How can I visualize the multivariable function z = f(x(t), y(t)) in 4 dimensions?

I am trying to visualize$$\ z = f(x(t), y(t))$$, and my model is that$$\ x, y, z$$ depend on$$\ t$$. The only way that I can visualize this is as a point moving in 3D space. However, wouldn't that be $$\ \vec p = \vec f(t)$$ ?

Is there any way to visualize the relationship between$$\ z$$ and$$\ t$$ as a surface that is changing in time, or is this wrong?

• You're not missing something. Both of these representations can work. – David G. Stork Apr 1 at 17:36
• Sorry David I realized that I had a more specific question instead of whether or not I was missing something. It's just that whenever I see f(x, y) I think surface, so I updated my question. – Benjamin Smus Apr 1 at 17:55

No, you should draw one surface $$z=f(x,y)$$ and think of parametrized curves on that surface. That is, given the parametrized curve $$(x(t),y(t))$$ in the $$xy$$-plane, you look up to the surface and watch the particle moving along the surface.
You could indeed end up in $$4$$-space by including the $$t$$-axis, i.e., plotting $$(t,x(t),y(t))$$ and then moving into one more dimension with the plot $$(t,x(t),y(t),f(x(t),y(t)))$$ — still a curve on the surface $$z=f(x,y)$$.