Unit Speed Curves Recall that a curve Α is said to be a unit speed curve if The reason that we like unit speed curves that the parameter t is equal to arc length; i.e. the value of t tells us how far along the curve we've traveled. For example, the curve r t cos t, sin t, the unit circle, is of unit speed and so t tells us how far around the circle we've gone; i.e. it is arc length.
2 Curve_EGs.nb Non-Unit Speed Curves If the norm of the tangent vector to the curve r t x t, y t, z t is not one, then the parameter does not equal arc length. In this case, if the curve is smooth (in the sense of our text), then we can "reparametrize" the curve, i.e. re-scale the parameter, so that the resulting function is of unit speed.
Curve_EGs.nb Reparametrizing by Arc Length Define the arc length function of the smooth curve r t, a t b, by With this function (that describes how far along the curve we are at "time" t), we can carry out the reparametrization. What we do is (when possible) evalute the integral to obtain an equation of the form Solve this equation for t in terms of s. Calling the solution t s, we obtain the new parametrization (or reparametrization). The "new" curve is One thing to note: reparametrizing has NOT changed the geometric object - the curve - but has only changed the speed that we move along the curve.
4 Curve_EGs.nb Example 1 of Reparametrizing by Arc Length Consider the helix r t a cos t, a sin t, b t, 0 t 4 Π. Note that r ' t a 2 b 2 and so This equation is easily solved for t: The reparametrized curve is
Curve_EGs.nb Example 1 Continued r' t The Unit Tangent: We can calculate the unit tangent vector in two ways. The first is to simply calculate T t. One r' t need not worry about the parameter being arc length when using this formula. For the helix, using the original parametrization, and it follows that The second way is to reparametrize by arc length as shown on the previous slide: r s a cos s, a sin s, b s. Then the same as the vector obtained above, when the relation between s and t is taken into account.
6 Curve_EGs.nb Curvature and Principal Normal For a curve r t parametrized by arc length, the curvature is defined by The curvature is simply the rate of change of the direction of the unit tangent vector when the curve is parametrized by arc length. One can think of the curvature as describing how the curve "bends". For a curve r t parametrized by arc length, the principal (unit) normal is defined as It is easy to show that T is perpendicular to N. (Indeed, 0 d T T T ' T T T ' 2 ΚT N and so the vectors T d s and N are orthogonal.) For a plane curve, these vectors form a "moving frame" that moves along the curve while always remaining orthogonal and so every vector in the plane can be written as a linear combination of these vectors at any point along the curve.
Curve_EGs.nb Curvature and Principal Normal for the Helix Returning to the helix from example 1, r t a cos t, a sin t, b t, we calculate the curvature to be using T s r ' s a cos s, a sin s, b. The principal normal is
8 Curve_EGs.nb Example 2 of Reparametrizing by Arc Length Consider the curve r t 2 e t 2 cos t, 2 e t 2 sin t. We have Clearly this curve is not parametrized by arc length as r ' t 5 e t. To parametrize by arc length, we integrate: 0 t t gives 5 e u u 5 2 2 t 2. Solving the equation s 5 2 2 t 2 for The reparametrized curve is therefore
Curve_EGs.nb Example 2 Continued The unit tangent is The curvature is, since we've reparametrized by arc length, the norm of T '. We get The principal normal is
10 Curve_EGs.nb Example 3 of Reparametrizing by Arc Length Let r t t 2, t 3, t 0. Once again, our goal is to reparametrize this curve by arc length and then calculate the unit tangent, curvature, and principal normal. To reparametrize by arc length, we have to calculate the arc length function. Note that r ' t 2 t, 3 t 2 and so r ' t 4 t 2 9 t 4 t 4 9 t 2. Integrating this norm gives The next step is to solve the equation s 1 27 8 4 9 t2 3 2 for t. We obtain The reparametrized curve is
Curve_EGs.nb Example 3: Unit Tangent Vector and Curvature The unit tangent vector is, after a little work, The curvature is the norm of the derivative of the unit tangent. Again, some calculus I and calculating the norm of the derivative, i.e. some algebra, leads to
12 Curve_EGs.nb Example 3: Principal Normal Vector The definition of the principal normal is N 1 T ' s. The curvature is calculated above and so all that is needed is the Κ s derivative of the unit tangent vector. The principal normal vector is the above vector divided by the curvature. A little algebra leads to