Understanding the compactification of elliptic curves I'm having trouble understanding the following example from my lecture notes:

Example: Let $E$ be an elliptic curve over a field $k$ given by the compactification of the affine equation $y^2=x^3+ax+b$ with $\Delta=4a^3+27b^2\neq 0$ inside $\mathbb{P}_k^2$, so $E=V(Y^2Z-X^3-aXZ^2-bZ^3)$. This curve has a point at $\infty\in E(k)$ given by $(0:1:0)$.

I understand that $Y^2Z-X^3-aXZ^2-bZ^3$ gives us the affine equation if we only look at the plane $z=1$ in $k^3$ and I would assume the point at $\infty$ to lie in $z=0$ (everything up multiplication by scalars). Plotting this with wolfram alpha yields this (for $a=b=1$) which doesn't look like $(0:1:0)$ at all. In fact, it even looks like we obtain two distinct points at infinity.
Where lies my misunderstanding?
 A: I'm going to elaborate on the explanations given in the two comments. The homogeneous form of your equation, $Y^2Z−X^3−XZ^2−Z^3=0$ (letting $a=b=1$) becomes $z-x^3-xz^2-z^3=0$ if you dehomogenize at $Y=1$. Let's call the original $(x,y)$ plane the "object plane" and the new $(x,z)$ plane the "picture plane" (borrowing the language of perspective). The line at infinity is represented in the picture plane by the line $z=0$. The curve looks like this in the picture plane:

You can see the curve crossing (or actually, tangent to) the line at infinity at one point.
Now, how can we make this geometrically intuitive? As the other comment indicates, points on the line at infinity represent directions in the object plane; equivalently, families of parallel lines in the object plane all intersect at the same point at infinity. (This is really evident in classical perspective--look up one-point and two-point perspective.) In this case, all lines parallel to the y-axis meet at the point (0:1:0) at infinity.
They meet at the same point whether you go to infinity upwards or downwards. This is an aspect of the topology of the projective plane. Remember, any two distinct lines in the projective plane meet in one point, so two parallel lines meet in a single point at infinity.
Another approach: projective points of $\mathbb{R}P^2$ correspond to lines through the origin in $\mathbb{R}^3$. Any such line intersects the unit sphere in two antipodal points. This pair can represent the "projective point". We can throw away one hemisphere and represent projective points with single points on the hemisphere, except for points on the equator. There we have to identify antipodal points. Suppose the equator represents the line at infinity; so a line doesn't have two points on it "at infinity", but just one, and topologically every projective line is a circle (when the field is $\mathbb{R}$).
Finally, you might be wondering why the curve is tangent to the line at infinity. Imagine a perspective drawing of an infinite chessboard. All the north-south lines, $x=c$, come together at a single point at infinity. In the picture plane, this looks like a pencil of lines all radiating from the same point. Now, what about the tangent lines to your curve? The tangent at $(x_0,y_0)$ has larger and larger slope (in absolute value) as $y_0$ becomes larger and larger. But also, $x_0\rightarrow\infty$ as $y_0\rightarrow\infty$. So for $|y_0|$ large, the tangent line is close to one of these $x=c$ lines, one making a small angle with the line at infinity in the picture plane.
You'll find an even more discussion of this sort of thing at my Algebraic Geometry Jottings.
