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Q: Formula for a rectangular solid ( No Answer,   2 Comments )
Subject: Formula for a rectangular solid
Category: Science > Math
Asked by: amsterdam-ga
List Price: $4.00
Posted: 17 Jun 2006 00:36 PDT
Expires: 17 Jul 2006 00:36 PDT
Question ID: 738899
I want to know the formula for a rectangular solid (in solid geometry)
that I can use in iMac Grapher. In the form x=???

I'm not looking for the volume - I want the generic formula to plot a
three dimensional rectangular solid in Cartesian space.
There is no answer at this time.

Subject: Re: Formula for a rectangular solid
From: brix24-ga on 17 Jun 2006 11:32 PDT
I'm no expert, but I don't think you can do this easily. If you could
cut off each of the twelve lines for the edges at the vertices, it
could be done - but I suspect that you cannot cut off a line in

Barring that, you could solve for the 8 vertices, but you'll still
have to arrange them in the correct order as a point set (including
duplicating some of the values for proper connection of the vertices).

Here is a simple example of drawing a rectangular solid once  you know
the 8 vertices and how to order them properly:

(Obviously, I choose a very easy rectangular solid; doing this
generically would take some work.)

You can generate a point set for this example by pasting the data
below into TextEdit and saving the file. Then go to Grapher > Equation
> New Point Set. Click on the "Untitled Set," then on "Edit Points."
Chose the "Import" button below the default values and import the
TextEdit file. Then highlight the top  rows of default data (4 rows on
my computer) and choose the pull-down "Delete" menu and delete those
rows, leaving only the data you imported. Click "Okay," then choose
the 3D view. (If you wish, you can eliminate the default frame under
"Format" > "Axes and Frame.")

Below are the 17 sample coordinates (8 vertices, with 9 repeats) to
paste into TextEdit:

Subject: Re: Formula for a rectangular solid
From: berkeleychocolate-ga on 21 Jun 2006 16:12 PDT
I'm not sure this is what you want, but here it is. We describe
conditions on A, B, C, D, E, F, G, H in order that they form the
coordinates of a rectangular solid. The first two points A and B are
arbitrary. Let v be the vector from C to B. Then v is perpendicular to
the vector from A to B. So v dot (B - A) = 0. So
v1*(b1-a)+v2*(b2-a2)+v3*(b3-a3)=0. This gives one condition on the
three numbers v1,v2,v3. Pick two of them as you choose and solve for
the third using this equation. Then C=B+v. Also D=A+v. Let w be the
vector from A to E (the point "directly above" A). Then w is an
arbitrary scalar multiple of the cross product of B-A and D-A. The
rest are obtained by adding w to B, C, and D. So F=B+w, G=C+w, H=D+w.

Hope tis helps.

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