Point in four dimensions To describe a point in $3$D:
Three parameters $ r,\phi, \theta $ are used in spherical coordinate system.  Taking them pairwise,two as $(r, \theta)$ in azimuth plane and two $(r,\phi )$ in meridian plane.
Similarly to describe a point in $4D$ 4-space, can we describe 4-tuple parameters with known $3$D geometrical meanings for some of six intersections of $3$-space ?
EDIT 1:
I like to see a description where at least 3 of 6 possibilities is convincingly demonstrated.
 A: Here are several coordinate systems which derive logically from the coordinate systems in lower dimension.
I'll give the coordinate systems labels which give a "grouping" of dimensions (I hope it will become clear what I mean; it's hard to explain before I show the coordinate systems).
(1,1,1,1) — Cartesian coordinates
These are the most obvious choice of coordinates. Each coordinate gives the position on one of the four orthogonal axes. These coordinates are typically labelled $(x,y,z,w)$, and so will I.
Setting one coordinate constant describes a hyperplane (3-dimensional affine subspace) orthogonal to the corresponding axis; for example, setting $w=0$ gives the $x$-$y$-$z$ space.
Setting two of the coordinates constant describes a plane orthogonal to both corresponding axes; for example, setting $z=w=0$ gives the $x$-$y$ plane.
Setting three of the coordinates constant describes a line parallel to the remining coordinate's axis. For example, setting $y=z=w=0$ gives the $x$ axis.
(2,1,1) — A generalization of cylinder coordinates
This is a natural extension of the cylinder coordinates in $3D$: We have one coordinate, which I'll call $r_{xy}$, that gives the distance from the $zw$ plane (that is, the radius of the projection into the $x$-$y$ plane), an angle $\phi_{xy}$ in the $x$-$y$ plane, and the remaining coordinates $z$ and $t$. They are related to the Cartesian coordinates by
$$\begin{align}
 x &= r_{xy} \cos\phi_{xy}\\
 y &= r_{xy} \sin\phi_{xy}\\
 z &= z\\
 w &= w
\end{align}$$
Setting $r_{xy}$ constant gives a sort of 4D cylinder around the $z$-$w$ plane, in the following sense: In 3D, a cylinder can be thought of as adding an orthogonal line to each point of a circle; in cylinder coordinates the circle is in the $x$-$y$ plane, and the lines are parallel to the $z$ axis. Similarly, this 4D cylinder is obtained by adding an orthogonal plane parallel to the $z$-$w$ plane to each point of the circle in the $x$-$y$ plane. Note that these planes are "completely orthogonal" to the $x$-$y$ plane in the sense that every direction on those planes is orthogonal to every direction in the $x$-$y$ plane. This is not possible in three dimensions, since there any two planes intersect in a line, that is, have a common direction. Note that in the case $r_{xy}=0$, that cylinder degenerated, and we get exactly the $z$-$w$ plane.
Setting $\phi_{xy}$ constant gives a half-hyperplane (three-dimensional half-space) starting at the $z$-$w$ plane and going away from it in the direction $\phi_{xy}$. That is again analogous to the three-dimensional case, where you have a half-plane starting at the $z$ axis and going away from it in the direction $\phi_{xy}$.
Setting $z$ or $w$ constant has, of course, the same effect as with Cartesian coordinates.
Setting both $r_{xy}$ and $\phi_{xy}$ constant has the same effect as setting $x$ and $y$ constant, since $r_{xy}$ and $\phi_xy}$ uniquely determine $x$ and $y$. Of course, setting both $z$ and $w$ constant also has the same effect as in Cartesian coordinates.
Since setting $w$ constant selects a $3$-dimensional hyperplane, in which the other coordinates just are cylinder coordinates, setting one of the other coordinates constant has the effect to select the corresponding subset of that 3D hyperspace. So both $w$ and $r$ constant select an ordinary 2D cylinder, and $w$ and $\phi$ constant selects a half-plane. Of course, having $z$ instead of $w$ constant doesn't qualitatively change anything.
Thus also the effect of setting three coordinates constant is obvious (I'm omitting combinations which are qualitatively equivalent to those given):


*

*$r_{xy}$, $\phi_{xy}$, $z$ constant gives a straight line parallel to $w$.

