Are circles and lines in two-space one-dimensional? Circles and lines are normally regarded as one-dimensional objects. However, when embedded in two-space, they require two coordinates $(x,y)$ to specify a point within them. Are they still considered one-dimensional, and why? This is somewhat related to a separate but very similar question of why a sphere embedded in three-space is (to my knowledge) considered two-dimensional, even though it has a height, a width and a length. To laymen, spheres are generally considered 3D objects. This is where a potential misunderstanding may lie. Therefore, I am also curious of what difference, if any, there is between a mathematician's conception of what a "dimension" is versus a layman's? 

The idea is that you can call objects "n-dimensional" if you can parameterise them using n distinct coordinates.
For example, the line is the image of the function $f(t)=(t,at+b)$, and the circle is the image of the function $f(t)=(a+R\cos(t),b+R\sin(t))$. The sphere is two-dimensional because it can be parameterised by $f(\theta,\varphi)=(R\sin\varphi\cos\theta,R\sin\varphi\sin\theta,R\cos\varphi)$.
