This first demonstration starts with the graph of the function in polar coordinates f(r,t) = r²cos(2t), and it shows the r- and t-slice curves at the orange hot spot P. Moving around the hotspot P=(r₀,t₀) in the Domain window, observe the two fundamentally different slice curves in the Function Graph: f(r,t) window. The green slice curve is the one determined by f(r₀,t) and the red slice curve is determined by f(r,t₀).

Note that the green slice curve is the intersection of the green cylinder centered around the z-axis, with radius r₀ and the function graph of f, which is depicted in blue. You can view the slice curve all flattened out in the r = r0 window. Furthermore, the red slice f(r,t₀) is a ray emanating from the z-axis, and it is the intersection of the red half-plane t = t0 with the graph of the function.