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Thread: how do you graph in 3d?

  1. #1
    Junior Member
    Oct 2008

    how do you graph in 3d?

    for example, these two functions:

    d = ax + by + cz

    and d = ax^2 + by^2 + cz^2

    I know the first one is a plane and i think the second one is an ellipse in 3d, so kind of like an egg, but how do you go about graphing them?
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  2. #2
    MHF Contributor

    Apr 2005
    Strictly speaking you don't "graph in 3D" unless you have some very special sculpturing equipment! What you do is draw a representation, in a 2 dimensional plane (paper or computer screen), of the 3 dimensional graph.

    For the plane, d= ax+ by+ cz, what I would do is note that when x= y= 0, d= cz so z= d/c. One point on the plane is (0, 0, d/c). Similarly, when x= z= 0, d= by so y= d/b. (0, d/b, 0) is a second point on the plane. Finally, when y= z= 0, d= ax so x= d/a. (d/a, 0, 0) is a third point on the plane and a plane is determined by three points.

    Draw two lines at right angles representing the x and z axes. Draw a third line at an angle, representing the y axis, through the intersection of those two lines. Mark the three points, (d/a, 0, 0), (0, d/b, 0), and (0, 0, d/c), on those axes. The plane, in the first octant, is the triangle having those points as vertices. The entire plane, of course, would cover the entire paper and really needs to be "imagined" more than drawn. Perhaps you could use some semi-opaque color to draw it.

    The angle at which you draw that y axis depends upon the "perspective" which, in turn, depends upon where you are imagining viewing the figure from. Three dimensional drawing depends much more upon "viewpoint" (where your eye is) than two dimensional drawing.

    For the ellipsoid (not "ellipse"- that is a two dimensional figure), draw the same x, y, and z axes as above. Mark the "limits" of the ellipsoid on those axes:
    When y= z= 0, $\displaystyle d= ax^2$ so $\displaystyle x= \pm\sqrt{\frac{d}{a}}$: the ellipsoid is inside the points $\displaystyle \left(\sqrt{\frac{d}{a}}, 0, 0\right)$ and $\displaystyle \left(-\sqrt{\frac{d}{a}}, 0, 0\right)$, etc. Sketch the ellipse $\displaystyle d= ax^2+ by^2$ in the xy-plane, the ellipse $\displaystyle d= ax^2+ cz^2$ in the xz-plane, and the ellipse $\displaystyle d= by^2+ cz^2$ in the yz-plane, with the correct perspective as a guiding frame work to the entire ellipsoid. Getting everything in the right perspective requires a certain amount of artistic, or at least "drafting", talent!
    Last edited by HallsofIvy; Oct 29th 2010 at 06:50 AM.
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