Parametric plots

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Tangent and normal vectors can help us plot make interesting parametric functions.

A sine curve on a circle

Suppose you wish to make a sine curve on a circle like this:

How do you do this? Well, a general method for placing one curve along another is to use unit tangent and unit normal vectors!

Plot the curve $y= \sin (5x)$ “wrapped” around a circle of radius $3$ .

To do this, start by setting $\vec {c}(t) = \vector {3\cos (t),\answer [given]{3\sin (t)}}$ This will produce our circle of radius $3$ .

xxxxxxxxxx
 
var('s t')
x(t) = 3*cos(t)
y(t) = 3*sin(t)
c(t) = (x(t),y(t))
circle=parametric_plot(c(t),(t,0,2*pi),color="black")
circle

Now we need to add our sine curve. To do this, we compute the unit tangent vector $\begin{align*} \utan(t) &= \frac{\vec{c}'(t)}{|\vec{c}'(t)|}\\ &=\vector{\answer[given]{-\sin(t)},\answer[given]{\cos(t)}} \end{align*}$ and the unit normal vector: $\begin{align*} \unormal(t) &= \frac{\utan'(t)}{|\utan'(t)|}\\ &=\vector{\answer[given]{-\cos(t)},\answer[given]{-\sin(t)}} \end{align*}$ We’ve plotted our circle of radius $3$ with some unit tangent and unit normal vectors for your viewing pleasure:

To add the sine curve, quite litterally, add it in: $\vec {f}(t) = \vec {c}(t) + \unormal (t)\cdot \sin (t)$ We can confirm our construction with SAGE:

xxxxxxxxxx
 
var('s t')
x(t) = 3*cos(t)
y(t) = 3*sin(t)
c(t) = (x(t),y(t))
dc=derivative(c,t)
ut= dc / dc.norm()
ddc = derivative(ut,t)
                                                                  
​
                                                                  
n = ddc / ddc.norm()
circle=parametric_plot(c(t),(t,0,2*pi),color="black")
curve=parametric_plot(c(t) + n*sin(5*t), (t,0,2*pi))
circle+curve

Thickening a curve

Suppose you have a vector-valued function $\vec {f}:\R \to \R ^2$ that defines a curve in space, and you want to build a parameterized surface that looks like a “thickened” version of the curve. In other words, we want to convert a curve like

into a thickened “tube” like

To plot a “tube” around a vector-valued function $\vec {f}$ , we need three handy vectors:

The unit tangent vector: $\utan (t) = \frac {\vec {f}'(t)}{|\vec {f}'(t)|}$
The unit normal vector: $\unormal (t) = \frac {\vec {t}'(t)}{|\vec {t}'(t)|}$
The unit binormal vector: $\ubinormal (t) = \utan (t) \cross \unormal (t)$

Let’s see these vectors in action with our next example.

You have been given a curve in space, say $\vec {f}(t) = \langle t^2-2, t^2-2, 1-t\rangle .$ We want to make a tube around it of radius $0.5$ .

Here is the idea, we want to take our curve and attach two orthogonal vectors to every point

We’ll start by computing the vectors. Write with me $\utan (t) = \frac {\vec {f}'(t)}{|\vec {f}'(t)|}$ From these vectors we can obtain the vectors by differentiating: $\unormal (t) = \frac {\utan '(t)}{|\utan '(t)|}$ Finally we can obtain the vectors via the cross product: $\ubinormal (t) = \utan (t) \cross \unormal (t)$ Looking again at

we see the unit normal vectors in red, and the unit binormal vectors in blue. So we now can plot our tube where $s$ runs from $0$ to $2\pi$ : $\begin{align*} \vec{F}(s,t) = \vec{f}(t) &+ 0.5 \unormal(\answer[given]{t})\cos(\answer[given]{s})\\ &+ 0.5 \ubinormal(\answer[given]{t})\sin(\answer[given]{s}) \end{align*}$ We can check our work with SAGE.

xxxxxxxxxx
 
var('s t')
x(t) = t^2-2
y(t) = t^2-2
z(t) = 1-t
f(t) = (x(t),y(t),z(t))
df=derivative(f,t)
ut = df / df.norm()
ddf = derivative(ut,t)
n = ddf / ddf.norm()
bn = ut.cross_product(n)
parametric_plot3d(f(t) + 0.5*n*cos(s) + 0.5*bn*sin(s), (t,-3,3), (s,0,2*pi))

Press...	...to do
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Type...	...to get
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arctan	$\arctan\left(\blue{[?]}\right)$
sin	$\sin\left(\blue{[?]}\right)$
cos	$\cos\left(\blue{[?]}\right)$
tan	$\tan\left(\blue{[?]}\right)$
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Phi	$\Phi$
Psi	$\Psi$
Omega	$\Omega$

A sine curve on a circle

Thickening a curve

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