Part 3: Parameterizations of Circles   

In example 2, we showed that the unit circle has a parameterization of

u(q) = á cos(q),  sin(q) ñ

where q is in radians. Consequently, if a circle of radius R centered at a point (p,q) has a counterclockwise orientation,

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then its standard parameterization is given by

r(q) =  p + R u(q)

where p = (p,q) is the position vector of the center. Straightforward vector arithmetic subsequently yields

r(q) = á p + R cos( q),  q + R sin( q) ñ

(2)

where q is in radians.       

EXAMPLE 6    What is the standard parametrization of a circle with radius 1 centered at ( 0,1) ?      

Solution: Substituting p = 0, q = 1, and R = 1 into (2) yields

r(q) = á cos(q),1+sin( q) ñ

To verify this, notice that x = cos( q) and y = 1+sin( q) implies that

x2 + ( y - 1)2 =  cos2( q)  + sin2(q)  = 1

An object is said to be in uniform circular motion if it is moving in a circle with a parameterization (2) in which

q  = w(t - t0)

where the constant w is the angular speed in radians per second and t₀ is the time at which q vanishes (i.e., is equal to 0). That is, the parameterization of the position of an object in uniform circular motion is given by

r( t) = á p + R cos( w(t - t0) ),  q + R sin( w(t - t0) ) ñ

Alternatively, uniform circular motion shows us that a curve (such as a circle) has infinitely many parameterizations. Examples 5 and 6 are also illustrations of this fact.

EXAMPLE 7    Two objects are moving on a circle with radius 2 centered at (3,4) with angular speeds of 2 radians per second and 3 radians per second, respectively. Parameterize the motion of each object and compare the two given that both are at q = 0 initially.       

Solution: To begin with, q = 0 initially means that t₀ = 0. Thus, if we denote the two motions by r₁(t) and r₂( t) , respectively, then

r1(t) = á 3 + 2cos(2t), 4 + 2sin(2t) ñ r2(t) = á 3 + 2cos(3t), 4 + 2sin(3t) ñ

Moreover the second object will traverse the circle 3 times for every 2 traversals of the circle by the first object. Indeed, r₁(t) and r₂(t) are shown below for ever increasing values of t.

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 The fundamental period of a uniform circular motion r(t) is the smallest positive number T for which

r( t + T) = r(t)  for all t

(3)

That is, T  is the time it takes an object in uniform circular motion to make one complete cycle around the circle. Since q = 2p radians is the angle corresponding to one complete cycle, it follows that

w =    2p T

(4)

Thus, if we are given the period of a uniform circular motion, we can use (3) to calculate its angular speed.         

EXAMPLE 8    The moon's orbit is nearly a uniform circular motion with a radius of 238,957 miles and a period of 27.321 days.  What is the parameterization of the moon's orbit about the earth?

Solution: Assuming the earth is centered at (0,0), the angular speed--in radians per day-- of the moon about the earth is

w =    2p 27.321   = 0.22998

To 4 decimal places, this is w=0.23, so that a parameterization of the moon's orbit about the earth is

r( t) =  á 238957 cos( 0.23 t),  238957sin( 0.23 t) ñ

where r(t) is in miles and t is in days.

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