Lecture Notes for 120 - UCLA Department of Mathematics

1.1. CURVES 9
Example 1.1.5. A far more subtle problem is Newton’s inverse square law:
d 2 q
dt 2 = q
|q| 3 = 1 q
|q| 2 |q|
Newton showed that the solutions are conic sections (intersections of planes and a
cone) and can be lines, circles, ellipses, parabolas, or hyperbolas.
Example 1.1.6. Finally we mention a less well known ancient example. This is
the conchoid (conch-like) of Nicomedes. It is given by a quartic (degree 4) equation:
x 2 + y 2 (y b) 2 R 2 y 2 =0
It is not clear why this curve looks like a conch. Descriptively it consists of two
curves that are given as points (x, y) whose distance along radial lines to the line
y = b is R. The radial line is simply the line that passes through the origin and
(x, y). Sowearemeasuringthedistancefrom(x, y) to the intersection of this radial
line with the line y = b. Asthatintersectionis
✓
x
x
y b ◆ 2
+(y b) 2 = R 2
⇣
x
y b, b ⌘
the condition is
which after multiplying both sides by y 2 easily reduces to the above equation.
The two parts of the curve correspond to points that are either above or below
y = b. Notethatnopointony = b solves the equation as long as b 6= 0.
A simpler formula appears if we use polar coordinates. The line y = b is
described as
and the point (x, y) by
(x, y) =(b cot ✓, b) =
b (cos ✓, sin ✓)
sin ✓
✓ ◆ b
(x, y) =
sin ✓ ± R (cos ✓, sin ✓)
This gives us a natural parametrization of these curves.
Another parametrization is obtained if we intersect the curve with the lines y =
tx and use t as the parameter. This corresponds to t = tan ✓ in polar coordinates.
Thus we obtain the parameterized form
✓ ◆
b
(x, y) =
t ± R
p (1,t)
1+t
2
As we have seen, what we consider the same curve might have several different
parametrizations.
Definition 1.1.7. Two parametrized curves q (t) , q ⇤ (t ⇤ ) are reparametrizations
of each other if it is possible to write t = t (t ⇤ ) as a function of t ⇤ and t ⇤ = t ⇤ (t)
such that
q (t) =q ⇤ (t ⇤ (t)) and q (t (t ⇤ )) = q ⇤ (t ⇤ )
If both of the functions t (t ⇤ ) and t ⇤ (t) are differentiable then it follows from the
chain rule that
dt dt ⇤
dt ⇤ dt =1
In particular, these derivatives never vanish and have the same sign. We shall almost
exclusively consider such reparametrizations. In fact we shall usually assume that