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March 7th, 2013, 03:49 PM   #1
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Spatial Analytic Geometry

Hello!

I'm stuck with trying to solve a 3rd dimensional geometry problem. Here it goes:

"A rectangle of base a and width b lies on a top plane, its base along x direction, its width along y direction, z exiting the plane of the rectangle. If the rectangle is rotated of an angle alpha, along its y axis, and of an angle beta along its x axis, which will be the general equation that describes the displacement of one of its vertices?"

I have managed to solve some parts of this problem, so far, and to understand better some details of this question. Like, if both angles alpha and beta are positive or negative, then also will be positive or negative the displacement. But I still didn't figure out how to compute the final value of the point in case alpha and beta have opposite signs. Would you please help me with that?

Thank you.
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March 7th, 2013, 06:00 PM   #2
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Is the first-mentioned rotation done first? Does that rotation move the axis for the second rotation? What is meant by "equation that describes the displacement"?
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March 8th, 2013, 03:48 AM   #3
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Re: Spatial Analytic Geometry

Ok, supposing that the rectangle is rigid, ie no changing on its sizes, the rotations are executed in the mentioned order. But as they are arbitrary angles (either positive or negative, between 0 and 90 deg, but preferably in a range below 45 degrees as it will be further considered) I wonder if it is really relevant to consider this order. I think not. Also, suppose there are two reference frames, if you like, one fixed in space, at the coordinate of the center of area of the rectangle, but not rotating along with it, and the other one rigidly connected with the rectangle, as if pointing out the directions of its plane and its normal vector.

By the words "displacement" and "rigid" I take it you already assumed this is a physics problem, but at a stage of its mathematical description. By "equation of displacement" I intend the equation that gives me the set of positions (trajectory) of one of its four vertices, as a function of its base and width lengths and these rotation angles. Supposing that I'm interested in the lower right vertex, and by looking at it from a front view (plane xz), I will see the true values of the alpha angle and of the rectangle's base. These values give me the delta_Z displacement of the vertex, with a sine equation. If at this instance I apply beta in a side view (plane yz), and depending on the signs that both alpha and beta have, this delta_Z value might both increase or decrease. If both alpha and beta have same sign, then the module of the resultant delta_Z displacement will be the Pythagorean equation for both single displacements of alpha and beta (supposing my mathematics worked out correctly so far =) ). But if alpha and beta have opposite signs, this is not true. And plus, it makes it more difficult to determine which will be the sign of the resultant displacement, because it depends on alpha and beta and their signs, and also in lengths b and w of the base and width. So, I will review my calculations in order to fix this problem. I wish you can give me some help with this.

Thank you.
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