Chapter 6

Parameterizing Curves

At the end of Part I, we described curves as sets of points satisfying equations. A parameterization gives us a different picture: imagine placing a moving point on the curve and recording its position as time passes. The path traced by the point is the geometric curve, while its motion tells us an orientation, a speed, and where to begin and end.

6.1Lines, Segments, and Curves

The simplest possible motion is to start at a point and move in a constant direction.

Definition 6.1 (Parametric Line). Let $𝑝$ be a point and let $⃗ 𝑣$ be a nonzero vector. The function

⃗ 𝑟 (𝑡) = 𝑝 + 𝑡 ⃗ 𝑣

parameterizes the line through $𝑝$ in the direction $⃗ 𝑣$ .

At $𝑡 = 0$ , the point is at $𝑝$ . Positive values of $𝑡$ move in the direction of $⃗ 𝑣$ , while negative values move in the opposite direction. Allowing $𝑡 \in ℝ$ traces the entire line. Restricting the parameter to a smaller interval traces only part of it.

Suppose instead that we want the line segment from a point $𝑝$ to a point $𝑞$ . The displacement from $𝑝$ to $𝑞$ is $𝑞 - 𝑝$ , so we can use that as the direction vector and stop when $𝑡$ reaches $1$ .

Theorem 6.2 (Line Segment Between Two Points). The line segment from $𝑝$ to $𝑞$ is parameterized by

⃗ 𝑟 (𝑡) = 𝑝 + 𝑡 (𝑞 - 𝑝) = (1 - 𝑡) 𝑝 + 𝑡 𝑞, 0 \leq 𝑡 \leq 1 .

The second form shows that the point is a weighted average of the two endpoints. When $𝑡 = 0$ all the weight is on $𝑝$ ; when $𝑡 = 1$ all the weight is on $𝑞$ ; and values between them interpolate along the segment.

Figure 6.1 Start at

𝑝

and add the vector

𝑡 ⃗ 𝑣

. As

𝑡

changes, the endpoint traces a line. For a segment the formula becomes

(1 - 𝑡) 𝑝 + 𝑡 𝑞

; the interval

0 \leq 𝑡 \leq 1

is what makes the motion begin at

𝑝

and stop at

𝑞

The same idea applies to any motion whose position is described by functions of one parameter.

Definition 6.3 (Plane Curve). A plane curve is a function $⃗ 𝑐 : ℝ \to ℝ 2$ . Written in the coordinates $(𝑥, 𝑦)$ of $ℝ 2$ , a plane curve can be expressed using two coordinate functions

⃗ 𝑐 (𝑡) = (𝑥 (𝑡), 𝑦 (𝑡))

We've seen examples of plane curves already, for instance parametric lines (like $ℓ (𝑡) = (2 𝑡 - 1, 3 𝑡 + 4)$ ) and parametric circles, (like $𝑐 (𝑡) = (2 c o s (𝑡) + 1, 2 s i n (𝑡) - 1)$ ).

Definition 6.4 (Space Curve). A space curve is a function $⃗ 𝑟 : ℝ \to ℝ 3$ . Written in the coordinates $(𝑥, 𝑦, 𝑧)$ of $ℝ 3$ , a space curve can be expressed using three coordinate functions:

⃗ 𝑟 (𝑡) = (𝑥 (𝑡), 𝑦 (𝑡), 𝑧 (𝑡)) .

We see curves like this in everyday life: watching a bird fly through the air, we see its position changing in time, so its $𝑥$ , $𝑦$ , and $𝑧$ components all change in time, $(𝑥 (𝑡), 𝑦 (𝑡), 𝑧 (𝑡))$ . Likewise, when watching your car on a GPS, we see the car's position changing—its latitude and longitude are functions of time, $c a r (𝑡) = (l a t (𝑡), l o n g (𝑡))$ . Indeed this is how we usually will use plane and space curves to trace out positions as a function of time. But it is also often useful to think of a curve as being traced out by a little arrow based at the origin (the vector picture, vs the position picture). This is particularly helpful when trying to build curves for yourself, as you can think about adding on terms, scalar multiplication, etc.

Parametric curves are used in animation, physics, and engineering. A single formula can control the motion of a bird, a car, or an animated character by giving its position at each time $𝑡$ .

6.2Parameterization Tips

Studying the properties of parametric curves falls squarely within mathematics, and we will soon spend significant time developing the calculus to do so. But creating parametric curves is more art than science: it helps to build up some intuition for a few basic examples and then learn how to combine and modify them to produce new and more interesting curves. I encourage you to follow along in the discussions below using the interactive figures throughout the chapter.

