Parametric Curves
(Com S 477/577 Notes)
Yan-Bin Jia
Oct 8, 2015
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Introduction
A curve in R2 (or R3 ) is a differentiable function α : [a, b] → R2 (or R3 ). The initial point is α[a]
and the final point is α[b]. The domain of the curve is the interval [a, b]. A portion of α defined
on an interval [c, d] ⊆ [a, b] is called a curve segment.
Example 1. Straight Line
linear. Explicitly, the curve
The line is the simplest curve in the plane as its coordinate functions are
α(t) = p + tv = (x0 + tvx , y0 + tvy ),
where v 6= 0,
(1)
is a straight line through the reference point p = α(0) = (x0 , y0 ) in the direction v = (vx , vy ). Here, t is the
signed distance from a point α(t) on the line to p as scaled by kvk.
As shown on the left, the vector from p to a point
(x, y) on the line must be either in the direction of
(x, y)
(vx , vy ) or in its opposite direction. Hence, the cross
(x0 , y0 )
product of the two vectors must be zero, that is,
(vx , vy )
(x − x0 , y − y0 ) × (vx , vy ) = 0.
Expansion of the above cross product yields an implicit
equation of the line that relates the x and y coordinates
of every incident point:
vy x − vx y − vy x0 + vx y0 = 0.
Example 2. Helix1
The curve t → (a cos t, a sin t, 0) travels around a circle of radius a > 0 in
the x-y plane. If we allow this curve to rise (or fall) at a constant rate, we obtain
a helix
α = (a cos t, a sin t, bt),
where a > 0 and b 6= 0.
Example 3. The curve α : R → R3 such that
√
α(t) = (et , e−t , 2t)
1
The figure is from [1, p. 16].
1
(2)
shares with the helix in Example 2 the property of rising constantly. However, it lies over the hyperbola
xy = 1 in the x-y plane instead of a circle.
A curve α(t) = (x(t), y(t)) is said to be smooth at t = t0 if its kth derivative
(k)
(k)
(k)
α (t) = x (t), y (t)
exists for any integer k > 0. A piecewise smooth curve α has a domain which is
the union of a finite number of subintervals over each of which α is smooth.
Example 4. A line α(t) = p + tq is a smooth curve. Here α′ (t) = q and α(k) = 0 for k > 1. A polygon,
on the other hand, is a piecewise smooth curve, where each edge determines a subdomain.
Example 5. Cuspidal cubic The curve α(t) = (t2 , t3 ) is smooth.
y
We have
α′ (t)
α′′ (t)
x
α′′′ (t)
α(k) (t)
= (2t, 3t2 ),
= (2, 6t),
= (0, 6),
= 0,
k ≥ 4.
Consider a plane curve α : [a, b] → R2 . It is called a closed parametric curve if α(a) = α(b). A
point of self-crossing is a point α(t1 ) for which there exist finitely many distinct values t1 , . . . , tn ∈
[a, b], n ≥ 2, which satisfy α(t1 ) = α(t2 ) = · · · = α(tn ), and in the case n = 2, [t1 , t2 ] 6= [a, b].
Example 6. A circle is closed. The other three curves all have self-crossings.
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Velocity, Speed, and Arc Length
Let α(t) be a curve. The velocity vector of α at t is α′ (t). The speed at t is the length kα′ (t)k. The
meaning is clear if we see α(t) as the location of a moving point at time t. The parametrization
α(t) is unit-speed if kα′ (t)k = 1 for all values of t. A point where α′ (t) = 0 is called a cusp on the
curve.
Example 7. The origin on the cuspidal cubic in Example 5 is a cusp.
The curve α(t) is regular if all velocity vectors are different from zero, that is, α′ (t) 6= 0 for all t.
Intuitively, a point moving on the curve with velocity α′ (t) will never come to a stop or reverse its
direction.
Example 8. Consider the curve α(θ) = (aθ cos θ, aθ sin θ). It has velocity
α′ (θ) = a(cos θ − θ sin θ, sin θ + θ cos θ),
and speed
p
p
kα′ (θ)k = |a| (cos θ − θ sin θ)2 + (sin θ + θ cos θ)2 = |a| 1 + θ2 6= 0.
