If ‖ r ′ ( t ) ‖ = 1 ‖ r ′ ( t ) ‖ = 1 for all t ≥ a, t ≥ a, then the parameter t represents the arc length from the starting point at t = a. The formula for the arc-length function follows directly from the formula for arc length:įurthermore, d s d t = ‖ r ′ ( t ) ‖ > 0. If a vector-valued function represents the position of a particle in space as a function of time, then the arc-length function measures how far that particle travels as a function of time. Let’s take this one step further and examine what an arc-length function is. We now have a formula for the arc length of a curve defined by a vector-valued function. This gives a formula for the length of a wire needed to form a helix with N turns that has radius R and height h. Recall Alternative Formulas for Curvature, which states that the formula for the arc length of a curve defined by the parametric functions x = x ( t ), y = y ( t ), t 1 ≤ t ≤ t 2 x = x ( t ), y = y ( t ), t 1 ≤ t ≤ t 2 is given by We have seen how a vector-valued function describes a curve in either two or three dimensions. We explore each of these concepts in this section. This is described by the curvature of the function at that point. Or, suppose that the vector-valued function describes a road we are building and we want to determine how sharply the road curves at a given point. We would like to determine how far the particle has traveled over a given time interval, which can be described by the arc length of the path it follows. For example, suppose a vector-valued function describes the motion of a particle in space. In this section, we study formulas related to curves in both two and three dimensions, and see how they are related to various properties of the same curve.
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