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
subject to the boundary condition
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( b^{2} + t^{2} ) y = b + b r / sqrt( b^{2} + t^{2} ) |
Witch of Agnesi
x = 2 * r * tan t
y = 2 * r cos( t ) ^{2} |