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## Multivariable calculus

### Unit 1: Lesson 4

Visualizing vector-valued functions# Parametric curves

When a function has a one-dimensional input, but a multidimensional output, you can think of it as drawing a curve in space. Created by Grant Sanderson.

## Want to join the conversation?

- What would happen if you have a function that has a parameter "t" and outputs 3 dimensional vectors? I can imagine that this would draw some weird 3D curve like a trail of smoke, but is this always the case or can you get surfaces with only one paremeter too?(15 votes)
- Think of the parameters as the coordinates ON the resulting shape. A curve will have a starting point and an ending point, no matter how many dimensions it takes (a good example of a 3 dimensional curve is a helix). The input parameter (t), tells you how far along the curve have you gone from the starting point. The parameter (t) doesn't care what the shape of the curve is, it sees the curve as an one dimensional object on which it can only move back and forth.

Analogically, a surface (in a 3D space) will always take two parameters. A surface represents a curved two dimensional plane. For example - any place on the Earth can be represented by two parameters (coordinates) - the latitude and longitude.(29 votes)

- how do we even interpret the rate of this function? for unctions in the normal coordinate system, we've got the derivative to tell us how fast something's changing, but what about here?(10 votes)
- Suppose we have a function f(t)=[t*sin(t)]

[t*cos(t)]

x=t*sin(t)

y=t*cos(t)

If you want to do some parametric graphing, use this link. https://www.desmos.com/calculator/ksjcpazwa9

In our example, we've got one-dimensional input, and we get the 2-dimensional output - we plot points as our output or vectors. So, we'll get a nice spiral graph with the above plot.

To find dy/dx,

we first get dy/dt for which we have to use product rule and we'll get (cost - t*sint) and divide that by dx/dt for which we'll get (sint + t*cost).

So,

dy/dx =

(dy/dt)

-------

(dx/dt)

So, let's take a point. For that, I will take the value of t as 1 (you can take any number). For x, I get 0.841, and for y, I get 0.540.

So, I will input the value of t in my derivative equation and I will get -0.301/1.382 = -0.218.

So, at point (0.841,0.540), dy/dx will be -0.218. If you've plotted the graph, you'll see it clearly there. I hope you understood. In case you didn't, Khan academy has videos on it by the name "Parametric equations differentiation".(6 votes)

- Yes, but how do I go from a function to a parametric equation? Everyone seems to assume that you already know what the parametric form is?(6 votes)
- What's is the software program used to get this kind of math animations? Or is it all programmed in python by Grant Sanderson?? Thanks(3 votes)
- I believe he's using the program Grapher, which comes as a stock app on all Mac computers. :)(5 votes)

- Hello, may I just ask in which computer language are all the simulations of transformations written?

Thank you :)(1 vote) - Would it be possible to graph the value of t in three dimensions on the z-axis so that you would have an understanding of what the value of t is?(3 votes)
- Whats really the difference between vector valued functions and other multivariable functions?(2 votes)
- What makes the graph slow or fast when it starts?(2 votes)
- What is the difference between vector valued functions and parametric curves? Are parametric curves a special type of vector valued function? If so, what defines them?(1 vote)
- Simply put, a parametric curve is a normal curve where we
**choose**to define the curve's x and y values in terms of another variable for simplicity or elegance. A vector-valued function is a function whose value is a vector, like velocity or acceleration(both of which are functions of time).(2 votes)

- Should be "and what curve do the tips of those vectors trace out" at3:52.(1 vote)

## Video transcript

- [Voiceover] More
function visualizations. So let's say you have a function that's got a single input t. And then it outputs a vector. And the vector's gonna depend on t. So the x component will be
t times the cosine of t. And then the y component will
be t times the sine of t. This is what's called
a parametric function. And I should maybe say
one-parameter parametric function. One parameter. And "parameter" is just kind
of a fancy word for input. Parameter. So in this case, t is
our single parameter. And what makes it a parametric function is that we think about
it as drawing a curve and its output is multidimensional. So you might think, when you
visualize something like this, ah, it's got, you know, a single input. And it's got a two-dimensional output. Let's graph it. You know, put those three
numbers together and plot them. But what turns out to be even better is to look just in the output space. So in this case, the output
space is two-dimensional. So I'll go ahead and draw
a coordinate plane here. And let's just evaluate this function at a couple different points and see what it looks like, okay. So I might, maybe the easiest place to evaluate it would be zero. So f of zero is equal to, and then, in both cases,
it'll be zero times something. So zero times cosine of zero is just zero. And then zero times sine
of zero is also just zero. So that input corresponds to the output. You could think of it as a
vector that's infinitely small or just the point at the origin, however you want to go about it. So let's take a different point, just to see what else could happen. And I'm gonna choose pi halves. Of course, the reason
I'm choosing pi halves, of all numbers, is that
it's something I know how to take the sine and the cosine of. So t is pi halves, cosine of pi halves. And you start thinking, okay, what's the cosine of pi halves? What's the sine of pi halves? And maybe you go off and
draw a little unit circle while you're writing things out. Whoops. See the problem with
talking while writing. Sine of pi halves. And you know it's, if you go off and scribble
that little unit circle. And you say pi halves is gonna bring us a quarter of the way around. Over here. And cosine of pi halves is measuring the x component of that. So that just cancels out to zero. And then sine of pi halves is the y component of that. So that ends up equaling one. Which means that the vector as a whole is gonna be zero for the x component and then pi halves for the y component. And what that would look like, you know, the y component of pi halves is about 1.7 up there. There's no x component. So you might get a vector like this. And if you imagine doing this at all the different input points, you might get a bunch of different vectors off doing different things. And if you were to draw it, you don't want to just
draw the arrows themselves. Because that'll clutter
things a whole bunch. So we just want to trace the points that correspond to the output,
the tips of each vector. And what I'll do here, I'll
show a little animation. Let me just clear the board a bit. An animation where I'll let t range between zero and 10. So let's write that down. So the value t is gonna start at zero. And then it's gonna go to 10. And we'll just see what values, what vectors does that output. And what curve does the tip
of that vector trace out? So there it goes. All of the values just kind
of ranging, zero to 10. And you end up getting this spiral shape. And you can maybe think about why this cosine of t, sine of t
scaled by the value t itself would give you this spiral. But what it means is that when you, you know, we saw that zero goes here. Evidently, it's the case
that 10 outputs here. And a disadvantage of
drawing things like this, you're not quite sure of
what the interim values are. You know, you could kind of guess. Maybe one goes somewhere here. Two goes somewhere here. And you're kind of hoping
that they're evenly spaced as you move along. But you don't get that information. You lose the input information. You get the shape of the curve. And if you just want, you know, an analytical way of describing curves, you find some parametric
function that does it. And you don't really care about the rate. But just to show where it might matter, I'll animate the same thing again, another function that
draws the same curve. But it starts going really quickly. And then it slows down as you go on. So that function is not quite
the t cosine t, t sine of t that I originally had written. And in fact, it would mean that, let's just erase these guys. When it starts slowly, you can interpret that
as saying, okay maybe... Well, actually, it started
quickly, didn't it? So one would be really far off here. And then two, we kind
of zipped along here. And three, you know,
still going really fast. And then maybe by the
time you get to the end, it's just going very slowly, just kind of seven is here, eight is here, and it's hardly making any
progress before it gets to 10. So you can have two different functions draw the same curve. And the fancy word here is "parameterize." So functions will parameterize a curve if, when you draw just
in the output space, you get that curve. And in the next video, I'll show how you can have functions with a two-dimensional input and a three-dimensional output draw surfaces in three-dimensional space.