Previous: Introduction
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. If you click on the picture at the right, you can watch 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. | |
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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 | |
We will always use coordinate representations when using our computer graphics system 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. You might want to compare this section with the description of Space Curves and Their Representation in Lab 3. | |
Demonstration 2: Curves in Two Dimensions The default function is a parabola with coordinate functions y(t)=t2 This demo displays plane curves described by coordinate functions. What follows is a series of exercises which will help to take advantage of the features of the demo.
Consider the curves
(t,tn)
for various integers n. Which curves have inflection points at the origin? | |
Consider the non-polynomial curve
(cos(3t)
,sin(4t))
where the domain is the interval from 0 to 2 pi. What are the
coordinates of the inflection points of this curve? | |
Change the curve you are viewing to
(tm,tn)
, where m and n are integers.
For various integer values of the exponents, in which quadrants is the graph
contained? | |
Describe the behavior of the curves (cosm(t),
sinn(t)) for various values of the
exponent m (not necessarily an integer) .
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