Example 6.5 (Parametrizing the Graph of a Function). If $𝑦 = 𝑓 (𝑥)$ is a function, its graph consists of the $𝑦$ value $𝑓 (𝑥)$ whenever the $𝑥$ -value is $𝑥$ . That means, the graph of $𝑓$ consists of the points $(𝑥, 𝑓 (𝑥))$ in the plane. Expressed yet a third way, a parametric equation that traces out the graph is given by

⃗ 𝑐 (𝑡) = (𝑡, 𝑓 (𝑡))

6.3Parameterizing Implicit Curves

Recall that an implicit equation gives a relationship between $𝑥$ and $𝑦$ . Some implicit equations describe functions, such as $𝑦 = 𝑥 2$ , but others do not; for instance, $𝑥 2 + 𝑦 2 = 1$ . Often, given an implicit equation, it is desirable to parameterize it: to find a way to trace out the curve as a function of $𝑡$ . There is no single way to do this, and it often takes some trial and error. Some tips appear below.

6.3.1The Circle

If we solve the circle equation for $𝑦$ , we obtain two functions,

𝑦 = \pm \sqrt 𝑅 2 - 𝑥 2 .

These give separate parameterizations for the upper and lower semicircles. To trace the entire circle in a single trip, it is better to build the defining equation into the coordinate functions.

Example 6.6 (Parametrizing Circles). The implicit equation for the unit circle is $𝑥 2 + 𝑦 2 = 1$ . Because the functions $c o s 𝑡$ and $s i n 𝑡$ satisfy the equation

c o s 2 (𝑡) + s i n 2 (𝑡) = 1

We see that if $𝑥 = c o s (𝑡)$ and $𝑦 = s i n (𝑡)$ then $(𝑥, 𝑦)$ must lie on the unit circle. Similarly, the equation $𝑓 (𝑡) = (𝑟 c o s (𝑡), 𝑟 s i n (𝑡))$ parameterizes a circle of radius $𝑟$ centered at $(0, 0)$ , and

𝑓 (𝑡) = (𝑟 c o s (𝑡) + ℎ, 𝑟 s i n (𝑡) + 𝑘)

parameterizes a circle of radius $𝑟$ centered at $(ℎ, 𝑘)$ .

6.3.2The Cusp $𝑦 2 = 𝑥 3$

Example 6.7 (Different Parameterizations of $y^2=x^3$). We can parameterize the implicit curve $𝑦 2 = 𝑥 3$ in several ways: taking the square root of both sides gives $𝑦$ as a function of $𝑥$ (with a plus and minus component), $𝑦 = \pm \sqrt 𝑥 3$ so one possible parameterization is

𝛼 \pm (𝑡) = (𝑡, \pm \sqrt 𝑡 3), 𝑡 \geq 0 .

This is not a single parameterization because of the $\pm$ sign; rather, it describes two parameterizations, one for each branch. Another option is to take the cube root. This causes no $\pm$ ambiguity and gives $𝑥$ as a function of $𝑦$ , namely $𝑥 = 3 \sqrt 𝑦 2$ , leading to the parametric curve

𝛽 (𝑡) = (3 \sqrt 𝑡 2, 𝑡)

A third option is to find a function for $𝑥 (𝑡)$ whose cube equals the square of the function chosen for $𝑦 (𝑡)$ . This is, of course, trickier, but one option is to take $𝑥 = 𝑡 2$ and $𝑦 = 𝑡 3$ . Then $𝑥 3 = 𝑡 6$ and $𝑦 2 = 𝑡 6$ , so $𝑥 3 = 𝑦 2$ , and our curve is

𝛾 (𝑡) = (𝑡 2, 𝑡 3)

The animation below compares these choices. Pay attention not only to whether the moving point satisfies the equation, but to how much of the implicit curve it actually reaches.

Figure 6.2 Solving for one variable may split an implicit curve into separate branches. The trigonometric circle and polynomial cusp parameterizations each reach their entire curve in a single pass.

6.3.3Intersections of Surfaces

Just as we parameterize an implicit plane curve by finding functions that satisfy its relation, we can parameterize curves that are intersections of two known surfaces.

