Coordinate Systems – 3D, part 1

Since we live in a 3 dimensional world, many problems we encounter in fields such as science and engineering, as well as others, are modelled mathematically using 3 variables, hence, 3D.

The first coordinate system introduced to students to handle 3 variables is an extension of the 2D Cartesian coordinate system. If another number line is added to the 2D system that is 90° t0 the previous 2 axes, with the origin coinciding with the other two origins, you have the 3D system. The third axis is called the z-axis. So a point now needs 3 numbers to place it in 3D space: (x,y,z). Frequently, to draw a 3D grid on a 2D surface, the y and z axes are drawn in he plane of the surface and the x axis is drawn in perspective to show that it is perpendicular to the surface. So placing a point in a 3D Cartesian frame is an artistic challenge for me but drawing dashed lines parallel to the axes helps:

There are other orientations of the 3 axes when showing them in 2D, but this is a very common one.

As with the 2D Cartesian coordinate system, equations relating the variables x, y, and z can be plotted, showing all the values of x, y, and z that make the equation true.

In 2D, a general equation of a line is ax + by = c, where the a, b, and c are specific numbers. For example, the set of points that satisfy the equation 2x -3y = 7, plot as a straight line. By extension, in 3D, the general linear equation is ax + by + cz = d. Though this is called a linear equation, it plots as a plane in 3D:

The 3D version of a circle in 2D is a sphere. The generic equation of a sphere of radius r centred at the origin is x2 + y2+ x2 = r2:

Very interesting shapes can be made using 3D graphs. Here are a few:

$z=5e^{-0.2(x^2+y^2)}\text{cos}(x^2+y^2)$
$z = \pm \sqrt{0.4^2-\left(2-\sqrt{x^2+y^2}\right)^2}$
$z=4 e^{-\frac{1}{4} y^2} \sin (2 x)$

As with 2D, there are other ways of locating points in 3D. I will present some of these in my next post.