Curves and Surfaces of the Second Order

Curves of the Second Order

General Equation

A second-order curve in the plane: $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$.

The discriminant $\Delta = B^2 - 4AC$ determines the type:

• $\Delta < 0 \rightarrow$ ellipse (or a point, empty set)
• $\Delta = 0 \rightarrow$ parabola (or degenerate case)
• $\Delta > 0 \rightarrow$ hyperbola (or a pair of lines)

Ellipse

Canonical equation: $x^2/a^2 + y^2/b^2 = 1$ ($a \geq b > 0$).

$a$ — major semi-axis, $b$ — minor. Foci: $F_1(-c, 0)$, $F_2(c, 0)$, $c^2 = a^2 - b^2$.

Definition: the locus of points, the sum of distances from which to two foci is constant $= 2a$.

Eccentricity $e = c/a \in [0, 1)$. For $e = 0$ — circle.

Hyperbola

$x^2/a^2 - y^2/b^2 = 1$. Foci: $F_1(-c, 0)$, $F_2(c, 0)$, $c^2 = a^2 + b^2$.

Definition: $|r_1 - r_2| = 2a$. Asymptotes: $y = \pm (b/a)x$.

Rectangular hyperbola ($a = b$): $xy = k$ — its equation in rotated axes.

Parabola

$y^2 = 2px$ ($p > 0$). Focus $F(p/2, 0)$, directrix $x = -p/2$.

Definition: distance to the focus $=$ distance to the directrix.

Parabolic antennas and mirrors: parallel rays are collected at the focus.

Surfaces of the Second Order

Ellipsoid: $x^2/a^2 + y^2/b^2 + z^2/c^2 = 1$.

Hyperboloids: one-sheeted ($x^2/a^2 + y^2/b^2 - z^2/c^2 = 1$), two-sheeted ($x^2/a^2 + y^2/b^2 - z^2/c^2 = -1$).

Paraboloids: elliptic ($z = x^2/a^2 + y^2/b^2$), hyperbolic (saddle: $z = x^2/a^2 - y^2/b^2$).

Cone: $x^2/a^2 + y^2/b^2 - z^2/c^2 = 0$.