Calculus of Parametric Curves
The calculus of parametric curves involves applying calculus techniques to curves defined by parametric equations. This includes finding derivatives, integrals, arc length, and curvature of parametric curves. It allows for the analysis of motion, such as the path of a moving object, and provides a powerful tool for solving problems in physics, engineering, and other fields.
3 Key excerpts on "Calculus of Parametric Curves"
- Stu Schwartz(Author)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...Taken together, the parametric equations create a graph where the points x and y are independent of each other and both dependent on the parameter t. When graphed, parametric curves do not have to be functions. Typically, it is necessary to take derivatives of parametrics. Since the study of vectors parallels the study of parametrics, in this section we will only analyze problems that are not associated with motion in the plane. If a smooth curve C is given by the parametric equations x = f (t) and y = g (t), then the slope of C at the point (x, y) is given by. The 2 nd derivative of the curve is given by. The arc length between t = a and t = b is given by. The curve must be smooth and may not intersect itself. EXAMPLE 9: A curve C is defined by the parametric equations x = t 3 – t 2 + 1 and y = t 2 –5 t –2. Find the equation of the line tangent to the graph of C at the point (5, –8). SOLUTION: Only t = 2 satisfies both equations. Tangent line: EXAMPLE 10: The curve C is defined by the parametric equations x = e –t +1 and y = sinπ t – t 2. a) At t = 2, is the curve increasing, decreasing or neither? Justify your answer. b) At t = 2, is the curve concave up, concave down or neither? Justify your answer SOLUTION: a) b) TEST TIP Your calculator can graph parametric equations. Change your MODE to PAR, input your equations and adjust your window. You need to set the minimum and maximum values for t as well as the Tstep. Note in Example 10 that if you trace the curve C, it will graph from right to left, which might make you think that the graph is decreasing. But, remember, this is not a particle motion problem. You are being asked about the shape of the graph and the behavior at t = 2. It is increasing and concave down. A particle moving along this curve would be moving left as, but the curve itself is increasing because. EXAMPLE 11: (Calculator Active) A curve C has the property that at every point (x, y) on C, x (t) = and y (t) is not specifically given...
eBook - ePub- Joy Ko, Kyle Steinfeld(Authors)
- 2018(Publication Date)
- Routledge(Publisher)
...In contrast, directly accessing the parametric form requires first deciding on both the properties desired and the best means to describe these properties before anything can be seen. If our only concern is how the curve looks, composing by parameterization would be unnecessary. However, in many design applications, the behavior of the curve is not a negligible factor. Rather than iteratively altering the shape of a curve until it looks right, this functional approach allows us to iteratively alter a description of what the curve does until it acts right. Because of these differences, it is worth becoming comfortable with creating and manipulating geometry in this extraordinarily powerful way. In the following three examples, we will become acquainted with this more direct access to the functional representation of curves. The first example demonstrates the parametric functions of a range of familiar two-dimensional curves, and presents some other more exotic mathematical curves in space as well. The second example focuses on just one class of curve, the helix, and presents an in-depth treatment of how to construct precise control mechanisms in order to achieve specific formal properties. A third example takes a closer look at a technique for producing forms called tweening, and shows how low-level control can overcome the undesired results that easily occur when working with curves in CAD. EXAMPLE El.18 A Gallery of Parametric Curves Previous examples have demonstrated the wealth of mathematical curves available to us, and the often confounding relationship between parametric equations and the forms of curves they produce...
eBook - ePubMesh Generation
Application to Finite Elements
- Pascal Frey, Paul Louis George(Authors)
- 2013(Publication Date)
- Wiley-ISTE(Publisher)
...The fundamental contribution of Gauss 2 consisted of using a parametric representation of the surfaces and of showing the intrinsic nature of the total curvature. The breakthrough came with Riemann 3 who gave a global mathematical definition of curves and surfaces, introducing the notion of n-dimensional manifold. The purpose of this chapter is to review the elementary notions of differential geometry necessary to understand the chapters related to the M 0 deling as well as the mesh generation of curves and surfaces (Chapters 12 to 15). If some of these notions may appear obvious to the initiated reader 4, we believe it advisable to recall these results so as to introduce the terminology and the notations that will be used subsequently. This chapter should not be considered as a substitute for the various references in this domain ([do CarM 0 -1976], [Lelong-Ferrand, Arnaudies-1977], [Berger-1978], [Farin-1997], aM 0 ng others). We limit ourselves here to the study of curves and surfaces embedded in the Euclidean space in two or three dimensions. The definitions require the use of the implicit function theorem. Therefore, we introduce the notion of a parameterized arc, and we conduct a local study. We also define surfaces and we introduce the two fundamental forms and the total curvature which plays an important role in the local or global behavior of the surfaces. Finally, the last section is devoted to practical aspects of the calculations related to curves and surfaces. 11.1 Metric properties of curves and arcs This section briefly recalls the different features used for the study of curves. In particular, we outline the notions of curvilinear abscissis, of arc length and we introduce the tangent and normal vectors as well as the Frenet frame. These definitions will allow us to compute quantities like the local curvature, the radius of curvature, the osculating circle as well as the local torsion and the relevant radius of torsion...
