Lecture Notes for 120 - UCLA Department of Mathematics

1.2. ARCLENGTH AND LINEAR MOTION 13
(c) Show that in polar coordinates we have
3R sin ✓ cos ✓
r =
sin 3 ✓ + cos 3 ✓
(12) Given two planar curves q 1 and q 2 we can construct a cissoid q as follows:
Assume that the line y = tx intersects the curves in q 1 =(x 1 (t) ,tx 1 (t))
and q 2 =(x 2 (t) ,tx 2 (t)), thendefineq (t) =(x (t) ,tx(t)) so that |q (t)| =
|q 1 (t) q 2 (t)|.
(a) Show that x (t) =± (x 1 (t) x 2 (t)).
(b) Show that the conchoid of Nicomedes is a cissoid.
(c) Show that the cissoid of Diocles is a cissoid.
(d) Show that the folium of Descartes is a cissoid.
(13) Let q be a cissoid where q 1 is the circle of radius R centered at (R, 0) and
q 2 averticallinex = b.
(a) Show that when b =2R we obtain the cissoid of Diocles
x x 2 + y 2 =2Ry 2
(b) Show that when b = R 2
we obtain the trisectrix (trisector) of Maclaurin
2x x 2 + y 2 = R 3x 2 y 2
(c) Show that when b = R we obtain a strophoid
y 2 (R x) =x 2 (x + R)
(d) Show that the change of coordinates x = u + v, y = p 3(u
the trisectrix of Maclaurin into Descartes folium.
1.2. Arclength and Linear Motion
v) turns
The arclength is the distance traveled along the curve. One way of measuring
the arclength geometrically is by imagining the curve as a thread that can be
stretched out and measured. This however doesn’t really help in formulating how
it should be measured mathematically. The idea of measuring the length of general
curves is relatively recent going back only to about 1600. Even Archimedes only
succeeded in understanding the arclength length of circles. Newton was the first to
give the general definition that we shall use below. As we shall quickly discover, it
is generally impossible to calculate the arclength of a curve as it involves finding
anti-derivatives of fairly complicated functions.
From a dynamical perspective the change in arclength measures how fast the
motion is along the curve. Specifically, if there is no change in arclength, then the
curve is stationary, i.e., you stopped. More precisely, if the distance traveled is
denoted by s (we can’t use d for distance as we’ll use that for differentiation), then
the relative change with respect to the general parameter is the speed
ds
dt = dq
dt
= |v|
This means that s is the anti-derivative of speed and is defined up to an additive
constant. The constant is determined by where we start measuring from. This
means that we should define the length of a curve on [a, b] as follows
ˆ b
L (q) b a = |v| dt = s (b)
a
s (a)