x = sin(π t (1-r)) / (cos(π t (1-r)) + cos(π r))
y = sin(π r) / (cos(π t (1-r)) + cos(π r))

The interesting features are better seen when you increase xmax and ymax to about 10 and set r step to 0.02. The curves bunch up around r=0.5

(For r=0, the curve is on the x-axis, for r -> 1 the curve vanishes to future infinity -- r is a compactified time coordinate, t is a compactified space coordinate)

Submitted by Tilman Vogel

The characteristics of the partial differential equation

uy + u ux = 0

subject to the boundary condition

u( x, 0 ) = cos π x

are given by

x = t cos(π r) + r
y = t

(Example 1.6 in P. G. Drazin & R. S. Johnson Solitons: an introduction)

Cycloids

x = r t - r sin(t)
y = r - r cos(t)

Epiycloids

x = (1 + r) cos(t) - r cos( (1+r) * t)
y = (1 + r) sin(t) - r sin( (1+r) * t)

Archimedian Spiral

x = t r cos(t)
x = t r sin(t)

Conchoids

b = 0.5
x = t + r t / sqrt( b2 + t2 )
y = b + b r / sqrt( b2 + t2 )

Witch of Agnesi

x = 2 * r * tan t
y = 2 * r cos( t ) 2