Example 6.8. Parameterize the twisted cubic, lying on $𝑦 = 𝑥 2$ and $𝑧 = 𝑥 3$ . Here, if $𝑥 = 𝑡$ , then $𝑦 = 𝑡 2$ and $𝑧 = 𝑡 3$ . This fully specifies a point in 3-dimensional space, so we have our parameterization:

𝑓 (𝑡) = (𝑡, 𝑡 2, 𝑡 3)

Example 6.9. Parameterize the intersection of the cylinder $𝑥 2 + 𝑦 2 = 3$ and the plane $𝑥 + 𝑦 + 𝑧 = 1$ . On the cylinder, $𝑥$ and $𝑦$ lie on a circle of radius $\sqrt 3$ , so we can write $𝑥 = \sqrt 3 c o s (𝑡)$ and $𝑦 = \sqrt 3 s i n (𝑡)$ . The cylinder equation does not tell us anything about $𝑧$ , so it is no help there. But - we can solve the plane for $𝑧$ in terms of $𝑥$ and $𝑦$ to get $𝑧 = 1 - 𝑥 - 𝑦$ . Now we can plug in what we know $𝑥$ and $𝑦$ to be to get the parameterization:

𝛾 (𝑡) = (\sqrt 3 c o s (𝑡), \sqrt 3 s i n (𝑡), 1 - \sqrt 3 c o s (𝑡) - \sqrt 3 s i n (𝑡))

Figure 6.3 Parameterize whichever surface is easiest, then use the other equation to supply the coordinate which is left. Here the cylinder supplies

𝑥

and

𝑦

, and the plane determines

𝑧

Example 6.10. Intersection of $4 𝑦 = 𝑥 2 + 𝑧 2$ and $𝑦 = 𝑥$ . Because $𝑦 = 𝑥$ , our points in space have the form $(𝑥, 𝑥, 𝑧)$ . We can substitute this into the first equation to see that $4 𝑦$ becomes $4 𝑥$ , and so

𝑥 2 - 4 𝑥 + 𝑧 2 = 0

This is the equation for a circle! We can find its center and radius by completing the square:

𝑥 2 - 4 𝑥 = 𝑥 2 - 4 𝑥 + 4 - 4 = (𝑥 - 2) 2 - 4

So, this is the circle

(𝑥 - 2) 2 + 𝑧 2 = 4

Which is a circle of radius $2$ centered at $(2, 0)$ . We can parameterize it as $𝑥 = 2 c o s (𝑡) + 2$ and $𝑧 = 2 s i n (𝑡)$ . So, in 3D along the plane $𝑦 = 𝑥$ this becomes

𝑓 (𝑡) = (2 c o s (𝑡) + 2, 2 c o s (𝑡) + 2, 2 s i n (𝑡))

6.4Reading a Parameterization

Different parameterizations can describe the same curve: since a parameterization is like an animation of the curve, you can think of this as tracing out the curve at different speeds.

Example 6.11 (Different Parameterizations of the Circle). All three of these parametric curves trace out the unit circle.

(c o s 𝑡, s i n 𝑡)

(c o s 2 𝑡, s i n 2 𝑡)

(c o s 𝑡, - s i n 𝑡)

The first traces it at unit speed, counterclockwise. The second at twice the speed in the same direction. And the third, at unit speed but backwards (clockwise).

The image of a parameterization $⃗ 𝑟 : 𝐼 \to ℝ 𝑛$ is the set of points it reaches:

i m a g e (⃗ 𝑟) = {⃗ 𝑟 (𝑡) ∣ 𝑡 \in 𝐼} .

The image is the geometric curve. The parameterization carries additional information: its formula and interval tell us the direction of travel, the speed, and how many times the image is traced.

Figure 6.4 Three motions, one circle. The rings record how much of the circle each parameterization has covered and in which direction. Changing the formula changes the traversal without changing the image.

The parameter interval matters just as much as the formula. For example, $(c o s 𝑡, s i n 𝑡)$ traces the full circle on $[0, 2 𝜋]$ , only the upper semicircle on $[0, 𝜋]$ , and the circle twice on $[0, 4 𝜋]$ .

6.5New Curves from Old

Once we know a few parametric curves—circles, lines, and some implicit curves—it is easy to find more by modifying the ones we already know! We have already used two of the simplest such transformations, scaling and translation, in the case of circles.

Theorem 6.12 (Scaling a Parametric Curve). If $𝑓 (𝑡) = (𝑥 (𝑡), 𝑦 (𝑡))$ is a parametric curve, then $𝑟 𝑓 (𝑡) = (𝑟 𝑥 (𝑡), 𝑟 𝑦 (𝑡))$ scales every coordinate by $𝑟$ . For $𝑟 > 0$ , the new curve is $𝑟$ times as large.

Theorem 6.13 (Translating a Parametric Curve). If $𝑓 (𝑡) = (𝑥 (𝑡), 𝑦 (𝑡))$ is a parametric curve, then $𝑓 (𝑡) + (𝑎, 𝑏) = (𝑥 (𝑡) + 𝑎, 𝑦 (𝑡) + 𝑏)$ is the result of shifting the curve over by $(𝑎, 𝑏)$ .

Of course, more interesting transformations are also possible - and it's easiest to see this through a couple examples!