Therefore the parametrization is regular.
The velocity and speed depend on its parametrization. Non-regularity at a point may be just
a property of the parametrization, and need not correspond to any special feature of the curve
geometry. For a different parametrization the curve may have a non-zero velocity at the same
point.
To formulate the length of α, we note that the portion
over [t, t + δt] is nearly a straight line when δt is very small.
α(t)
So the length over [t, t + δt] can be approximated by
α(t + δt)
kα(t + δt) − α(t)k,
which again is approximated by
kα′ (t)kδt.
We divide α up into segments, each of which corresponds to a small increment δt. As δt tends to
zero, we will obtain the exact length. The arc length of α from t = a to t = b is thus defined as
Z b
kα′ (t)k dt.
a
Example 9. Logarithmic spiral The curve
α(t) = (et cos t, et sin t),
has a spiral motion. We obtain that
α′ (t) =
kα′ (t)k
=
et (cos t − sin t), et (sin t + cos t) ,
√ t
2e .
3
y
x
Figure 1: Logarithmic spiral (et/20 cos t, et/20 sin t) over [0, 50].
Hence the arc length of α starting at α(0) = (1, 0), for instance, is
Z t√
√
s=
2eu du = 2(et − 1).
0
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Reparametrization
Let I and J be intervals. Let α : I → R3 be a curve and h a differentiable function. Then the
composite function β = α ◦ h is a curve called the reparametrization of α by h.
β
I
J
s
Example 10.
β(s) = α(h(s))
α
h
t
√ √
Suppose α(t) = ( t, t t, 1 − t) on (0, 4). If h(s) = s2 on (0, 2), then
β(s) = α(h(s)) = α(s2 ) = (s, s3 , 1 − s2 ).
The curve α has been reparametrized by h to yield the curve β.
At each time s in the interval J, the curve β is at the point β(s) = α(h(s)) reached by the
curve α at time h(s) in the interval. Thus β does follow the route of α, but it reaches a given point
on the route at a different time than α does.
Sometimes one is interested only in the route followed by a curve and not in the particular
speed at which it traverses its route. One way to ignore the speed of a curve α is to reparametrize
to a curve α̃ which has unit speed kα̃′ k = 1.
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Theorem 1 If α is a regular curve, then there exists a reparametrization α̃ that has unit speed.
Proof
Consider the arc length function
s(t) =
Z
t
kα′ (u)k du,
c
where c is a number in the domain of α. It then follows that
s′ (t) = kα′ (t)k;
namely, the derivative of s is the speed function kα′ (t)k. Since α is regular, α′ 6= 0 everywhere;
hence ds
dt > 0 always holds. By a standard theorem of calculus, the function s has an inverse
function t(s), and
1
1
dt
= ds =
.
′
ds
kα (t)k
dt
Now we let α̃(s) = α(t(s)) be the reparametrization of α. Then
α̃′ (s) = α′ (t(s))
Hence, the speed of α̃ is
kα̃′ (s)k = kα′ (t(s))k
dt
.
ds
1
kα′ (t(s))k
= 1.
The unit-speed curve α̃ is said to have arc-length parameterization, since the arc length of α̃
from s = a to s = b, a < b, is just b − a.
Example 11.
Let us consider the helix α = (a cos t, a sin t, bt) in Example 2 again. It has velocity
α′ (t) = (−a sin t, a cos t, b).
Hence
kα′ (t)k2 = α′ (t) · α′ (t) = a2 sin2 t + a2 cos2 t + b2 = a2 + b2 .
Thus α has constant speed:
p
a2 + b 2 .
c = kα′ k =
The arc length from t = 0 is then
s(t) =
Z
t
c du = ct.
0
Hence, t(s) = sc . Substituting this into the formula for α, we get the unit-speed reparametrization
s s bs
s
= a cos , a sin ,
.
α̃(s) = α
c
c
c c
Although every regular curve has a unit-speed reparametrization, this may be very complicated,
or even impossible to write down explicitly, as the following examples show.
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Example 12.
The logarithmic spiral
α(t) = (et cos t, et sin t),
has speed
√ t
2e > 0.
