1.1: Plane Curves and Their RepresentationMany familiar curves can be described as the graph of a function of one variable, y=f(x), for x defined over an interval. Some curves are generated kinematically, that is, traced out in time by a point in motion. The demonstration below shows the curve traced out by a point fixed with respect to a wheel which is rolling without slipping on the circumference of another wheel.While some curves are naturally defined by a point in motion, we can always represent a curve parametrically. This means that each point on the curve corresponds to a value of a parameter, usually called t, and the curve is traced out as t runs through an interval. By a parametrized plane curve, we mean a continuous mapping of an interval into ordinary two-dimensional Euclidean space. We will consider the points of two-dimensional space as vectors emanating from the origin. A plane curve is usually represented as a vector function X(t), which can be written X(t) = x(t)E1 + y(t)E2 where {E1, E2} is a basis in two-space. The most common basis is the standard orthogonal pair (1,0), (0,1), but any pair of linearly independent vectors can be used. When using the standard basis, the functions x(t) and y(t) are called the coordinate functions of the curve. We will always use coordinate representations when using java applets to display geometric phenomena. However, at other times working directly with a single vector symbol will be a more powerful approach. Later labs will show that the basis vectors we use to represent a vector do not have to be constant vectors. For example, it is possible to represent the acceleration vector at a point of a curve with respect to the unit tangent and unit normal vectors at the point, a "moving frame" of vectors that change as we move along the curve. Methods similar to those used in the representation of plane curves are used to represent space curves. This demo lets you define a parametric curve X(t) over a domain t in the controls window. You can then study the graph of the curve in the graph window. If you find that the resolution of the of the curve does not provide enough detail, you can increase it by changing the number of steps for t.
1. Graph the curve X(t)=(t,t^n) for various integers n. For which values of
n does the curve have an inflection point at the origin?
2. Graph the non-polynomial curve X(t)=(cos(3*t),sin(4*t)) where t ranges from 0 to 2*pi. What are the coordinates of the inflection points of this curve? 3. More generally, consider the curve X(t) = (cos(m*t),sin(n*t)) where m and n are integers. Describe the shape of these curves and identify the location of their inflection points. These curves are called Lissajous Figures. If m = n, the curve is a circle covered m times. What happens when m and n have no common factor? How many self-intersection points will the curve have? 4. Consider the curve X(t) = (t^m,t^n). For various integer values of m and n, in which quadrants does the graph lie? 5. Describe the behavior of the curve X(t) = (cos(t)^m,sin(t)^n) for various values of m and n (not necessarily integers). |