6.6Case Study: Spirals

We will make and understand a collection of spirals starting with the basic equation of the unit circle

(c o s (𝑡), s i n (𝑡))

Example 6.14 (Archimedean Spiral). For $𝑡 \geq 0$ , the Archimedean spiral rotates about the origin at unit angular speed but, after rotating through angle $𝑡$ , lies not at unit distance, as on a circle, but at distance $𝑡$ from the origin. To parameterize it, we multiply the circle by $𝑡$ :

𝛾 (𝑡) = (𝑡 c o s (𝑡), 𝑡 s i n (𝑡))

Example 6.15 (Logarithmic Spiral). The logarithmic spiral moves away from the origin exponentially fast, instead of linearly. This has radius at time $𝑡$ equal to $𝑒 𝑡$ , so

𝛾 (𝑡) = (𝑒 𝑡 c o s (𝑡), 𝑒 𝑡 s i n (𝑡))

Different functions $𝑟 (𝑡)$ for the radius multiplied by the circle give spirals that move outwards (or inwards) at different speeds. Try making some of these in the interactive figure below!

Exercise 6.16 (Whirlpool). Can you make a spiral that rotates about the origin at unit speed, but whose radius asymptotes to 2, never getting any larger?

Example 6.17 (Helix). A helix is a curve where $𝑥, 𝑦$ travel around a circle, and $𝑧$ increases at unit speed. For example, the unit helix is

𝛾 (𝑡) = (c o s (𝑡), s i n (𝑡), 𝑡)

Example 6.18 (Slinky-Like Helix). What if we want a helix like curve to move vertically at an uneven rate? Replace the $𝑧$ component with a more interesting function of $𝑡$ ! For instance, if $𝑧 = 𝑒 𝑡$ then the curve bunches up as $𝑡 \to - \infty$ along the $𝑥 𝑦$ plane:

𝛾 (𝑡) = (c o s (𝑡), s i n (𝑡), 𝑒 𝑡)

Example 6.19 (Spiral on a Cone). The surface $𝑧 = \sqrt 𝑥 2 + 𝑦 2$ traces out a cone—the height equals the radius! How can we draw a spiral on the surface of the cone? If we know what we want the spiral to do in its $𝑥$ and $𝑦$ components, we can calculate the $𝑧$ component using the formula above. For instance, given the Archimedean spiral $(𝑡 c o s (𝑡), 𝑡 s i n (𝑡))$ , we see For $𝑡 \geq 0$ ,

𝑧 = \sqrt (𝑡 c o s 𝑡) 2 + (𝑡 s i n 𝑡) 2 = 𝑡 .

Thus, the curve is

𝛾 (𝑡) = (𝑡 c o s 𝑡, 𝑡 s i n 𝑡, 𝑡)

The final figure places these constructions side by side. The same circular motion becomes a spiral when its radius changes and a helix when its height changes.

Figure 6.5 The graphs of

𝑟 (𝑡)

and

𝑧 (𝑡)

control the geometry of the curve. Changing the radius turns a circle into a spiral; changing the height lifts it into a helix. On the cone, the surface equation forces

𝑧 (𝑡) = 𝑟 (𝑡)

, so choosing one determines the other.

6.7A Parameterization Checklist

There is no formula which constructs every parameterization automatically, but the following checklist catches most of the important choices:

Identify the geometry. Is the curve a line, a graph, a circle, an implicit curve, or an intersection of surfaces?
Choose a useful parameter. Use one coordinate directly when possible; use sine and cosine for circular behavior; or choose functions which make the defining equation hold automatically.
Specify the parameter interval. The interval determines which part of the curve is traced and how many times it is covered.
Verify the image. Substitute the coordinate functions into the original equations, and check that the parameterization reaches every desired point.
Check endpoints and orientation. Evaluate the endpoints of the interval and follow increasing values of $𝑡$ to see which direction the curve moves.

Remark 6.20 (Optional: Eliminating the Parameter). The reverse problem is to start with a parameterization and find an equation for its image. For example, if

⃗ 𝑟 (𝑡) = ⟨ 𝑡, 𝑡 2 ⟩,

then $𝑥 = 𝑡$ and $𝑦 = 𝑡 2$ , so eliminating $𝑡$ gives $𝑦 = 𝑥 2$ . Likewise, $𝑥 = c o s 𝑡$ and $𝑦 = s i n 𝑡$ imply $𝑥 2 + 𝑦 2 = 1$ .

Eliminating the parameter recovers the geometric path, but it usually loses the parameter interval, orientation, speed, and number of traversals. It therefore answers a different question from constructing a parameterization.

We now know how to construct and interpret vector-valued functions. In the next chapter, we bring calculus back into the story: limits, derivatives, and integrals of curves will all be computed one component at a time.