Good Sunday!
This blog will show how to transform polar equations, in the form of r(θ) to a pair of parametric equations, x(t) and y(t).
Tools We Need
x = r * cos θ
y = r * sin θ
Let θ = t.
For each example, we will change each polar equation and display a graph for each form. The polar form, colored blue, is on top; the parametric form, in red, is on the bottom. (A TI nSpire-CX is used for the pictures)
Example (I):
r = 2*θ
Then θ = r/2 and let θ = t.
x = r * cos t
x = 2 * t * cos(r/2)
x = 2 * t * cos((2*t)/2)
x = 2 * t * cos t
Similarly,
y = r * sin t
y = 2 * t * sin(r/2)
y = 2 * t * sin t
To summarize:
r = 2*θ
is equivalent to
x = 2 * t * cos t
y = 2 * t * sin t
Example (II):
r = e^(2 * θ)
Then θ = 1/2 * ln r and let θ = t.
And...
x = r * cos t
x = e^(2*t) * cos(1/2 * ln r)
x = e^(2*t) * cos(1/2 * ln(e^(2*t)))
x = e^(2*t) * cos(1/2 * 2 * t)
x = e^(2*t) * cos t
y = r * sin t
y = e^(2*t) * sin(1/2 * ln r)
y = e^(2*t) * sin(1/2 * ln(e^(2*t)))
y = e^(2*t) * sin t
To summarize:
r = e^(2*θ)
is equivalent to:
x = e^(2*t) * cos t
y = e^(2*t) * sin t
Example (III):
r = 2 cos θ
Then:
θ = arccos(r/2) = cos⁻¹(r/2)
Then, with θ=t...
x = r * cos t
x = 2 * cos t * cos(cos⁻¹(r/2))
x = 2 * cos t * cos(cos⁻¹(2*cos(t)/2))
x = 2 * cos t * cos t
x = 2 * cos² t
And... (some trigonometric identities are required)
y = r * sin t
y = 2 * cos t * sin(cos⁻¹(r/2))
Note: sin(cos⁻¹ x) = √(1 - x²)
y = 2 * cos t * √((1 - (r/2)²)
y = 2 * cos t * √(1 - r²/4)
y = 2 * cos t * √(1 - (4 cos² t)/4)
y = 2 * cos t * √(1 - cos² t)
Note: sin² x + cos² x = 1
y = 2 * cos t * sin t
Note: sin(2*x) = 2 * cos x * sin x
y = sin (2*t)
To Summarize:
r = 2 cos θ
is equivalent to:
x = 2 * cos² t
y = sin (2*t)
Until next time, take care!
Eddie
This blog is property of Edward Shore. 2013