09 - Conics and Quadrics

Circle

Equation:

$$x^2+y^2=a^2$$

Parametrisation: $(x,y)=(a\cos \theta ,a\sin \theta )$>
Area $\pi a^2$

Ellipse

Equation:

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(b<a)$$

Parametrisation: $(x,y)=(a\cos \theta ,b\sin \theta )$
Eccentricity: $e=\sqrt{1-\frac{b^2}{a^2}}$
Foci: $(\pm ae,0)$
Directrices: $x=\pm \frac{a}{e}$
Area: $\pi ab$

Parabola

Equation:

$$y^2=4ax$$

Parametrisation: $(x,y)=(a{t}^2,2at)$
Eccentricity: $e=1$
Focus: $a=0$
Directrix: $x=-a$

Hyperbola

Equation:

$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$

Parametrisation: $(x,y)=(a\sec t,b\tan t)$
Eccentricity: $e=\sqrt{1+\frac{b^2}{a^2}}$
Focus: $(\pm ae,0)$
Directrix: $x=\pm \frac{a}{e}$
Asymptotes: $y=\pm \frac{bx}{a}$

Quadrics

Sphere:

$$x^2+y^2+z^2+a^2$$

Ellipsoid

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$$

Hyperboloid of One Sheet

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$$

Hyperboloid of Two Sheets

$$\frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$$

Paraboloid

$$z=x^2+y^2$$

Hyperbolic Paraboloid

$$z=x^2-y^2$$

Cone

$$z^2=x^2+y^2$$