Parametric Equations to Polar Converter — Theory
1. Parametric equations
A parametric curve describes the coordinates of a point using a parameter:
\[
x=x(t),
\qquad
y=y(t).
\]
As \(t\) changes, the point \((x(t),y(t))\) moves along a curve.
2. Polar coordinates
A point can also be described using polar coordinates:
\[
(r,\theta),
\]
where \(r\) is the distance from the origin and \(\theta\) is the angle from the positive \(x\)-axis.
3. Parametric to polar conversion
Given \(x(t)\) and \(y(t)\), compute
\[
r(t)=\sqrt{x(t)^2+y(t)^2}.
\]
The angle is
\[
\theta(t)=\operatorname{atan2}(y(t),x(t)).
\]
The function \(\operatorname{atan2}\) is used because it places the angle in the correct quadrant.
4. Polar to parametric conversion
If a curve is given in polar form
\[
r=f(\theta),
\]
then we can use \(\theta\) as a parameter:
\[
x(\theta)=f(\theta)\cos\theta,
\qquad
y(\theta)=f(\theta)\sin\theta.
\]
5. Example: circle
Consider
\[
x=2\cos t,
\qquad
y=2\sin t.
\]
Square and add:
\[
x^2+y^2
=
(2\cos t)^2+(2\sin t)^2.
\]
Simplify:
\[
x^2+y^2
=
4\cos^2t+4\sin^2t
=
4.
\]
Since \(r^2=x^2+y^2\), we get
\[
r^2=4,
\qquad
r=2.
\]
6. Example: ellipse
For
\[
x=a\cos t,
\qquad
y=b\sin t,
\]
we have
\[
\frac{x^2}{a^2}+\frac{y^2}{b^2}=1.
\]
In polar form, use \(x=r\cos\theta\) and \(y=r\sin\theta\):
\[
\frac{r^2\cos^2\theta}{a^2}
+
\frac{r^2\sin^2\theta}{b^2}
=
1.
\]
Solving for \(r\) gives
\[
r=
\frac{ab}{\sqrt{b^2\cos^2\theta+a^2\sin^2\theta}}.
\]
7. Example: line through the pole
If
\[
x=at,
\qquad
y=bt,
\]
then
\[
\frac{y}{x}=\frac{b}{a}.
\]
This is a line through the origin. In polar form, the angle is constant:
\[
\theta=\operatorname{atan2}(b,a).
\]
8. Example: parabola
If
\[
x=t,
\qquad
y=t^2,
\]
then \(t=x\), so
\[
y=x^2.
\]
In polar form, substitute \(x=r\cos\theta\) and \(y=r\sin\theta\):
\[
r\sin\theta=(r\cos\theta)^2.
\]
9. Symmetry detection
The calculator numerically checks whether reflected points are also on the curve.
The common tests are:
10. Why symbolic conversion can fail
Not every parametric curve has a simple polar equation. A curve may only be expressible as
\[
r(t)=\sqrt{x(t)^2+y(t)^2},
\qquad
\theta(t)=\operatorname{atan2}(y(t),x(t)).
\]
This is still a valid polar description, but it is not always an explicit equation of the form
\(r=f(\theta)\).
12. Common mistakes
- Using \(\arctan(y/x)\) alone: this can give the wrong quadrant. Use \(\operatorname{atan2}(y,x)\).
- Expecting every curve to simplify: many parametric curves only have numerical polar descriptions.
- Ignoring the parameter interval: different intervals can trace different parts of the same curve.
- Confusing \(t\) and \(\theta\): in polar-to-parametric conversion, \(\theta\) becomes the parameter.
- Missing negative radius behavior: polar curves may plot points opposite the angle when \(r<0\).
- Relying only on formulas: side-by-side graphs help confirm the conversion.
