Circular cubics
These cubics are created beginning with the product of a circle (2nd degree) and a line (1st degree). On these pages we add other lines examine the cubics created thereby. The line at infinity remains a useful analytical tool.

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.

These graphs can be obtained using an implicit plotting program. We used Grapher, the new program that comes with all macs.

Circular points at infinity.

The circle and the line at infinity do not cross, therefore their intersections are complex numbers. All circles go through these two points, hence the term: circular points at infinity. Circular cubics go through these same points.

In this figure the cubic is created from a circle C2 and a line L1 that intersects the circle.

C2 · L1 = 1.

L1 is the asymptote.

We can insert the line at infinity as

C2 · L1 = (infinity)^3.

Hence L1 crosses the line at infinity at a triple point or point of inflection. for this reason L1 never crosses the cubic again. The circular points at infinity are also triple points.

Since the triple points for a cubic are colinear in 3's (we show this below), the line through the visible points of inflection

Since the cubic is formed with the product of the functions representing a circle and a line, only the regions where both function are positive or both negative host the curve.

For this graph a second line is added to the first. circle · lin1 = line2. With the line at infinity this looks like.

circle · line1 = line2 · infinity^2

Hence the cubic curve is tangent to line1 at infinity, making line1 an asymptote.

The cubic goes through the intersections of line2 with both line1 and the circle.

Here circle · line1 = (line2)^2 · infinity

Here line 1 is no longer an asymptote since the cubic is not tangent to it at infinity. The cubic is tangent to line2 where line2 meets the circle and line1.

Here circle · line1 = line2 · line3 , making the cubic go through the intersections of line2 with both line 3 and line 3 as well as the intersections of the circles with these lines.

Here circle · line1 = line2 · (line3)^2 , making the cubic go through the intersections of line2 with both line 3 and line 3 as well as the intersections of the circles with these lines. Further it is tangent to both lines 2 and three where line 1 and the circle cross them.

This cubic does not go through the circlular points at infinity and is not a circular cubic.

From here try the quartics slideshow.