Parametric Equations Plotter and Converter — Theory
1. What is a parametric curve?
A parametric curve describes a point whose coordinates depend on a parameter \(t\).
Instead of writing \(y\) directly as a function of \(x\), we write
\[
x=x(t),\qquad y=y(t).
\]
The moving point is
\[
\mathbf r(t)=\langle x(t),y(t)\rangle.
\]
As \(t\) changes, the point traces the curve.
2. Orientation of a parametric curve
A Cartesian equation often shows the shape of a curve, but a parametric equation also shows direction.
The direction is the path followed as \(t\) increases.
This is why animation and direction arrows are useful: they show how the point moves along the curve.
3. Tangent vector
The tangent vector is found by differentiating both coordinate functions:
\[
\mathbf r'(t)=\langle x'(t),y'(t)\rangle.
\]
This vector points in the instantaneous direction of motion.
4. Slope of a parametric curve
If \(x'(t)\ne0\), then the slope of the tangent line is
\[
\frac{dy}{dx}
=
\frac{dy/dt}{dx/dt}.
\]
This formula comes from the chain rule:
\[
\frac{dy}{dt}
=
\frac{dy}{dx}\frac{dx}{dt}.
\]
5. Eliminating the parameter
To convert from parametric form to Cartesian form, solve one equation for \(t\) and substitute into
the other equation.
For example, if
\[
x=t,\qquad y=t^2,
\]
then \(t=x\), so
\[
y=x^2.
\]
6. Example: \(x=t^2,\ y=t^3\)
This is a common example because the parameter cannot be removed by a single-valued formula for all
\(t\).
\[
x=t^2,\qquad y=t^3.
\]
From \(x=t^2\),
\[
t=\pm\sqrt{x}.
\]
Squaring \(y=t^3\) gives
\[
y^2=t^6.
\]
Since \(t^2=x\), we get
\[
y^2=x^3,\qquad x\ge0.
\]
If \(t\ge0\), then \(y=x^{3/2}\). If \(t\le0\), then \(y=-x^{3/2}\).
7. Circle and ellipse parametrizations
A circle of radius \(R\) centered at the origin can be written as
\[
x=R\cos t,\qquad y=R\sin t.
\]
Using \(\cos^2t+\sin^2t=1\), we get
\[
x^2+y^2=R^2.
\]
An ellipse can be written as
\[
x=a\cos t,\qquad y=b\sin t.
\]
Its Cartesian equation is
\[
\frac{x^2}{a^2}+\frac{y^2}{b^2}=1.
\]
8. Arc length of a parametric curve
For a smooth parametric curve from \(t=a\) to \(t=b\), the arc length is
\[
L=
\int_a^b
\sqrt{\left(\frac{dx}{dt}\right)^2+
\left(\frac{dy}{dt}\right)^2}\,dt.
\]
The calculator estimates this length numerically from sampled points on the curve.
9. Common parametric curves
10. Why some curves cannot be converted simply
Some parametric curves involve trigonometric combinations, repeated tracing, self-intersections, or
inverse functions. In these cases, a simple Cartesian equation may not exist or may not describe the
orientation and tracing behavior.
For such curves, the parametric form is often the best description.
11. Common mistakes
- Ignoring the \(t\)-interval: the same formulas can trace different parts of a curve on different intervals.
- Losing branch information: \(x=t^2\) gives \(t=\pm\sqrt{x}\), so the sign of \(t\) matters.
- Forgetting orientation: Cartesian form does not show the direction of motion.
- Using \(\frac{dy}{dx}\) incorrectly: for parametric curves, use \(\frac{dy/dt}{dx/dt}\).
- Forgetting equal-axis scale: without equal scaling, circles may appear as ellipses on the screen.
