## Point on a Line, a Line on a Plane, and a Plane in Space

Each point has a specific location. Two points determine a line. Three points determine a plane. Let us consider some simple math derivations to arrive at a format for each. For simplicity’s sake, we will use the familiar x, y, z Cartesian coordinate system. We begin with a point on a line. First, Point on a Line In space, a single point has an x value, a y value, and a z value. If the coordinate system chosen for the point is a simple 1-D line, then only one variable – say x – is needed to describe it. Then, since there is no y or z to consider, the mathematical description of the point is x = c But let us, for reasons that will be understood later, write…

## Understanding the Spherical Polar Coordinate System

[caption id="attachment_9496" align="alignright" width="480"] A cone in the spherical polar coordinate system. CC-SA 3.0 Unported by Lantonov[/caption] Do you have a basic knowledge of the spherical polar coordinate system? A coordinate system provides a way to describe and plot math functions using two or three variables. If there are two variables the graph is 2D. If there are three variables, the graph is 3D. The Cartesian Coordinate System The most familiar coordinate system is the Cartesian coordinate system. Typical variable names are x and y in 2D (although variables can have any name), and x, y, and z in 3D. Every point of every 2D function has a unique value in (x, y). Every 3D function similarly has a unique value in (x, y, z). The Polar Coordinate System This…

## Generate 2D Math Objects by Collapsing 3D Math Objects

I had some excellent high school mathematics instructors. They both loved their field and took an interest in their students. Since high school, I have had a deep interest in collapsing 3D mathematical equations to derive equations for 2D mathematical objects or modifying 2D objects into other 2D, or 2D objects into 1D. A 3D sphere becomes a 2D circle. A 2D parabola becomes a 1D line. The 2D hyperbola shown, if collapsed along the x-axis, becomes two 1D line segments stretching at one end to infinity. The same hyperbola collapsed along the y-axis becomes a complete line. A 2D circle becomes a single 1D line segment of a length equal to the diameter. An Example of Collapsing 3D into 2D What can be obtained by collapsing 3D math objects…