Parametric Curves

Explore parametric curves: see the tracing point and tangent vector as t varies.

Worked example — Parametric Curves

t	3.1416
(x, y)	(-1.000, 0.000)
Speed \|dr/dt\|	2.000
dx/dt, dy/dt	(0.000, 2.000)

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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The same picture can be drawn at many different speeds 🖖

A parametric curve carries more information than the shape it leaves behind. Compare (cos t, sin t) with (cos 2t, sin 2t): both trace the identical unit circle, and no drawing could tell them apart — yet the second one finishes in half the time. Infinitely many parametrisations produce exactly the same picture. That is why arc length is a property of the trace, while speed and direction belong only to the schedule that produced it, and why physics reaches for parameters rather than graphs: a planet’s orbit is a shape, but its motion along that shape is the part that matters.

Two rules steer one pen 🖖

Instead of asking what y is for each x, a parametric curve runs a clock t and lets two separate rules steer a pen: x(t) sets how far right it moves, y(t) how far up. The trace you see is only half the story — the moving dot and its tangent arrow carry the rest: direction and speed. Since the arrow's length is the speed, the same shape can be drawn lazily or in a rush and look identical, while the motion behind it is completely different.

The clock that ignores its swing 🖖

Turn the cycloid x = t − sin(t), y = 1 − cos(t) upside down and it becomes the tautochrone: a frictionless bead reaches the lowest point in exactly the same time no matter where you release it. Christiaan Huygens used this in 1673, curving pendulum clocks with cycloidal cheeks so the bob kept perfect time however wide it swung. One parametric curve handed horology its first amplitude-proof pendulum.

Problem solved in full

Position and speed of the Lissajous curve at t = π 6 steps
The Lissajous curve x = cos 3t, y = sin 2t at t = π. Find the position and speed the panel prints, then find the one thing it does not: how sharply the curve is turning.
1. Two independent functions of the same parameter. Differentiating them is the whole of the velocity, and the chain rule supplies the factors 3 and 2.
  
  $x (t) = cos 3 t, y (t) = sin 2 t; \overset{x}{˙} = - 3 sin 3 t, \overset{y}{˙} = 2 cos 2 t$
2. Evaluate the position first. Both trigonometric functions land on exact values, which is why π is worth choosing.
  
  $t = π : x = cos 3 π = - 1, y = sin 2 π = 0$
3. Now the velocity. One component vanishes and the other is at its maximum — the motion at this instant is purely vertical.
  
  $\overset{x}{˙} = - 3 sin 3 π = 0, \overset{y}{˙} = 2 cos 2 π = 2 ⟹ ∣ \dot{r} ∣ = 2.000$
4. That vanishing is not an accident of arithmetic: x is at a turning point, and the curve touches its leftmost extreme here.
  
  $\overset{x}{˙} = 0 with \overset{y}{˙} \neq = 0 : the curve is moving straight up, and x is at a turning point$
5. Curvature needs one more derivative of each. Both are as easy as the first, which is the payoff of the parametric form.
  
  $\overset{x}{¨} = - 9 cos 3 t = 9, \overset{y}{¨} = - 4 sin 2 t = 0$
6. Assemble the standard formula. The numerator is a cross product and the denominator is the cube of the speed, which is why fast points look flat unless they really are bending.
  
  $κ = \frac{∣ x ˙ y ¨ - y ˙ x ¨ ∣}{( x ˙ ^{2} + y ˙ ^{2} ) ^{3/2}} = \frac{∣0 - 18∣}{8} = 2.25 ⟹ ρ = \frac{1}{κ} = 0.444$
Answer

Radius of curvature 0.444 — the curve is turning inside a circle of well under half a unit. Speed and curvature are independent, and this point makes the distinction visible: the speed is 2, the largest either component reaches, while the path is bending as tightly as anywhere on the figure. A car at that point is going fast and cornering hard, which is exactly the combination that produces lateral load. Nothing in the position or velocity readout hints at it, because curvature needs the second derivative and the panel stops at the first. That is also why parametric form is worth the trouble: as a relation between x and y this curve is a sixth-degree mess, and as two functions of t every derivative is one line.

$(x, y) = (- 1, 0), ∣ \dot{r} ∣ = 2.000, κ = 2.25, ρ = 0.444$

References (1)

Insight block 3 — the tautochrone, and the clock built on it: C. Huygens, Horologium Oscillatorium sive de motu pendulorum. F. Muguet, Paris, 1673 — the cycloidal cheeks that make a pendulum's period independent of its swing.

By the 42Math team · Reviewed against published sources. Corrections welcome.