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Q: Formula for a rectangular solid ( No Answer,   2 Comments )
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 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.```
 ```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 Grapher. 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: 0,0,0 5,0,0 5,10,0 0,10,0 0,0,0 0,0,8 5,0,8 5,0,0 5,0,8 5,10,8 5,10,0 5,10,8 0,10,8 0,10,0 0,10,8 0,0,8 0,0,0```
 ```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.```