Is there a method for graphing three dimensions on a Cartesian plane? With x being the horizontal and y being the vertical, I would only assume that z is the third dimension. Is this ever done in mathematics, and if so, where can I learn how to do it?
The first one. What is the algebraic formula for a three-dimensional Cartesian system? Keep in mind that while I’m a math student, this isn’t homework help…just wanting to learn a little more than I do now.
Well, you can have different formulas, depending on what you’re graphing. As you may know, y=x^2 forms a parabola (U-shape), y=5x-2 forms a line, y=sin x forms a wave pattern, etc. Similarly, x^2 + y^2 + z^2 forms a sphere, z = 6y-4x forms a plane, z = tan(x+y) forms… um… some kind of shape (I honestly don’t know what it would look like) as does x^3-1/y=z. x represents how far a point is left-right wise from the origin, y represents how far it is up-down wise, and z represent front-back wise. Points are represented as, for example, (3, -1, 5), a logical extension of the way points in a 2-d system are represented.
Any equation that relates x, y, and z to each other will be the equation of some kind of 3-dimensional shape. Make sense?
When graphing on paper, you usually draw the z-axis at a 45-degree angle to the other axes, and sometimes include various dotted lines to help people looking at your drawing visualize what the 3-d shape looks like.
In 2-D Cartesian coordinates, every point in the plane is associated with a pair of numbers, (x, y).
In 3-D Cartesian coordinates, every point in space is associated with a triple of numbers, (x, y, z). The z-corrdinate tells how far above the xy-plane your point is (or below, if it’s negative). If you imagine the x and y axes drawn as normal on a sheet of paper, the z-axis would be sticking up out of the paper at right angles to each of the other axes. Points that are actually on the sheet of paper would have a z-corrdinate of 0. z would be positive above the paper, negative below it.
In 2-D, equations involving x and y describe curves in the plane (parabolas, circles, straight lines, etc.). In 3-D, equations in x, y, and z describe surfaces in space (spheres, planes, paraboloids (like a satellite dish), “saddle surfaces,” etc.).
There are, of course, ways of representing what a 3-D image would look like from a particular point of view on a flat surface (like a piece of paper or a computer screen).
Probably the first place anyone would study 3-D coordinate systems in any detail would be the third semester of Calculus.
What I meant by “algebraic formula” is the formula to draw a system on a Cartesian plane. For example, the system in slope-intercept form, y1=2x+3 and y2=-½x+1, would create two perpendicular lines on a Cartesian plane. What would be the equation be to create a 2x2x2 cube centered on the origin? The coordinates are (0,2,2), I know, but what is the formula?
I like the formulae you gave me, TJdude825. I’m fiddling with them now on graph paper.
I’m still not sure what you’re looking for, but if it’s what I think you want, you might want to look at gnuplot. Free, open-source, available for Linux, Windows, etc., and used in academia all over the place. (I’ve also found this gnuplot tips page to be extremely helpful.)
A simple algebraic “formula”, or equation, will determine either a single point, an infinite line (staight or curved), or an infinite plane (or curved surface), or even a solid. This depends on how many of x, y, and z are defined by the equation.
Using your example of a cube, it can be defined as the intersection of 6 planes, each with a different formula, which would define the surface of the cube only, not its interior. The equations for the 6 planes are:
Note that would define 6 infinite planes; the idea that the intersection of them is the interesting part is nowhere in the formulas.
Or you can define the cube as a solid including all the interior points. That would be defined by a different set of 3 equations:
This does define just the cube itself and does not extend out to infinity like your 2D lines or my cube surface did.
In general, the simple Y=mX+b single equation format you’re familiar with falls apart in higher dimensions. Those equations are still defined, but the really interesting things require more complicated algebraic expressions that do NOT reduce to a single “formula”. A cube is an exmaple of something that does NOT reduce to a single formula. A sphere, however does: 0 = X^2 + Y^2 + Z^2 - k^2 is the formula for a sphere of radius k. A paraboloid is just 0=aX^2 - bY + k for constants a, b & k
Parametric equations are common, where in you have a separate “formula” for X, Y, and Z each depending on other variables (“parameters”) like a time T or a field strength z or … i.e. graph this:
for t ranging from -10 to 10 by steps of 0.10, where f(t) = t^1.6 + 7, g(t) = ln(t), and h(t) = sin^2(t)
It’s also common to use non-cartesian coordinates for 3D (or higher-D) work. For example, in polar coordinates a sphere’s equation is R=k for some constant k.
If you can try explaining more about what you’re trying to learn we could provide more assistance. Right now the lack of a common vocabulary is getting in the way. I suspect you’ve asked a much larger question than you recognized at the time.
It’d also be helpful to know if you’re trying to learn the concepts for their own sake or are just trying to do some specific task, and whether you’re wanting to do the task by hand or via computer.
For example, there are established fairly simple techniques for translating a 3D point into a 2D plane for rendering. Whether you’re trying to draw an accurate perspective drawing of a box by hand or trying to program a computer to draw a scene on a screen, the concepts are the same but the process would be very different.
Interestingly enough, curves (circles, hyperbolas, etc.) are much easier to define than some other shapes that you might think of as “simpler” like squares. The same is true for 3-d graphs. I don’t know the formulas off the top of my head, but if you’re enjoying “playing with formulas on graph paper” perhaps you’d like to challenge yourself with finding the formulas for a cone, a torus (basically, a donut shape), an elongated sphere (um, like a stretched-out sphere… would you call that a spheroid?), a cylinder, etc.
By the way, in addition to traditonal “rectangular coordinates” you can also express 3-d points and shapes using two different types of polar coordinates. Check mathworld.com or Wikipedia and they’ll probably have a fairly decent description of how that works. Hope this thread has answered your questions somewhat well.