Graphs in Space

In this section we will introduce some basic graphs in space.

Lines

We have three basic methods to express a line in three space:

• direction vectors
• parametric equations
• symmetric equations

We will introduce them one by one.

Direction Vectors

This is the most basic way to draw a line in space, with a point and its direction.

For example, a line through ${\displaystyle (x_0,y_0,z_0)}$, have the same direction as ${\displaystyle <a,b,c>}$ has the equation:

${\displaystyle (x,y,z)=(x_0,y_0,z_0)+t(a,b,c)}$

Parametric Equations

This is actually the split of the vector equation.

For each independent directions, we have:

${\displaystyle x=x_0+at}$
${\displaystyle y=y_0+bt}$
${\displaystyle z=z_0+ct}$

Symmetric Equations

For equations like:

${\displaystyle \frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}}$

We call it the symmetric equation of the line.

Planes

We mainly introduce two form of the quations of planes here, the standard form and the general form. Remember that the equation of planes is actually the linear combination of three variables.

Standard Form

We use a point ${\displaystyle (x_0,y_0,z_0)}$ and a direction vector that perpendicular to the plane ${\displaystyle <A,B,C>}$ to fix a plane.

The standard equation of this plane is ${\displaystyle A(x-x_0)+B(y-y_0)+C(z-z_0)=0}$.

General Form

The general form of the plane:
${\displaystyle Ax+By+Cz=D,A^2+B^2+C^2\ne 0}$ Can be translated to the standard form.

Spheres

Similar to the circle in plane, we expand the equation to:

${\displaystyle (x-h{)}^2+(y-k{)}^2+(z-l{)}^2=r^2}$

And that is the equation of spheres in three space.