Precalculus preliminaries

The interior angles of a convex \(n\)-gon sum to \((n-2)\pi\), so each interior angle of a regular \(n\)-gon is \(\frac{n-2}{n}\pi\).

(Standard forms of conic sections)

• Circle: \(x^2 + y^2 = r^2\), or \(\abs{z-a}=r\). For \(a\neq b\) and a constant \(c>0\) with \(c\neq 1\), the locus \(\abs{z-a} = c\abs{z-b}\) of points whose distances \(d_1, d_2\) to the two points \(a, b\) have constant ratio \(d_1/d_2 = c\) is also a circle:

• Ellipse: \(\qty{\frac x a}^2 + \qty{\frac y b}^2 = 1\), or \(\abs{z-a} + \abs{z-b} = c\) with \(c > \abs{a-b}\), the locus of points whose distances to \(a\) and \(b\) have constant sum \(d_1 + d_2 = c\):

• Hyperbola: \(\qty{\frac x a}^2 - \qty{\frac y b}^2 = 1\), or \(\abs{\abs{z-a} - \abs{z-b}} = c\) with \(0 < c < \abs{a-b}\):

The rectangular hyperbola is \(xy = \frac{c^2}{2}\). - Parabola: \(y^2 = 4ax\).

(Classification by the discriminant)

For a nondegenerate real conic \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\) with at least one real point and discriminant \(\Delta \coloneqq B^2 - 4AC\):

• \(\Delta < 0\) if and only if the conic is an ellipse, and it is a circle if and only if moreover \(A=C\) and \(B=0\);
• \(\Delta = 0\) if and only if the conic is a parabola;
• \(\Delta > 0\) if and only if the conic is a hyperbola.

(Parameterizations)

• The ellipse \(\qty{x\over a}^2 + \qty{y\over b}^2 = 1\) is parameterized by \(t\mapsto (a\cos(t), b\sin(t))\):

• In complex form, \(\gamma(t) = pe^{it} + qe^{-it}\) with \(p \coloneqq \frac{a+b}{2}\) and \(q \coloneqq \frac{a-b}{2}\), since \(pe^{it} + qe^{-it} = a\cos(t) + ib\sin(t)\).

• For \(a\geq b\) the foci are \((\pm c, 0)\) with \(c\coloneqq \sqrt{a^2-b^2}\), and the vertices are \((\pm a, 0)\) and \((0, \pm b)\):

  

• The circle \((x-h)^2 + (y-k)^2 = r^2\) is parameterized by \(t\mapsto (r\cos(t) + h, r\sin(t) + k)\), in complex form \(\gamma(t) = z_0 + re^{it}\) with \(z_0\coloneqq h + ik\).

• With \(u\coloneqq \tan(t/2)\), \[ \cos(t) = {1-u^2\over 1+u^2}, \qquad \sin(t) = {2u\over 1+u^2}. \] This gives the rational parameterization of the ellipse \[ u\mapsto a {1-u^2\over 1+u^2} +ib{2u\over 1+u^2}, \] and, for \(a=b=R\), of the circle of radius \(R\); each misses only the point \(-a\).

• For \(a\in\CC\setminus\ts{0}\) and \(c\in\RR\), the set \(\ts{z \st \bar{a}z + a\bar{z} + c = 0}\) is a line, and every line has this form.

(Numerical values)

\(\sqrt{2} \approx 1.4142\), \(1/\sqrt{2} \approx 0.7071\), \(\sqrt{3} \approx 1.7321\), \(1/\sqrt{3} \approx 0.5774\), \(e\approx 2.7183\), \(\pi \approx 3.14159\).

(Completing the square)

For \(a\neq 0\), \[ ax^2 + bx + c = a\qty{x + d}^2 + e, \qquad d \coloneqq \frac{b}{2a}, \quad e\coloneqq c - \frac{b^2}{4a}. \] In particular \(x^2 + bx + c = \qty{x+ {b\over 2}}^2 + c - \qty{b\over 2}^2\).