Surfaces

Basic Surfaces

Circular Cylinder

Equations of the form

$$(x-x_0{)}^2+(y-y_0{)}^2=r^2$$

define a circular cylinder with central line $(x_0,y_0,z)$ and radius $r$.

In 3D, any closed curve of the form $f(x,y)=0$ will give a cylinder with a different cross-section.

Planes

$$ax+by+cz=d$$

for some $a,b,c,d\in \mathbb{R}$.

Sphere

$$(x-x_0)+(y-y_0)+(z-z_0{)}^2=r^2$$

The sphere will be centered at $(x_0,y_0,z_0)$ with some radius $\mid r\mid$.

Paraboloids

$$z=a(x-x_0{)}^2+b(y-y_0{)}^2+c$$

where $a,b$ have the same sign. Paraboloids are basically just 3D parabolas (kinda look like a parachutes)

---

Complex Surfaces

In 2D, a quadratic form is an equation of the form $a(x-x_0{)}^2+b(y-y_0{)}^2=c$, which describe circles/ellipses, parabolas, and hyperbolas, depending on the relative values of the parameters of the equation.

In 3D there are more classifications:

For more, see Quadric Surfaces