Equation of a Rectangle?Date: 11/11/2012 at 07:26:41 From: Ahmet Subject: What is the Equation of a Rectangle in a Cartesian Plane Simply put, I want to know if there is an equation for a rectangle. Perhaps it would be similar to that for a circle, which is (x - h)^2 + (y - k)^2 = r^2 There must be some such equation out there. But because the graph of a rectangle is not continuously smooth like it is for conic sections, it cannot exist in the way we think of conic equations. I imagine it will be somewhat implicit in nature. If such an equation exists, can it be rotated using the same (x', y') formulas for conics? Unfortunately, I have done zero work in an attempt to derive the equation of a rectangle -- way too advanced for me. Date: 11/11/2012 at 11:28:09 From: Ahmet Subject: What is the Equation of a Rectangle in a Cartesian Plane One more thing: is there a way to define the equation from the center of the rectangle, and perhaps use transformations to plot it at a location other than the origin? And what about rotating the rectangle? Date: 11/11/2012 at 11:36:12 From: Ahmet Subject: What is the Equation of a Rectangle in a Cartesian Plane I did find an equation on a Web site. Denoting the absolute value with "|": |x/p + y/q| + |x/p - y/q| = 2 I'm not sure how to solve this equation for "y," so I can't graph it to verify it. Also, how would I transform this to another point or rotate it? Date: 11/12/2012 at 18:45:33 From: Doctor Peterson Subject: Re: What is the Equation of a Rectangle in a Cartesian Plane Hi, Ahmet. This is the sort of equation that requires some ingenuity to invent! I briefly thought about doing it with absolute values when I first saw your question, but didn't have the time to put into it. It is NOT an equation that you can solve for y. To graph it, you have to consider cases. For example, |x/p + y/q| will be equal to x/p + y/q wherever the latter is positive, namely in the region above the line y = -(q/p)x. Similarly, |x/p - y/q| will be equal to x/p - y/q, where the latter is positive, that is, in the region below the line y = (q/p)x. (Each absolute value will be the opposite in the opposite region.) So in the region that is both above y = -(q/p)x and below y = (q/p)x, your equation becomes (x/p + y/q) + (x/p - y/q) = 2 2x/p = 2 x = p This is the vertical line forming the right side of the rectangle: ^ | . + . . | . | . | . | . | . | --+---------------+---------------+--> . | . | . | . | . | . | . + . | You can verify that in the other three regions determined by the two lines, you get the other three sides of the rectangle. You could translate this to another center as usual, replacing x with x - h and y with y - k. Rotation would mean replacing each of x and y with an expression involving x and y, which perhaps you have seen. It would get ugly. In general, not every shape can be reduced to a single, simple equation. Without the absolute value trick, you would simply write this as four separate lines (which is what you would get if you did manage to solve the equation for y): x = p for -q <= y <= q (the part we just did) x = -p for -q <= y <= q y = q for -p <= x <= p y = -q for -p <= x <= p This is usually easier to work with than the "elegant" but unwieldy single equation. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 11/12/2012 at 22:25:05 From: Ahmet Subject: Thank you (What is the Equation of a Rectangle in a Cartesian Plane) I very much appreciate your time. I guess this was the answer I was expecting the entire time. Thanks again. |
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