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…

Collapsing then Expanding the Equation for a Sphere

[caption id="attachment_8626" align="alignright" width="480"] How simple is a sphere?[/caption] Equation for the Simplest Sphere The equation for a sphere with its center at the origin is: x2 + y2 + z2 = c2 Where c is a positive constant. For simplicity, let's choose a positive constant, k, such that k = c2. Equation for a Circle by Collapsing a Sphere Collapsing it in one dimension generates the equation of one of three circles: x2 + y2 = k x2 + z2 = k y2 + z2 = k Equation for a Point by Collapsing a Circle Collapsing the three circles in one dimension generates two equations representing precisely two points for each of them: For x2 + y2 = k, x²2 = k y2 = k For x2 + z2…

Not Infinitesimal Point Coordinates

[caption id="attachment_7388" align="alignright" width="480"] A coordinate system.[/caption] Infinitesimal point coordinates or not? Do you occasionally enjoy speculating? Even if one has technical training, if he speculates outside his field of expertise, he is opening himself up for possible difficulty. What he speculates, if he voices it, could result in his being labeled someone who doesn’t know what he is talking about. But I never was bright enough to avoid speculation, though I always acknowledge it for what it is. So humor me here… So if you know something of all this and what I'm writing either doesn't jive, or you can 'supercharge' it, please let me know. Mathematicians’ Points – Reality? Generally, a point in space is seen as a dot in space, having infinitesimal point coordinates, that is, no…

Discrete Mathematics: What is a Point and What a Line?

[caption id="attachment_7234" align="alignright" width="440"] Discrete points? An abstract.[/caption] Most technically minded people will probably take me to task over what I am going to say in this article. That is OK, though. I’m used to it. Not only are my writings quirky—I am quirky. Hey! This is QuirkyScience. I want to talk about points and lines in the real world—in other words, discrete mathematics. What is Reality, What Fantasy? To a mathematician, the point may be a dimensionless object in 3D space, a mathematical object with an x, y, z coordinate in Euclidean space. A line would be a collection of such dimensionless points lined up all in a row. But that is in the world of the mathematician. In the real world, there can be no such thing. Rather,…