Parametric Equations
A curve C in plane can be represented by parametric equations
x = f (t), y = g(t),
t ∈ I,
where f and g are functions on an interval I. Each
value of t determines a point (x, y) = (f (t), g(t)) in
the plane. As t varies over I, the point (f (t), g(t))
varies and traces out the curve C.
y
C
(f(t),g(t))
x
O
1
Parametric Equations of Lines
Let L be the line in plane passing through a
point P0 (x0 , y0 ) with a direction vector v = (a, b).
Then L has parametric equations
−∞ < t < ∞.
x = x0 + at, y = y0 + bt,
It also can be written in the symmetric form:
x − x0
y − y0
.
=
b
a
y
P0
P1
x
O
2
If the direction vector v = (a, b) has a 6= 0, then
m = b/a is the slope of the line and the line has the
point-slope equation: y − y0 = m(x − x0 ).
In particular, if x0 = 0, then the line intercepts the
y-axis at the point (0, y0 ). So the line has the slopeintercept equation: y = mx + y0 .
If the direction vector v = (a, b) has a = 0, then
the line is vertical and it has an equation x = x0 .
If a line passes through two points P0 (x0 , y0 )
and P1 (x1 , y1 ), then it has v = (x1 − x0 , y1 − y0 )
as a direction vector. Hence, the line has parametric
equations x = x0 +(x1 −x0 )t, y = y0 +(y1 −y0 )t for
−∞ < t < ∞. The line segment P0 P1 corresponds
to 0 ≤ t ≤ 1.
3
The Unit Circle
The distance from the origin O(0, 0) to a point
p
P (x, y) is x2 + y 2 . Thus, the equation x2 +y 2 = 1
represents the unit circle with center at O.
y
(0,1)
(-1,0)
O
t
P(cost, sin t)
(1,0)
(0,-1)
4
x
Let t be the angle (measured in radians) from
Ox counterclockwise to the ray OP . Then x = cos t
and y = sin t. Thus, the unit circle is represented by
parametric equations
x = cos t, y = sin t,
0 ≤ t ≤ 2π.
We have the following observation:
t = 0 7→ (1, 0)
t=
π
2
⇒
cos 0 = 1,
sin 0 = 0;
⇒
cos π2 = 0,
sin π2 = 1;
⇒
cos π = −1,
sin π = 0;
7→ (0, −1) ⇒
cos 3π
2 = 0,
sin 3π
2 = −1.
7→ (0, 1)
t = π 7→ (−1, 0)
t=
3π
2
5
Arcs
An arc of the unit circle is represented by parametric equation
x = cos t, y = sin t,
α ≤ t ≤ β,
where α and β are two real numbers such that
0 < β − α ≤ 2π. In particular, the upper half of the
unit circle is represented by parametric equations
x = cos t, y = sin t,
0 ≤ t ≤ π.
y
(0,1)
(-1,0)
O
(1,0)
6
x
Circles
Let C be the circle with radius r and center at
(a, b). Then C is described by the equation
(x − a)2 + (y − b)2 = r2 .
We choose a parameter t such that (x − a)/r = cos t
and (y − b)/r = sin t. Thus, the circle C has parametric equations
x = a + r cos t, y = b + r sin t,
0 ≤ t ≤ 2π.
y
r
(a,b)
x
O
7
Example. Let C be the curve given by parametric
equations
x(t) = t2 − 1, y(t) = t3 − t,
−∞ < t < ∞.
Sketch the curve and find the intersection points of
the curve with the line x = 3.
Solution. We have
t = −2 7→ (3, −6),
t = −1 7→ (0, 0),
−0.7 7→ (−0.51, 0.357), −0.4 7→ (−0.84, 0.336),
t = 0 7→ (−1, 0),
0.4 7→ (−0.84, −0.336), 0.7 7→ (−0.51, −0.357),
t = 1 7→ (0, 0),
t = 2 7→ (3, 6).
We plot these points and connect them to produce
a continuous curve.
We have x(t) = t2 − 1 = 3. This equation has solutions t = 2 or t = −2. But y(2) = 6 and y(−2) = −6.
Hence, the intersection points are (3, 6) and (3, −6).
8
The following is the graph of the curve given by
parametric equations
x(t) = t2 − 1, y(t) = t3 − t,
−∞ < t < ∞.
y
(3,6)
(-1,0)
x
O
(3,-6)
9