 So now we've reached the fundamental problem of higher-dimensional graphing. How can you represent a three-dimensional surface on a two-dimensional map without losing vital information? This became important during the Napoleonic era when Napoleon was invading the rest of Europe. It was important to know what was on the other side of the mountain. And so during the Napoleonic Wars, contour maps began to be used. And the basic idea is that every point along a contour line had the same height. Now in mathematics, we call these contour lines level curves. And these represent the portion of a surface that is at a constant height. And stacking these level curves produces the surface. For example, let's consider z equals x squared plus y squared. Let's draw the level curves for z equals negative one, zero, one, and two. And then let's get to the surface. So if z equals negative one, the equation becomes x squared plus y squared equals negative one. And we try and find a point that solves this equation. And there isn't any. If z equals zero, the equation becomes x squared plus y squared equals zero. And we try and find a point that satisfies this equation. And the only point that does is the point zero, zero. And so this level curve is just the point zero, zero. So notice that right now we're graphing in two dimensions. So let's look down at the x, y plane and plot the point zero, zero. If z is equal to one, the equation becomes x squared plus y squared equals one. And we recognize this as a circle with radius one centered at the origin. And we can graph it. If z equals two, the equation becomes x squared plus y squared equals two. And this is a circle with radius square root of two centered at the origin. And we'll graph that. So what does our surface look like in three dimensions? So since zero, zero is the only point where z is equal to zero, then the surface includes the point zero, zero, zero. When z was equal to one, the level curve is a circle. And because z is equal to one, this circle is at height z equal to one. So we'll raise that circle up a bit. When z is equal to two, the level curve is a circle. And again, because z equals two, this circle is at a height of z equals two. And so we'll raise our circle up a bit. Now the important thing to keep in mind is that these level curves are on the surface itself. And as long as z is greater than zero, the level curve will be a circle. And if z is negative, there are no points on the curve. And so our surface will look something like this. And we call this type of surface a paraboloid. Now you might be a little bit suspicious. We introduced this curve on the surface, but this is not the only type of figure that could have a circle as a cross-section. And so you might ask, well, how do we know the surface curves? And the answer to that is we can sketch level curves by choosing values for other variables. And in fact, we've already looked at this curve. For example, we found that if x is equal to zero, then we're on the yz plane. And the equation will be z equals y squared, which is a parabola. And so we know that our surface has to curve when it goes through the yz plane. So in contrast, we might try to sketch the graph of z squared equals x squared plus y squared. So if we let z equals negative two, then we get a circle centered at the origin with radius two. If we let z equals negative one, we get a circle centered at the origin with radius one. If we let z equals zero, we get a circle centered at the origin with radius zero. Well, this is also known as a point. And if we let z equals one or z equals two, we get a couple more circles. Now, that's if we choose value for z. If we choose values for, say, x, if we let x equals zero, our equation becomes, and these are going to be straight lines through the origin in the yz plane. So remember right now we're looking down on the xy plane. If we want to graph in the yz plane, we're going to shift our perspective and graph. And so if we put our observations together, for any value of z, the equation z squared equals y squared plus x squared corresponds to a circle at the height of z. And if we let x equals zero, we're on the yz plane, and we get the straight lines z equals plus or minus y. And so what our surface has to look like, the circles are raised or lowered until they catch on these straight lines. And we connect the dots or rather the curves to form our surface. And we get a cone.