Parametric and polar curves

A point whose coordinates are written in terms of t is a parametric curve, drawn over Desmos’ own range of t - 0 to 1 - unless a domain says how far it runs. An equation in r and theta is polar.

examples/11-parametric-and-polar.axis

config {
    showGrid: true
}

"Parametric curves"

folder "Parametric" {
    "The unit circle, once round: t from 0 to tau."
    (cos(t), sin(t)) @ color: RED, lineWidth: 3, domain: 0..tau
    "An ellipse: different radii per axis."
    (3cos(t), 2sin(t)) @ color: BLUE, lineWidth: 3, domain: 0..tau
    "An Archimedean spiral - its domain runs t through three whole turns."
    (0.3t * cos(t), 0.3t * sin(t)) @ color: GREEN, lineWidth: 2, domain: 0..6pi
    "A Lissajous figure."
    (2sin(3t), 2sin(4t)) @ color: PURPLE, lineWidth: 2, domain: 0..tau
}

folder "Polar" {
    "Turn on polarMode in config for polar grid lines."
    r = 3 @ color: RED, lineWidth: 3
    "A four-petal rose."
    r = 3cos(2theta) @ color: BLUE, lineWidth: 3
    "A cardioid."
    r = 1 + cos(theta) @ color: GREEN, lineWidth: 3
    "A logarithmic spiral."
    r = exp(0.2theta) @ color: ORANGE, lineWidth: 2
}

folder "A point travelling a curve" {
    s = 0 @ slider: 0..2pi step 0.01, playing
    (3cos(s), 2sin(s)) @ color: RED, pointSize: 18
    "Its position vector."
    V = [(0, 0), (3cos(s), 2sin(s))] @ color: RED, lines, points: false, lineStyle: DASHED
}

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