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#1




3D Cartesian graphing
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?
Adam 
#2




Well, there's the graphing of level sets, where you show all the x and yvalues on the surface for a given fixed zvalue. Is that what you're thinking of?
There are also other ways of representing the third dimension on a twodimensional graph, e.g., by using a spectrum of color values. 
#3




Are you referring to simple Cartesian coordinates, where the zaxis is orthogonal to the x and yaxes? If it's a righthanded coordinate system, the zaxis will point "out" of the plane towards you.
Or did you mean something strictly in a twodimensional system, like a contour plot? 
#4




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Thanks. Adam 


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Any equation that relates x, y, and z to each other will be the equation of some kind of 3dimensional shape. Make sense? When graphing on paper, you usually draw the zaxis at a 45degree angle to the other axes, and sometimes include various dotted lines to help people looking at your drawing visualize what the 3d shape looks like. I hope that explains it fairly well. 
#6




What, do you mean like video games mapping a 3d scene onto the 2d topology of your computer screen?

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#8




In 2D Cartesian coordinates, every point in the plane is associated with a pair of numbers, (x, y).
In 3D Cartesian coordinates, every point in space is associated with a triple of numbers, (x, y, z). The zcorrdinate tells how far above the xyplane 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 zaxis 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 zcorrdinate of 0. z would be positive above the paper, negative below it. In 2D, equations involving x and y describe curves in the plane (parabolas, circles, straight lines, etc.). In 3D, 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 3D 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 3D coordinate systems in any detail would be the third semester of Calculus. 
#9




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I like the formulae you gave me, TJdude825. I'm fiddling with them now on graph paper. Adam 


#10




The Agent FoxtrotMississippienne Effect stikes again. That was my post, sorry.
Adam 
#11




A 2x2x2 cube centered at the origin would consist of the set of points (x, y, z) in which all three coordinates are between 1 and 1.
You may find this fun to play around with. (I don't know anything about the page; it came up when I googled.) 
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#13




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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: X=1 X=1 Y=1 Y=1 Z=1 Z=1 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: 1>=Abs(X) 1>=Abs(Y) 1>=Abs(Z) 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: X=f(t) Y=g(t) Z=h(t) 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 noncartesian coordinates for 3D (or higherD) 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.
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Cynicism is Evil. It provides an excuse to expect nothing but the worst from others while simultaneously excusing all the worst in yourself. It's lazy, selfish, and ignorant all tightly wrapped into a toxic ball of nasty. 
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If you don't get it, don't worry: it was really bad. 


#15




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 3d 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 stretchedout sphere... would you call that a spheroid?), a cylinder, etc.
By the way, in addition to traditonal "rectangular coordinates" you can also express 3d 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. 
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