Visualising the ladder formed by two paths on Exercise Sheet 1, Question 9
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The following picture shows a pair of helices that form a ladder. The precise definition is that we can parametrise these curves by \(p_1(t)\) and \(p_2(t)\), so that the principal normal lines through \(p_1(t)\) and \(p_2(t)\) coincide. Equivalently, there exists some scalar function \(d(t)\) such that \(p_2(t) = p_1(t) + d(t) N_1(t)\), where \(N_1(t)\) is the principal normal vector of the curve \(p_1(t)\).
In the question you will prove that the distance between \(p_1(t)\) and \(p_2(t)\) is the same for all \(t\). This is illustrated in the diagram below, where the dashed line is the principal normal line through \(p_1(t)\) and \(p_2(t)\), and you can see that the length of this line segment is constant (to visualise this clearly, you can rotate the picture to view it from above so that the helices become circles).
