Lines to cubics
The equation of a line is of first degree. The product of three lines is of third degree and is some form of cubic. This product set to zero would be the degenerate cubic formed of three lines. The the product is not zero, we usually get some form of non-degenerate cubic.

A general property of a cubic curve is that a line intersects it three times, counting multiplicities and complex roots. A tangent is counted as a double interstions and a tangent at a point of inflection is counted as a triple intersection.

There are many more possibilites for cubics than for quadratics.

Good to read first: oriented lines. and the page on the line at infinity.

In this figure there are three oriented lines and a cubic created by setting the product of these three lines equal to zero.

For this cubic the three lines are asymptotes.

Since the cubic is formed with the product of the functions representing three lines being –1, only the regions where the signs of the three lines multiply to a negative are occupied.

The lines divide the plane into seven regions, distinguished by the signs of the lines in each region. The signs are given with respect to (line1, line3, line4).

2. The graph of line1 . line3 . line4 - n = 0, where n is any number, is a cubic with the three lines as asymptotes. This can be analyzed using the line at infinity (see here). We add the lines at infinity to the constant term, adding 3 factors so that each term will contain the same number of lines. (Note: since the line at infinity is only a factor of 1, we can add them whereever they are useful).

line1 . line3 . line4 - n (infinity )^3 = 0

At the point where line1 meets the line at infinity, both terms are zero, so the cubic goes through this point, which because of the cube is a triple point, meaning that line1 meets the line at infinity at a point of inflection. Among other things, this means that line1 is tangent to the cubic at infinity. Same logic for the other two lines, they are the three asymptotes of this cubic. Since each line hits the cubic at a triple point, none of the lines can intersect the cubic again.

The graph of line1 . line3 . line4 - n line2 = 0 is a cubic for which the first three lines are asymptotes and line4 goes through the three intersection points of the asymptotes with the cubic.

We analyze this by adding two lines of infinity in the last term, so that all terms will have the same number of lines, we have

line1 . line3 . line4 - n line2 . (infinity )^2 = 0

At the point where line1 meets the line at infinity, both terms are zero, so that the cubic goes through this point, a double point. Hence line1 is tangent to the cubic at infinity, making it an asymptote. Similarly line3 and line4 are asymptotes.

The points at infinty are double points, so each asymptote will hit the cubic once again.

We now get an interesting result for those points. Now look at P24, the point of intersection of line2 and line4. At this point line1 = line4 = 0, making both terms zero, so this point is on the cubic. Similarly P23 and P12 are on the cubic. P12, P24, P23 are the three points where the asymptotes cross the cubic. line4 goes through them all.

Theorem: the finite points where three asymptotes cross a cubic are colinear.

line1 . line3 . line4 - n (line2)^2 . infinity = 0

Again the point where line1 meets the line at infinity, both terms are zero, so that the cubic goes through this point. Since it is not a double point, line1 is not tangent to the cubic.

line2 (the blue one) meets the cubic at the point where the cubic is tangent to the other three lines.

Same properties as the last one, just the sign of (line2)^2 is changed.

After this move on to circular cubics.