Algebraic Plane Curves

Section 2.1 Algebraic Plane Curves

In Section 1.2, I defined implicit derivatives to understand tangents to loci. These implicit derivatives relied on the assumption that, at least locally, the variable \(y\) was a function of the variable \(x\text{.}\) When this assumption failed, the tangent line wasn’t defined. So far, the examples had vertical tangents when the assumption failed. This made some sense, since a vertical line doesn’t have a well-defined slope. However, there are a variety of ways, in addition to vertical tangents, that implicit derivative can fail. This week, I’m going to investigate the places where these loci don’t have well defined derivatives.

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First, however, I want to define a new class of loci. Doing the theory for this week in full generality for any loci in \(\RR^2\) isn’t really feasible. I’m going to restrict loci with the following definition.

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Definition 2.1.1.

Another name for loci in \(\RR^2\) is plane curves. An algebraic plane curve is a locus where the expression is a polynomial. The degree of an algebraic plane curve is the highest polynomial degree involved in the equation of the curve.

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Figure 2.1.2. An off centre, rotated ellipse

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There are two classes of algebraic loci which you should be pretty familiar with by now.

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An algebraic plane curve of degree one is a line. There are several ways to write equations of lines, but the most general form is the following, where \(a\text{,}\) \(b\) and \(c\) are real constants.
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\begin{equation*} ax + by + c = 0 \end{equation*}

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An algebraic plane curve of degree two is a conic. I’ve already shown in Calculus 1 that all conics have equations of degree two (they only involve squares of the variable). What is new, here, is the fact that all algebriac plane curves of degree two are, in fact, conics. (I’m not going to prove this fact, just state it.) The forms of the equation can differ from what I’ve shown you before, but all possible forms can be manipulated into the equation of one of the four conics. This is another nice reason to put the conics in a family with a unified name. The most general form of a conic is the following, where \(a_1\text{,}\) \(a_2\text{,}\) \(a_3\text{,}\) \(a_4\text{,}\) \(a_5\) and \(a_6\) are real constants.
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\begin{equation*} a_1 x^2 + a_2 xy + a_3 y^2 + a_4 x + a_5 y + a_6 = 0 \end{equation*}

This form does include some equations which are not familiar, such as \(-4x^2 + 4xy - 2y^2 + 4x - 3y + 2 = 0\text{.}\) This particular equation produces an ellipse at an angle: the major and minor axes are no longer oriented along the \(x\) or \(y\) axes. The ellipse is also not centred at the origin. Figure 2.1.2 shows a graph of this ellipse. The general form, as indicated in this example, can produce conics with any angle of rotation or shift of position compared with the familiar forms centred at the origin.
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Non-vertical lines have no undefined tangents, since the slope of the line is the slope of the tangent at all points. Conics can have vertical tangents, as shown in previous examples. With one important exception (which I’ll talk about in the next section), vertical tangents are the only problems with tangent slopes to conics. However, when I consider higher degree algebraic plane curves, all sorts of things can happen.

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For now, just to show the variety of shapes, let me give you a few examples of interesting algebraic plane curve in degrees higher than two.

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Example 2.1.3.

Figure 2.1.4. The curve \(xy^4 - 4xy^3 + x^2 - y^3 + x^2 - 4 = 0\)

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This is an algebriac plane curve of degree five. Note that it has two separate pieces, one of which is bounded and one of which is unbounded. Its graph is shown in Figure 2.1.4.

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\begin{equation*} xy^4 - 4xy^3 + x^2 - y^3 + x^2 - 4 = 0 \end{equation*}

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Example 2.1.5.

Figure 2.1.6. The curve \(\frac{x^3y^3}{8} - \frac{x^5}{4} + \frac{y^6}{6} - 2xy - 3y - 3 = 0\)

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This is an algebriac plane curve of degree six. Now there are three separate pieces, two of which are unbounded and one of which is bounded. Its graph is shown in Figure 2.1.6.

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\begin{equation*} \frac{x^3y^3}{8} - \frac{x^5}{4} + \frac{y^6}{6} - 2xy - 3y - 3 = 0 \end{equation*}

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Example 2.1.7.

Figure 2.1.8. A complicated degree eight curve.

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This is an algebriac plane curve of degree eight, and quite a complicated one. It has several components, many of which intersect at various places. Its graph is shown in Figure 2.1.8. If you study this carefully, you might notice that this looks somewhat like the amalgamation of one line and two hyperbolae.

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\begin{align*} \amp - x^2y^6 + x^3y^5 - x^5y^2 + x^4y^3 + 4x^2y^5 - 4x^4y^2 - x^2y^4 + xy^5 + x^4y \\ \amp - 4x^3y^2 + 3x^2y^3 - 4xy^4 + 4x^3y - 12x^2y^2 + 3x^2y - 3xy^2 + 12 xy = 0 \end{align*}

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