2 - Points at Infinity
Summary
The examples in the last chapter reveal a wide range of behavior of algebraic curves in the real plane.
Some are bounded, others are not.
Some form one piece — that is, they're connected, having just one topological component — while others are not.
For a curve having two or more topological components, there can be a mixture of bounded and unbounded components. For example, in Figure 1.10 on p. 19, all components are bounded. The middle right graph of Figure 1.3 on p. 8 has a bounded and an unbounded component together. In a hyperbola, both branches are unbounded.
Even the dimension may not be 1. In the real plane, for example, the locus of x2 + y2 = 0 consists of the origin which has dimension 0, while x2 + y2 + 1 = 0 defines the empty set which has dimension −1.
The curve's dimension can be mixed. For example, in Figure 1.3 on p. 8 we see a cubic having both a one-dimensional component and a zero-dimensional component.
If the curve's defining polynomial has degree n, there are times when a line intersects the curve in n points, but there are other times when there are fewer than n intersections. As one example, there are always lines completely missing any bounded curve. As another, two distinct lines usually intersect in 1 point, but parallel lines intersect in no points. And the zero set of the degree-two polynomial x2 + y2 is just a point, so no line intersects the locus in 2 points.
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- A Guide to Plane Algebraic Curves , pp. 29 - 44Publisher: Mathematical Association of AmericaPrint publication year: 2011