By Thomas F. Banchoff, Stephen T. Lovett
Preface Acknowledgements airplane Curves: neighborhood houses Parameterizations place, speed, and Acceleration Curvature Osculating Circles, Evolutes, and Involutes usual Equations airplane Curves: international homes easy homes Rotation Index Isoperimetric Inequality Curvature, Convexity, and the Four-Vertex Theorem Curves in area: neighborhood houses Definitions, Examples, and Differentiation Curvature, Torsion, and the Frenet body Osculating aircraft and Osculating Sphere common Equations Curves in house: worldwide homes simple homes Indicatrices and overall Curvature Knots and hyperlinks Re. Read more...
summary: Preface Acknowledgements airplane Curves: neighborhood houses Parameterizations place, speed, and Acceleration Curvature Osculating Circles, Evolutes, and Involutes normal Equations aircraft Curves: international houses simple homes Rotation Index Isoperimetric Inequality Curvature, Convexity, and the Four-Vertex Theorem Curves in area: neighborhood homes Definitions, Examples, and Differentiation Curvature, Torsion, and the Frenet body Osculating airplane and Osculating Sphere normal Equations Curves in area: international houses simple homes Indicatrices and overall Curvature Knots and hyperlinks Re
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1. Calculate the velocity, the acceleration, the speed, and, where deﬁned, the unit tangent vector function of the following parametric curves: (a) The circle x(t) = (R cos ωt, R sin ωt). 10. 1. 2. What can be said about a parametrized curve x(t) that has the property that x (t) is identically 0? 3. Find the arc length function along the parabola y = x2 , using as the origin s = 0. 4. 13 given by x(t) = (t − sin t, 1 − cos t). Prove that the path taken by a point on the edge of a rolling wheel of radius 1 during one rotation has length 8.
Since T is a unit vector, we always have T · T = 1. Therefore, (T · T ) = 0 =⇒ 2T · T = 0 =⇒ T · T = 0. Thus, T is perpendicular to T . Just as there are two unit tangent vectors at a regular point of the curve, there are two unit normal vectors as well. 3. Curvature 21 parametrization, there is no directly preferred way to deﬁne “the” unit normal vector, so we make a choice. 1. Let x : I → R2 be a regular parametrized curve and T = (T1 , T2 ) the tangent vector at a regular value x(t). We deﬁne the unit normal vector U by U (t) = (−T2 (t), T1 (t)).
1 .. .. ... .. . . . .... 0 .... ..... . . .... ........... 5. Arc length segment. ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 14 1. ) Taking the limit of this Riemann sum, we obtain the following formula for the arc length of C: b (x (t))2 + (y (t))2 dt. 4), we deﬁne the arc length function s : [a, b] → R related to x(t) as the arc length along C over the interval [a, t]. Thus, t t (x (u))2 + (y (u))2 du. 5) a By the Fundamental Theorem of Calculus, we then have s (t) = x (t) = (x (t))2 + (y (t))2 , which, still following the vocabulary from the trajectory of a moving particle, we call the speed function of x(t).