*$r_{xy}$, $z$, $w$ constant gives a circle with center on the $z$-$w$ plane.

*$\phi_{xy}$, $z$, $w$ constant gives a ray originating on the $z$-$w$ plane.


(2,2) — "Toric coordinates"
This is a genuinely 4D coordinate system which has no obvious 3D equivalent. Basically, we introduce polar coordinates both on the $x$-$y$ and the $z$-$w$ plane, giving us the four coordinates $(r_{xy},\phi_{xy},r_{zw},\phi_{zw})$ (now it should be clear why I introduced the index before):
$$\begin{align}
 x &= r_{xy} \cos\phi_{xy}\\
 y &= r_{xy} \sin\phi_{xy}\\
 z &= r_{zw} \sin\phi_{zw}\\
 w &= r_{zw} \cos\phi_{zw}
\end{align}$$
Clearly there's a symmetry between coordinates $(x,y)$ and $(z,w)$, which allows us to cut down the essentially different cases we need to consider. Moreover, the cases where only one of the coordinates is set constant is exactly equivalent to the case of setting $r_{xy}$ or $\phi_{xy}$ constant in the case (2,1,1). The same is true for setting both coordinates of one of the sets constant.
Setting $r_{xy}$ and $r_{zw}$ constant gives a "flat" torus, that is, a torus whose inner curvature vanishes (yet another thing that only exists in four dimensions). Setting $\phi_{xy}$ and $\phi_{wz}$ constant gives a quadrant of a plane selected by those two angles. Setting $r_{xy}$ and $\phi_{zw}$ constant gives a half-infinite cylinder of radious $r_{xy}$ going into a direction orthogonal to the $x$-$y$ plane determined by $\phi_{zw}$.
(3,1) — Another generalization of cylinder coordinates.
Here we introduce spherical coordinates in the $x$-$y$-$z$ hyperplane (3D space) and add the $w$ coordinate:
$$\begin{align}
 x &= r_{xyz} \cos\theta \cos\phi\\
 y &= r_{xyz} \cos\theta \sin\phi\\
 z &= r_{xyz} \sin\theta\\
 w &= w
\end{align}$$
Obviously setting $w$ constant again selects a hyperplane parallel to the $x$-$y$-$z$ hyperplane, and setting any other single coordinate constant gives the result of setting that coordinate constant in 3D spherical coordinates and extruding into the fourth dimension. In particular, $r_{xyz}$ constant gives a spherical cylinder, $\theta$ constant gives an "extruded halfcone" and $\phi$ constant gives a half-hyperplane.
The analogue happens also for two coordinates fixed: If $w$ is among them, we get the normal 3D results (sphere, halfcone, halfplane) embedded in 4 space, otherwise we get an extruded version of the result of setting the two variables fixed in spherical coordinates (that is, spherical cylinders or halfplanes). Similar for three coordinates (including $w$: circle, ray; not including $w$: straight line parallel to $w$ axis)
(4) — "Hyperspherical coordinates"
This is the straightforward generalization of spherical coordinates to 4D:
$$\begin{align}
 x &= r \cos\eta \cos\theta \cos\phi\\
 y &= r \cos\eta \cos\theta \sin\phi\\
 z &= r \cos\eta \sin\theta\\
 w &= r \sin\eta
\end{align}$$
A constant $r$ gives a hypersphere ($^3S$, 3D surface that is the 4D equivalent of the 2D sphere in 3D). A constant $\eta$ gives a sort of spherical hypercone. For constant $\theta$ this gives some complicated shape which I'm currently not able to imageine, and for constant $\phi$ this gives a half-hyperplane.
Determining the shapes for more variables constant is now too complicated for me, especially that here it's already past midnight. ;-)
