Mathematics.

curves

Parametric Curves

Analytic Geometry30 minDifficulty5 out of 10

You should know: parametric equations

Overview

A parametric curve is the actual geometric trace — the set of points (x(t), y(t)) in the plane — produced by a parametric equation, viewed as an object of study in its own right rather than as a computational tool. Where 'parametric equations' emphasizes the algebraic machinery (differentiating, eliminating the parameter, computing arc length), 'parametric curves' emphasizes the resulting shapes: circles and ellipses traced at varying speeds, cycloids traced by a rolling wheel, Lissajous figures, and spirals. The same curve can arise from many different parametrizations, and classic named curves — the cycloid, the astroid, the cardioid — are most naturally defined and studied only in parametric form, since they generally have no simple equation of the form y = f(x).

Intuition

Think of a parametric curve as the ink trail left by a pen whose x and y position are each controlled independently by a clock hand t. Because x(t) and y(t) don't have to be inverses of each other, the pen is free to loop, retrace, cross itself, or pause — freedoms a plain function y=f(x) never has, since it can only assign one height to each horizontal position. Classic curves like the cycloid (traced by a point on a rolling wheel) or an ellipse traced at non-constant speed are natural in this language precisely because their motion, not their shape as a graph, is the defining feature.

Interactive Graph

Trace named parametric curves (cycloid, astroid, ellipse) as t increases

Loading visualization…

Formal Definition

Definition

A parametric curve in the plane is the image of a continuous map from a parameter interval into ℝ²:

γ(t)=(x(t),y(t)),t∈[α,β]\gamma(t) = (x(t), y(t)), \qquad t \in [\alpha, \beta]

The curve is the set of points {γ(t) : t ∈ [α,β]}, traced in the direction of increasing t

x=acos⁡t, y=bsin⁡t,t∈[0,2π)x = a\cos t,\ y = b\sin t,\quad t\in[0,2\pi)
Ellipse (semi-axes a, b)
x=r(t−sin⁡t), y=r(1−cos⁡t)x = r(t - \sin t),\ y = r(1-\cos t)

Path traced by a point on a circle of radius r rolling along the x-axis

Cycloid
x=acos⁡3t, y=asin⁡3tx = a\cos^3 t,\ y = a\sin^3 t
Astroid

Properties

Multiple parametrizations

A single curve can be traced by infinitely many maps γ(t); e.g. (cos⁡t,sin⁡t) and (cos⁡3t,sin⁡3t) both trace the unit circle\text{A single curve can be traced by infinitely many maps } \gamma(t); \text{ e.g. } (\cos t,\sin t) \text{ and } (\cos 3t,\sin 3t) \text{ both trace the unit circle}

Self-intersection

A parametric curve may cross itself: γ(t1)=γ(t2) for t1≠t2 is possible, unlike the graph of a function\text{A parametric curve may cross itself: } \gamma(t_1)=\gamma(t_2) \text{ for } t_1\neq t_2 \text{ is possible, unlike the graph of a function}

Orientation

The direction of increasing t gives the curve a traversal direction; reversing t (e.g. t→−t) reverses orientation\text{The direction of increasing } t \text{ gives the curve a traversal direction; reversing } t \text{ (e.g. } t \to -t\text{) reverses orientation}

Speed

∣γ′(t)∣=x′(t)2+y′(t)2 is the instantaneous speed of the trace, generally not constant\left|\gamma'(t)\right| = \sqrt{x'(t)^2+y'(t)^2} \text{ is the instantaneous speed of the trace, generally not constant}

Applications

Bézier and spline curves used throughout vector graphics, fonts, and animation are parametric curves built from control points.

Worked Examples

  1. Use cos²t + sin²t = 1 to eliminate t.

    (x3)2+(y3)2=cos⁡2t+sin⁡2t=1  ⟹  x2+y2=9\left(\frac{x}{3}\right)^2 + \left(\frac{y}{3}\right)^2 = \cos^2 t + \sin^2 t = 1 \implies x^2+y^2=9

  2. At t=0, the point is (3,0); as t increases slightly, y becomes positive while x stays near 3, so motion is counterclockwise.

    γ(0)=(3,0),γ(π/2)=(0,3)\gamma(0)=(3,0),\quad \gamma(\pi/2)=(0,3)

Answer: The circle x² + y² = 9 of radius 3, traced counterclockwise starting at (3, 0).

Practice Problems

Difficulty 4/10

What curve is traced by x = 2 + 5cos t, y = -1 + 5sin t, t ∈ [0, 2π)?

Difficulty 5/10

Find the point traced by the astroid x = 4cos³t, y = 4sin³t at t = π/2, and verify it lies on the astroid's boundary curve x^(2/3)+y^(2/3)=4^(2/3).

Difficulty 6/10

An oscilloscope displays x(t) = sin(2t), y(t) = sin(3t), producing a Lissajous figure. Find the point at t = π/2 and state whether the curve is periodic, and if so with what period.

Common Mistakes

Common Mistake

Assuming that eliminating the parameter t always yields the full picture of a parametric curve, including its orientation and speed.

Eliminating t recovers the geometric shape (e.g. x²+y²=9) but discards information about direction of travel, starting point, and speed — all of which are only visible in the original (x(t), y(t)) form.

Common Mistake

Treating a self-intersecting parametric curve as if it were the graph of a function, and expecting the vertical line test to apply.

The vertical line test applies to graphs y=f(x), not to general parametric curves, which are perfectly entitled to cross themselves or trace loops.

Quiz

Which feature can a parametric curve exhibit that the graph of a function y = f(x) never can?
The cycloid x = r(t − sin t), y = r(1 − cos t) reaches its maximum height y = 2r when:
The same geometric curve (e.g. the unit circle) can be produced by:

Summary

  • A parametric curve is the geometric trace {(x(t), y(t))} of a parametric equation — the shape itself, not just the algebra used to study it.
  • Named curves like the cycloid, astroid, and Lissajous figures are naturally defined parametrically and generally lack a simple y=f(x) form.
  • The same curve can come from infinitely many parametrizations, differing in speed, direction, or starting point.
  • Unlike function graphs, parametric curves can self-intersect, loop, or retrace, because x(t) and y(t) need not be inverses of one another.

References

  1. WebsiteWikipedia — Parametric equation
  2. WebsiteWikipedia — Cycloid