In the case of a curve other than a circle, it is often useful first to inscribe a circle to the curve at a given point so that it is tangent to the curve at that point and “hugs” the curve as closely as possible in a neighborhood of the point ( Figure 3.6). In this case you would need to turn more sharply to stay on the road. In this case you would barely have to turn the wheel to stay on the road. Suppose the road lies on an arc of a large circle. The smaller the radius of the circle, the greater the curvature. The concept of curvature provides a way to measure how sharply a smooth curve turns. CurvatureĪn important topic related to arc length is curvature. Then, use the relationship between the arc length and the parameter t to find an arc-length parameterization of r ( t ). The arc-length parameterization also appears in the context of curvature (which we examine later in this section) and line integrals, which we study in the Introduction to Vector Calculus. One advantage of finding the arc-length parameterization is that the distance traveled along the curve starting from s = 0 s = 0 is now equal to the parameter s. 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 ) r ( t ) by substituting the expression for t back into r ( t ). Suppose that we find the arc-length function s ( t ) s ( t ) and 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 0 ≤ t ≤ π / 2 to traverse the circle once. For example, if we have a function r ( t ) = 〈 3 cos t, 3 sin t 〉, 0 ≤ t ≤ 2 π r ( t ) = 〈 3 cos t, 3 sin t 〉, 0 ≤ t ≤ 2 π that parameterizes a circle of radius 3, we can change the parameter from t to 4 t, 4 t, obtaining a new parameterization r ( t ) = 〈 3 cos 4 t, 3 sin 4 t 〉. 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. 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 Arc Length of a Parametric Curve, 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. 3.3.3 Describe the meaning of the normal and binormal vectors of a curve in space.3.3.2 Explain the meaning of the curvature of a curve in space and state its formula.3.3.1 Determine the length of a particle’s path in space by using the arc-length function.
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