√
So it is regular. The arc length starting at (1, 0) was found in Example 9 to be s = 2(et − 1). Hence,
t = ln( √s2 + 1), so a unit-speed reparametrization of α is given by the rather unwieldy formula
α̃(s) =
s
s
s
s
√ + 1 cos ln √ + 1
, √ + 1 sin ln √ + 1
.
2
2
2
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Example 13. Twisted cubic2 This is the space curve given by
α(t) = (t, t2 , t3 ),
−∞ < t < ∞.
We have
α′ (t) =
kα′ (t)k
=
(1, 2t, 3t2 ),
p
1 + 4t2 + 9t4 .
Since the speed kα′ (t)k is not zero everywhere, α is regular. And the
arc-length starting at α(0) = 0 is
Z tp
1 + 4u2 + 9u4 du.
s=
0
The above integral has a horrendous closed form not in terms of familiar functions.
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Tangent and Normal
The standard method of studying the geometry
of a curve at a point is to attach orthonormal
tangent line
vectors to the point and see how the directions of
t increasing
these vectors change as the point moves on the
curve for an infinitesimal distance. We choose
α
tangent and normal vectors at a regular point.
(x′ (t), y ′ (t))
(−y ′ (t), x′ (t))
Let α(t) = (x(t), y(t)) be a curve. At a regular point α(t) there exists a (non-zero) tangent
vector α′ (t) = (x′ (t), y ′ (t)). It represents the velocity of the curve at the point. The normal vector (−y ′ (t), x′ (t)) at α(t) is given by rotating
the tangent vector counterclockwise through an angle π2 . Note that (x′ (t), y ′ (t)) × (−y ′ (t), x′ (t)) =
(x′ (t))2 + (y ′ (t))2 > 0.
If α(t) is a unit-speed curve, then both the tangent vector and the normal vector are unit vectors.
By convention they are denoted as T and N , respectively, with the cross product T × N = 1.
normal line
2
The figure originally appears in [3, p. 14].
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For a parametric curve we have a tangent line and a normal line at each regular point α(t).
The tangent line to the curve at α(t) passes through α(t) and is parallel to α′ (t) 6= 0. So it has
the parametric equation
x(s), y(s) = α(t) + sα′ (t),
s ∈ (−∞, ∞),
or equivalently, the algebraic equation
(x, y) − α(t) · −y ′ (t), x′ (t) = 0.
The normal line at α(t) passes through the point and is parallel to (−y ′ (t), x′ (t)). So its equations
are of the form
x(s), y(s) = α(t) + s −y ′ (t), x′ (t) ,
s ∈ (−∞, ∞),
or equivalently,
x(s), y(s) − α(t) · α′ (t) = 0.
Example 14. Crunodal cubic is described as
α(t) = t2 − 1, t(t2 − 1) .
Find its tangent and normal lines of the curve at the points t = ±1, 0.
We obtain
α′ (t) =
α′ (1) =
α′ (−1) =
α′ (0) =
α(±1) =
y
(2t, 3t2 − 1),
(2, 2),
(−2, 2),
(0, −1),
(0, 0).
x
Here α = (0, 0) is referred to as a double point since it is attained at both
t = 1 and t = −1. The tangent lines at this double point are respectively
(x, y) = s(1, 1),
or equivalently,
y = x,
and
(x, y) = s(−1, 1),
or equivalently,
The normal lines at the double point are respectively
(x, y) = s(−1, 1),
or equivalently,
y = −x.
y = −x,
and
(x, y) = s(−1, −1),
′
or equivalently,
y = x.
At t = 0, we have α (0) = (0, −1), and the tangent line at α(0) is
(x, y) = (−1, 0) + s(0, −1),
or equivalently,
x = −1.
or equivalently,
y = 0.
The normal line at α(0) is
(x, y) = (−1, 0) + s(1, 0),
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References
[1] B. O’Neill. Elementary Differential Geometry. Academic Press, Inc., 1966.
[2] J. W. Rutter. Geometry of Curves. Chapman & Hall/CRC, 2000.
[3] A. Pressley. Elementary Differential Geometry. Springer-Verlag London, 2001.
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