One advantage of finding the arc-length parameterization is that the distance traveled along the curve starting from s = 0 Since the variable s represents the arc length, we call this an arc-length parameterization of the original function r ( t ). The vector-valued function is now written in terms of the parameter s. We can then reparameterize the original function r ( t )īy substituting the expression for t back into r ( t ). Suppose that we find the arc-length function s ( t )Īnd are able to solve this function for t as a function of s. The new parameterization still defines a circle of radius 3, but now we need only use the values 0 ≤ t ≤ π / 2 Obtaining a new parameterization r ( t ) = 〈 3 cos 4 t, 3 sin 4 t 〉. That parameterizes a circle of radius 3, we can change the parameter from t to 4 t , For example, if we have a function r ( t ) = 〈 3 cos t, 3 sin t 〉, 0 ≤ t ≤ 2 π Recall that any vector-valued function can be reparameterized via a change of variables. Ī useful application of this theorem is to find an alternative parameterization of a given curve, called an arc-length parameterization. Then the parameter t represents the arc length from the starting point at t = a. Recall, 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 2įurthermore, d s d t = ‖ r ′ ( t ) ‖ > 0. 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.

Describe the meaning of the normal and binormal vectors of a curve in space.Explain the meaning of the curvature of a curve in space and state its formula.Determine the length of a particle’s path in space by using the arc-length function.

Arc length function