Representing 2D objects in 4D What is the general quation of a 2D object, e.g. a surface in 4D?
I read, that while a line is representable in 2D using the general equation of degree one ($ax + by + c = 0$), in 3D, one needs a system of equations.
In 3D, the general equation of one degree is a plane ($ax + by + cz + d = 0$).
The general equation of second degrees are curves in 2D, and surfaces of revolution in 3D. 
I wonder, how does one represent a 2D object in 4D? Do we need a set of equations? How do they look like? 
PS : is there a list of objects which are solutions of general equations of first, second and third degree in 4D?
 A: In general, every time you introduce an equation, you go down 1 dimension. So in 4D space, to represent a 2D object you need 2 equations.
For example, if you take two linear equations: $$\begin{cases} Ax+By+Cz+Dw=E\\Fx+Gy+Hz+Iw=J \end{cases}$$then you generally get a 2D plane. (I say generally, because it could also be a 3D space, for example of both equations are identical, or absolutely nothing, for example if one of the equations is $0=1$... But for "generic" choices of coefficients you'll get a plane.)
However, this is not the only case of a 2D surface. Another example might be the 2D sphere embedded in 4D space, one possible instance of which is the following:$$\begin{cases} x^2 + y^2 + z^2 + w^2 = 1 \\ w = 0 \end{cases}$$
This is a sphere, which is obtained by 2 equations, one of which is 2nd degree and another is linear. Another thing you could look at is the following $$\begin{cases} x^2 + y^2 = 1 \\ z^2 + w^2 = 1 \end{cases}$$which algebraically is two "independent" circles. This surface is known as the Clifford torus and is topologically a 2-dimensional torus.
Those examples above are called "algebraic surfaces", because they are obtained as the solution set of algebraic (polynomial) equations. The first one (a plane) is a degree 1 surface, the others are degree 2. However, not all surfaces are algebraic. For example, consider the following: $$\begin{cases}\sin (\pi x) = 0 \\ \sin (\pi y) = 0\end{cases}$$
This surface is a union of infinitely many planes, and contains all points $(x,y,z,w)$ for which $x, y$ are integers.
The most general case is obtained by the definition of a 2-dimensional manifold inside 4-dimensional space. For some manifolds, it is hard or not fruitful to give equations that describe them.
A: For a line in 4D, it needs to be represented by parametric equations, like a 3D space curve does. A 4D space curve would have 4 parametric equations:
$\frac {x-t}{a}=\frac {y-t}{b}=\frac {z-t}{c}=\frac {w-t}{d} $
Where $<a, b, c, d> $ is a 4D vector
