words if it doesn’t change when we replace a parameterized curve by another one, equivalent to it. e. 1. Let (I, r = r(t)) and ρ = ρ(u) be two equivalent parameterized curves with the parameter change λ : I → J, u = λ(t). Then, at the corresponding points t and u = λ(t), their Frenet frame coincide if λ (t) > 0 . If λ (t) < 0, then the origins and the unit principal normals coincide, while the other two pairs of versors have the same direction, but opposite senses.