What is parametrization of a function with more than one parameter?

[soandos]

could someone explain to me what exactly parametrization of a function in more than parameter means?
so i know that for f(x)=x^2
there are two parameters,
x
x^2

but how does that lead to a circle being
Sin(x)
Cos(y)?

what does this actually mean??
i.e. in the first example, i get it, in the second, i really don't.

is there a general way to take a parametrization and turn it into a "regular" function?

 

[tiny-tim]

could someone explain to me what exactly parametrization of a function in more than [one?] parameter means?

Hi soandos!

Parametrization means that instead of y = f(x), we use another variable t (the parameter), and write x = g(t), y = h(t).

For example, a circle can be y = ±√(1 - x²),

or it can be x = cos(t), y = sin(t).

One parameter means one dimension (a curve), two parameters means two dimensions (a surface), and so on.

so i know that for f(x)=x^2
there are two parameters,
x
x^2

No, that's not a parametrisation.

but how does that lead to a circle being
Sin(x)
Cos(y)?

sorry, but a circle isn't that … that's just wrong …

see the equations above.

 

[soandos]

sorry for all the errors, but how is it ploted in terms of two functions?

 

[tiny-tim]

Two parameters would give you a surface …

for example, a sphere z = ±√(1 - x² - y²) could be described by using the usual latitude and longitude as parameters:

x = cosθcosφ, y = cosθsinφ, z = sinθ.

 

[soandos]

sorry, i meant that with one parameter what is actually happening,
ex: Sin(t),Cos(t) plots as a circle. where are the points coming from?

are values of t plugged in and then both Sin(x) and Cos(x) plotted?
because that does not give a circle.

 

[tiny-tim]

Yes, you plug in values of t, and plot sin(t) and cos(t) …

and that must be a circle, because sin²t +cos²t = 1.

 

[soandos]

I must be missing something because if i plot y=sin(x) and y=Cos(x) on the same set of axes i do not get a circle

 

[tiny-tim]

… if i plot y=sin(x) and y=Cos(x) on the same set of axes …

What are you doing?

y = sin(t), x = cos(t).

 

[soandos]

Oh. got it.

 

[Tac-Tics]

Any function f can be easily "parametrized" this way by setting:

y(t) = f(t)
x(t) = t

But you can't always "undo" this parametrization. For example, a graph of a circle is not a function (because f(0) would have to take one two values, 1 and -1).

Parametrization is very useful because it corresponds to what you imagine when you think about a car or a guy walking around on a map. "t" represents time and x(t), y(t) represent his x,y coordinates at time t.

 

[soandos]

Unrelated, I read somewhere that all objects of rotation are easily parametrized. how so?

 

[tiny-tim]

Unrelated, I read somewhere that all objects of rotation are easily parametrized. how so?

Use θ as one of the parameters …

so lines of constant θ are all the same shape, adn going parallel to the axis of rotation.

 

[slider142]

Another way of thinking about it is giving a coordinate system to the shape. As a curve is 1-dimensional, you only need one parameter to locate a point on the curve. Ie., that parameter may be distance from some point (an origin) on the curve.
For example, the curve generated by the graph of y = f(x) = x^3 can be parametrized by letting the point (0,0) be the origin of our 1-dimensional coordinate system and then noting that x(t) = t and y(t) = t^3 when describing the curve using the Cartesian xy-coordinate system.
Another valid parametrization is to choose the origin arbitrarily, ie., let the point (1,1), the coordinates of the point on the curve with respect to the Cartesian plane, be the point 0 with respect to our 1-dimensional curve coordinate system. Then x(t) = t + 1 and y(t) = (t + 1)^3.
You are now free to embed curves in any dimension, ie., curves in R^3, which would be rather clumsy to define in terms of functions. Any 1-parameter family of functions x(t), y(t), z(t) should describe a curve, as long as they're well-behaved. Try to describe a right-circular helix about the z-axis. (Hint: x(t) and y(t) should describe a circle, while z(t) just stretches it out. Drawing a picture always helps).
Surfaces are by definition 2-dimensional objects, so we will need 2 coordinates to locate a point on a surface. We usually go for the variables (u,v). That is to say, if we are talking about the parametrization of a surface as a subset of R^3, we would have 3 functions x(u,v), y(u,v), and z(u,v). Just imagine laying a grid down on the surface you want to parametrize; one of the directions will be the u-direction, the other will be v.

 

[soandos]

OK so the circle part is easy,
x(t)=Sin(t)
y(t)=Cos(t)
z(t)=t
for z(t) doesn't any linear function work (assuming that the space between "coils" remains a constant)?

 

[slider142]

OK so the circle part is easy,
x(t)=Sin(t)
y(t)=Cos(t)
z(t)=t
for z(t) doesn't any linear function work (assuming that the space between "coils" remains a constant)?

Yep. By varying z(t), you can make the coils do any wacky thing along the z-axis that you want. With 3-d software that renders curves from parametrizations and a little bit of physics knowledge, you should be able to animate a bouncing spring.
You should even be able to parametrize a closed ring-like helix, where the "axis" is a circle.
As for surface parametrizations, a simple but non-trivial one would be parametrizing a torus in R^3. One common parametrization would be to let one coordinate (u) represent angle from the center of the torus and let the second coordinate (v) represent angle from the center of the tube of the torus. If we place our torus around the origin in the xy-plane with inner radius r and outer radius R, we then have x(u, v) = ((r + R)/2 + (R - r)sin(v))cos(u) by applying some trigonometry. You may get a slightly different form depending on exactly how you defined your u and v. You can then derive the expressions for y(u, v) and z(u, v).
For a sanity check, you will want to use some graphing software that accepts parametrizations. http://blog.viamentis.com/articles/2008/01/18/plotting-mathematical-functions-with-gnuplot/ raytracer for aesthetic manipulation/animation. If you are already experienced in POV-Ray scripts, you can input parametric equations right into it, and even visualize curves in 3d using a #while loop inside a sphere_sweep object.

 

[soandos]

Just curious, if a helix is a circle that is "extended" what is a cylinder?

 

[slider142]

Just curious, if a helix is a circle that is "extended" what is a cylinder?

A helix is a curve, while a cylinder is a surface. It can only be described by 2 coordinates. One easy parametrization is having height be the second parameter: x(u,v) = sin(u), y(u, v) = cos(u), z(u,v) = v.

 

[Tac-Tics]

A helix is a curve, while a cylinder is a surface. It can only be described by 2 coordinates. One easy parametrization is having height be the second parameter: x(u,v) = sin(u), y(u, v) = cos(u), z(u,v) = v.

You'd also need to put restrictions on z, so it isn't infinitely long.

 

[HallsofIvy]

Why shouldn't it be infinitely long?

 

[Tac-Tics]

Why shouldn't it be infinitely long?

Because an infinitely long can of soda would be expensive and would go flat before you ever finished =-)

 

[soandos]

is it possible to parametrize a double helix? triple?
besides the easy way of taking a single helix and then rotating that by a given amount of radians.

 

[HallsofIvy]

The double helix is not a connected set. The obvious thing to do is to set up a parameterization for each helix separately.