Parametric Equation Calculator & Plotter – Lines & Curves
A parametric curve in the plane is described by two functions of a parameter \(t\):
\[
x=x(t),\qquad y=y(t).
\]
As \(t\) varies over an interval \([t_{\min},t_{\max}]\), the point \(\big(x(t),y(t)\big)\) traces a curve.
This tool plots the curve, shows the direction of increasing \(t\), and computes tangents, curvature, and more.
1) From parametric to Cartesian (when possible)
Sometimes you can eliminate \(t\) and obtain an equation in \(x\) and \(y\).
The easiest case is when both components are linear in \(t\):
\[
x(t)=a+bt,\qquad y(t)=c+dt.
\]
If \(b\neq 0\), then \(t=\dfrac{x-a}{b}\) and substituting into \(y\) gives a line:
\[
y=c+d\left(\frac{x-a}{b}\right)=\left(\frac{d}{b}\right)x+\left(c-\frac{da}{b}\right).
\]
If \(b=0\), then \(x(t)=a\) is a vertical line \(x=a\).
2) Tangent line at a parameter value \(t_0\)
Differentiate both components with respect to \(t\):
\[
\frac{dx}{dt},\qquad \frac{dy}{dt}.
\]
If \(\dfrac{dx}{dt}(t_0)\neq 0\), the slope of the tangent line is:
\[
\left.\frac{dy}{dx}\right|_{t_0}=\frac{\left.\dfrac{dy}{dt}\right|_{t_0}}{\left.\dfrac{dx}{dt}\right|_{t_0}}.
\]
The tangent line through \(P_0=\big(x(t_0),y(t_0)\big)\) can be written in point-slope form:
\[
y-y(t_0)=\left(\left.\frac{dy}{dx}\right|_{t_0}\right)\big(x-x(t_0)\big).
\]
If \(\dfrac{dx}{dt}(t_0)=0\) (and \(\dfrac{dy}{dt}(t_0)\neq 0\)), the tangent is vertical:
\[
x=x(t_0).
\]
3) Normal line at \(t_0\)
The normal line is perpendicular to the tangent.
If the tangent slope is \(m\neq 0\), the normal slope is:
\[
m_\perp=-\frac{1}{m}.
\]
If the tangent is vertical, the normal is horizontal \(y=y(t_0)\).
If the tangent is horizontal, the normal is vertical \(x=x(t_0)\).
4) Speed and (approximate) arc length
The derivative vector is:
\[
\mathbf{r}'(t)=\langle x'(t),y'(t)\rangle.
\]
The speed is its magnitude:
\[
|\mathbf{r}'(t)|=\sqrt{\big(x'(t)\big)^2+\big(y'(t)\big)^2}.
\]
The arc length of a smooth curve is:
\[
L=\int_{t_{\min}}^{t_{\max}} \sqrt{\big(x'(t)\big)^2+\big(y'(t)\big)^2}\,dt.
\]
This calculator approximates \(L\) by sampling the curve and summing straight-line distances between consecutive points (polyline length).
5) Curvature \(\kappa(t)\)
Curvature measures how strongly a curve bends. For a planar parametric curve \(x(t),y(t)\),
curvature is:
\[
\kappa(t)=\frac{|x'(t)y''(t)-y'(t)x''(t)|}{\left(\left(x'(t)\right)^2+\left(y'(t)\right)^2\right)^{3/2}}.
\]
The numerator \(x'y''-y'x''\) is related to the “turning direction” of the curve.
If \(|\mathbf{r}'(t_0)|=0\) (the curve is not moving at \(t_0\)), curvature is not defined in this formula,
so the tool reports curvature as unavailable at that \(t_0\).
6) Osculating circle (circle of curvature)
Near a point \(P_0=\mathbf{r}(t_0)\), a smooth curve is well-approximated by a circle called the
osculating circle. Its radius is the radius of curvature:
\[
\rho(t_0)=\frac{1}{\kappa(t_0)}.
\]
The center of this circle lies along the normal direction from the point \(P_0\).
In vector form, one can write:
\[
C = P_0 + \rho\,\hat{\mathbf{n}},
\]
where \(\hat{\mathbf{n}}\) is a unit normal pointing toward the center of curvature.
When \(\kappa(t_0)\) is very small, \(\rho(t_0)\) is very large and the curve is locally almost a straight line.
7) Worked example (line)
Example:
\[
x=3+2t,\qquad y=1-t.
\]
Since both are linear in \(t\), eliminate \(t\):
\[
t=\frac{x-3}{2}
\quad\Rightarrow\quad
y=1-\frac{x-3}{2}
=-\frac{1}{2}x+\frac{5}{2}.
\]
A line has \(\kappa=0\) (no bending), so the osculating circle radius is effectively infinite.
Graph note: The plot uses square units (equal scaling) so angles and shapes are not distorted,
and tick labels are drawn near their axes for readability while panning/zooming.
