Circles, Ellipses, and Lissajous Curves
Author: Jamie Mulholland

Here are some simple examples of parametric curves.  All curves have the form ?? ...

Example

x(t) = cos(t)

y(t) = sin(t)

c_1(t) = ( cos(t) , sin(t) )

Circle

Circle (change of speed)

Ellipse

Lissajous

b1 = 1
a1 = 1
T = 0.6 (slider from 0 to 6.28, step 0.1)
b4 = 5
g1(x) = sin(b1 * x)
a4 = 3
f1(x) = cos(a1 * x)
tab = 1 (slider from 1 to 4, step 1)
texty4: "y(t) = sin(" + b4 + "t)"
textx4: "x(t) = cos(" + a4 + "t)"
f4(x) = cos(a4 * x)
g4(x) = sin(b4 * x)
c4 = CurveCartesian(f4(t), g4(t), t, 0.0, T)
a2 = 2
f2(x) = cos(a2 * x)
g2(x) = sin(a2 * x)
a3 = 3
b3 = 2
f3(x) = a3 * cos(x)
g3(x) = b3 * sin(x)
c1 = CurveCartesian(f1(t), g1(t), t, 0.0, T)
c2 = CurveCartesian(f2(t), g2(t), t, 0.0, T)
c3 = CurveCartesian(f3(t), g3(t), t, 0.0, T)
P1: c1(T)
P2: c2(T)
P3: c3(T)
P4: c4(T)
textx2: "x(t)=" + "cos(" + a2 + "t)"
textC2: "c_2(t) = ( cos(" + a2 + "t) , sin(" + a2 + "t) )"
texty2: "y(t)=" + "sin(" + a2 + "t)"
textx3: "x(t) = " + a3 + "cos(t)"
texty3: "y(t) = " + b3 + "sin(t)"
textC3: "c_3(t) = ( " + a3 + "cos(t), " + b3 + "sin(t) )"
textC4: "c_4(t) = (cos(" + a4 + "t) , sin(" + b4 + "t))"