Curves in Space

In previous chapters, we have seen that the curvature κ2[α] of a plane curve α measures the failure of α to be a straight line. In the present chapter, we define a similar curvature κ[α] for a curve in ℝ ⁿ ; it measures the failure of the curve to be a straight line in n-dimensional space. The function κ[α] reduces to the absolute value of κ2[α] when n = 2. For curves in ℝ³, we can also measure the failure of the curve to lie in a plane by means of another function called the torsion, and denoted τ[α].