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Problems

  1. Prove that in PZ geometry, every PZ line has an equation of the form ax+by=c, where a, b, c are all rational numbers and a and b are not both zero.

  2. Prove that every equation of the form ax+by=c, where a, b, and c are all rational numbers and a and b are not both zero, is in fact a PZ line.

  3. Prove that if two distinct PZ lines m and n are not parallel in the usual Euclidean sense, then they have a PZ point in common.

  4. In terms of the possible parallel axioms for geometry, which geometry does PZ geometry most resemble: projective, Euclidean, or hyperbolic geometry?

Progress

PZ means Pixel + Zoom geometry. The PZ points are $(x,y)$ of the coordinate plane such that $x$ and $y$ are rational numbers and the $PZ$ lines are the lines of Euclidean geometry which pass through (at least) two $PZ$ points.

I have tried using geometer's sketchpad to come up with PZ lines, but I'm not sure how to start the proof. Should I just use random variables to represent ratios for a and b? How do I show that a and b cannot be zero?

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What is "pixel + zoom geometry"? This very question is the first Google hit -- and the only hit for an exact phrase search. It could be some teacher's quaint personal name for projective plane, except that the question explicitly asks for a comparison between "PZ" and projective ... There are some Google hits for "PZ geometry" but they are about materials science and seem to unfold to "plastic zone" rather than "pixel and zoom". – Henning Makholm Mar 5 '12 at 19:32
    
Pixel + Zoom geometry is what PZ geometry does stand for. I did not mean to list all of the questions. I can clarify that the PZ points are (x,y) of the coordinate plan such that x and y are rational numbers and the PZ lines are the lines of Euclidean geometry which pass through (at least) two PZ points. – Nahida Mar 7 '12 at 22:14
    
I have tried using geometer's sketchpad to come up with PZ lines, but I'm not sure how to start the proof. Should I just use random variables to represent ratios for a and b? How do I show that a and b cannot be zero? – Nahida Mar 7 '12 at 22:17
    
Thanks for the clarification. For future reference, it's better to edit your original post or put more information about a problem in comments than it is to post the new information as a solution. – Brett Frankel Mar 7 '12 at 22:18

Hint for the first question: if $(x_1, y_1)$ and $(x_2,y_2)$ are on the same line, then either:

a) $x_1=x_2$ (what is the equation for the line in this case?)

or

b)this line has the form $y-y_1=\frac{y_1-y_2}{x_1-x_2}(x-x_1)$ And all the numbers in sight are rational. I'll leave it to you to manipulate this equation into the appropriate form.

For the second question: What does it mean, in terms of $a$, $b$, and $c$ for the lines to be parallel? When the lines are not parallel, can you find a common solution to the two equations? Keep track of your steps to show that the common solution has rational coordinates.

The last question is pretty much answered by the second.

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