I am struggling with the concept of parameterizing curves. I am not ev

chezmarylou1i 2021-12-14 Answered
I am struggling with the concept of parameterizing curves. I am not even sure if I know what it means so I tried to look some things up.
Since I didnt
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

Jeffery Autrey
Answered 2021-12-15 Author has 35 answers
The idea of parameterization is that you have some equation for a subset X of a space (often Rn), e.g., the usual equation
for the unit circle C in R2, and you want to describe a function γ(t)=(x(t),y(t)) that traces out that subset (or sometimes, just part of it) as t varies.
With a parameterization in hand, you can then specify a point on X just by giving a single value of t, which corresponds to the point γ(t) on X. One can still give points on X, say, (x,y), directly, of course, but this has the disadvantage that often one needs to check that a given point (x,y) is on X, that is, that it satisfies the equation defining X, whereas by construction a point γ(t) is always on X. Provided that the function γ(t) traces out all of X, we say that X is the image of γ.
In your example, we can parameterize the unit circle C by the parametric function
We can check that the points specified by γ(t) really do lie on C just by substituting cost for x and sint for y; indeed:
It's not too hard to show that γ actually traces out the full circle t (in fact, this is an immediate consequence of the usual geometric definitions of cos and sin). Note too that this parameterization traces over the circle infinitely many times, and in particular, there is more than one t value corresponding to any point on the circle. In fact, since the components cost and sint of γ(t) have period 2π, we have γ(t+2π)=γ(t) for all t.
There are many other parameterizations for all or part of the circle, too, and which is best depends on the context. Substituting the components in x2+y2=1 (try this!) shows that for all t,
is on the unit circle, and with some more work we can show that (1) α traces out the full circle with the single exception of the point (0,1) (because we have t21t2+1<1 for all t), and (2) it is injective, that is, it only traces over the (punctured) circle once. This parameterization looks qualitatively different from the trigonometric parameterization γ(t) above, but they are related by a clever and important change of variable related to Pythagorean triples and which proves to be extremely helpful in evaluating certain integrals.
One can, by the way, also parameterize surfaces (and even higher-dimensional objects); the most important difference is that (at least sensible) parameterizations of surfaces require two parameters, as a consequence of the fact that on surfaces one can move in two independent directions. A simple example is the parameterization r(ϕ,θ) of the unit sphere
by latitude ϕ and longitude θ:
When giving latitude an longitude of a point on Earth, we typically specify points with latitude 90ϕ90 and longitude 180θ180. (Here, points with ϕ=0 comprise the equator, and points with θ=0 the "prime meridian").
Many common shapes (lines, circles, other conic sections, planes, spheres, etc.) have well-known parameterizations, and graphs of functions RmRn have canonical parameterizations that are easy to write down, but like you say, for sufficiently complicated shapes parameterization can be a very difficult problem.

We have step-by-step solutions for your answer!

Answered 2021-12-16 Author has 46 answers
Often a curve γ in the plane is defined as the set of points (x,y) satisfying a certain geometric or algebraic condition. An example is
γ:={(x,y)x2a2+y2b2=1}, (1)
whereby the values of a>0 and b>0 are given. Such a description is implicit; it just provides a quick test whether a trial point (x,y) is lying on γ or not.
When we really want to geometrically analyze the curve γ, which means calculating its length or the enclosed area, etc., then we need a parametric representation. This is a production scheme that produces all points of γ in a systematic and analytically manageable way. In this way the points of the curve γ in (1) are produced by the vector-valued function
in a 1:1 way, whereby t=0 and t=2π produce the same point.
Since an implicit representation of a curve γ:F(x,y)=0 does not determine a parametric representation (a "timetable") t(x(t),y(t)) in an unique way there is no automatic procedure (like multiplication of two polynomials, or calculating a derivative) that generates a parametric representation from an equation F(x,y)=0; whence experience and intuition is asked for. Sometimes you can solve such an equation for y and obtain a representation as a graph in the form x(x,y(x)).
That deep mathematics is involved here can be seen from the following simple example: The innocent looking equation x2+y2=1 determines a curve γ that encloses an area of π and whose length is 2π.

We have step-by-step solutions for your answer!

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2021-05-14
Use the given graph off over the interval (0, 6) to find the following.

a) The open intervals on whichfis increasing. (Enter your answer using interval notation.)
b) The open intervals on whichfis decreasing. (Enter your answer using interval notation.)
c) The open intervals on whichfis concave upward. (Enter your answer using interval notation.)
d) The open intervals on whichfis concave downward. (Enter your answer using interval notation.)
e) The coordinates of the point of inflection. (x, y)=
asked 2021-12-06
Let P be the vector-valued function P(t)=(sint+,cost).
(a) What curve is traced by P? Draw a picture of this curve.
(b) Draw the vector P(π3) at the point corresponding to t=π3 on the picture you produced in part (a).
(c) Find a vector equation for the line tangent to the curve at t=π3.
asked 2021-12-20
If a curve has the property that the position vector r(t) is always perpendicular to the tangent vector r(t), show that the curve lies on a sphere with center the origin.
asked 2022-04-30

Determine if the two vectors are skew lines or if they intersect each other. !D = h2;3;1i+d h5;3;6i and −!F = h−5;−3;−5i+ f h15;3;−3i


asked 2021-08-31
Convert the following coordinates between rectangular and polar form
asked 2021-12-03

Find a path that traces the circle in the plane y=5 with radius r=2 and center (1, 5, 5) with conctant speed 8.


asked 2021-08-30
Polar coordinates:(8,π3) convert into retangular coordinates

New questions