Parametric Curve Plotter Math

t: ---, (x, y): (---, ---)
0.00 6.28
50%

What is a Parametric Curve Plotter?

Quick Answer: A tool that draws a curve by evaluating two functions x(t) and y(t) as a parameter t varies. It visualizes motion, Lissajous figures, cycloids, and any path impossible to describe as y=f(x). Essential for calculus, physics, and JEE Advanced.

Theory Behind Parametric Curves

Parametric equations express the coordinates of a curve as functions of an independent parameter t. For example, a circle radius 5 is x(t)=5cos(t), y(t)=5sin(t) with t in [0, 2π]. This representation allows loops, self‑intersections, and motion paths. The plotter uses Math.js to compile the expressions, then samples points at many t values. A moving dot shows the position at the current t, and animation sweeps t from start to end, leaving a colored trail.

Formulas: Point (x(t), y(t)) computed for each t. View transform: canvasX = x * scale + offsetX, canvasY = offsetY - y * scale.

Step-by-Step Examples

Example 1: Ellipse (Lissajous)

  1. Set x(t) = 5 * cos(t), y(t) = 3 * sin(t).
  2. t range 0 to 6.28 (2π). Move trace slider to see point position.
  3. Click Animate to watch the point travel around the ellipse.

Example 2: Cycloid

  1. Enter x(t) = t - sin(t), y(t) = 1 - cos(t).
  2. Set t from 0 to 12.56 (4π) to see multiple arches.
  3. Observe the characteristic "rolling wheel" shape.

Example 3: Spiral

  1. Use x(t) = t * cos(t), y(t) = t * sin(t).
  2. t range 0 to 20. The curve winds outward.
  3. Zoom out if needed (scroll wheel).

Frequently Asked Questions

What functions can I use in x(t) and y(t)?

Any valid math expression using t, numbers, arithmetic, sin, cos, tan, log, sqrt, exp, abs, etc. Examples: sin(2*t), t^2.

How do I see the motion direction?

Click Animate; a moving dot shows current position and a small arrow indicates direction. The trace gradually builds.

Can I plot a circle?

Yes, x(t)=r*cos(t), y(t)=r*sin(t) with t from 0 to 2π (6.28). Use r=5 for a circle of radius 5.

How do I zoom and pan?

Use the mouse wheel to zoom, and drag while holding Shift to pan. Reset View button returns to default window.

Why does the curve disappear at some t values?

If the expression returns an undefined or infinite value (e.g., tan(t) at π/2), that point is skipped and the curve breaks momentarily.