Parametric Equations Calculator

A parametric curve describes both coordinates through a third variable, the parameter $t$:

$x = f(t), \qquad y = g(t), \qquad t \in [a, b]$

As $t$ runs over its interval the point $(f(t), g(t))$ traces a path. Unlike $y = F(x)$, this can double back, cross itself or close into a loop — a circle needs two half-functions in rectangular form but only one parametrisation:

$x = r\cos t, \quad y = r\sin t, \quad t \in [0, 2\pi]$

A line through $(x_0, y_0)$ with direction $\langle a, b \rangle$ is $x = x_0 + at$, $y = y_0 + bt$, the component form of the vector equation $\mathbf{r}(t) = \mathbf{r}_0 + t\mathbf{v}$.

The parameter also carries orientation and timing, which rectangular form throws away.

First derivative

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}, \qquad \frac{dx}{dt} \neq 0$

This is the chain rule rearranged. Where $dx/dt = 0$ but $dy/dt \neq 0$ the tangent is vertical; where both vanish the point may be a cusp and the formula says nothing.

Second derivative

$\frac{d^2y}{dx^2} = \frac{\dfrac{d}{dt}\!\left(\dfrac{dy}{dx}\right)}{dx/dt}$

Differentiate the first derivative with respect to $t$, then divide by $dx/dt$ again. It is emphatically not $\frac{d^2y/dt^2}{d^2x/dt^2}$.

Speed and arc length

$\text{speed} = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}, \qquad L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2}\,dt$

The arc-length integral counts every retraced piece, so it equals the geometric length only when the curve is traversed once.

Eliminating the parameter

Solve one equation for $t$ and substitute, or use an identity such as $\cos^2 t + \sin^2 t = 1$ for trigonometric pairs. Always carry the domain restriction across.

• Dividing the second derivatives: $\frac{d^2y}{dx^2}$ requires differentiating $dy/dx$ with respect to $t$ and dividing by $dx/dt$ once more.
• Losing the domain when eliminating $t$: $x = t^2$, $y = t$ gives $x = y^2$, but only the branch with $x \ge 0$ — and every restriction on $t$ transfers.
• Ignoring $dx/dt = 0$: those parameter values are exactly where vertical tangents and cusps hide.
• Confusing speed with $dy/dx$: $dy/dx$ is the tangent slope, a shape property; speed measures how fast the point moves and depends on the parametrisation.
• Assuming the parametrisation is unique: the same curve has infinitely many, differing in orientation and rate.