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Topic: split a rectangle. (Read 1234 times) |
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bbskill
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split a rectangle.
« on: Nov 14th, 2007, 11:20pm » |
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1. Cut off an angle from a rectangle. then how to split the rectangle into two parts of equal size using one line.
2. Cut off a rectangle from a rectangle.then how to split the rectangle into two parts of equal size using one line.
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towr
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Re: split a rectangle.
« Reply #1 on: Nov 15th, 2007, 12:05am » |
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The latter is easiest, cut through the center of both rectangles
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bbskill
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Re: split a rectangle.
« Reply #2 on: Nov 15th, 2007, 12:29am » |
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Yes, And I guess that there is something relative in the two cases.
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| « Last Edit: Nov 15th, 2007, 12:29am by bbskill » |
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Ghost Sniper
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Re: split a rectangle.
« Reply #3 on: Nov 15th, 2007, 4:58am » |
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If you cut off an angle from the first rectangle, you wouldn't have a rectangle anymore, so the first question is invalid. 
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bbskill
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Re: split a rectangle.
« Reply #4 on: Nov 15th, 2007, 5:35am » |
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on Nov 15th, 2007, 4:58am, Ghost Sniper wrote:| If you cut off an angle from the first rectangle, you wouldn't have a rectangle anymore, so the first question is invalid. |
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-_-|||. The problem should be splitting the remaining polygon into to 2 parts of equal size.
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Grimbal
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Re: split a rectangle.
« Reply #5 on: Nov 15th, 2007, 7:47am » |
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While we are at it:
3) punch a hole in a rectangle. How to split the rectangle remaining shape in 2 equal parts with a straight cut?
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| « Last Edit: Nov 15th, 2007, 8:45am by Grimbal » |
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bbskill
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Re: split a rectangle.
« Reply #6 on: Nov 15th, 2007, 8:28am » |
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3) cut through the center of the hole and the orignal rectangles again. The same soluction with problem 2.
And I have no idea about problem 1. Any suggestion is appreciated.
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| « Last Edit: Nov 15th, 2007, 8:30am by bbskill » |
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Grimbal
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Re: split a rectangle.
« Reply #7 on: Nov 15th, 2007, 8:51am » |
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For #1, I first thought: cut through the center of the initial rectangle and the center of the missing corner. But indeed, it is not that easy.
To cut the triangle into 2, you need to cut along a line tangent to one of 3 hyperbolas, each hyperbola joining 2 of the corners of the triangles.
[edit]And it is even more tricky than that. I think the curves are not hyperbolas after all, and they don't touch the corners. [/edit]
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| « Last Edit: Nov 16th, 2007, 12:46am by Grimbal » |
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towr
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Re: split a rectangle.
« Reply #8 on: Nov 15th, 2007, 9:35am » |
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on Nov 15th, 2007, 8:51am, Grimbal wrote:| To cut the triangle into 2, you need to cut along a line tangent to one of 3 hyperbolas, each hyperbola joining 2 of the corners of the triangles. |
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That sounds rather complicated; why not just find the center of gravity by crossing the lines from a corner to the midpoint of the opposite side?
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bbskill
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Re: split a rectangle.
« Reply #9 on: Nov 15th, 2007, 10:41am » |
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on Nov 15th, 2007, 9:35am, towr wrote:
That sounds rather complicated; why not just find the center of gravity by crossing the lines from a corner to the midpoint of the opposite side?
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I think finally the problem will become to crossing a given point, draw a line to split a vertical triangle into 2 parts of equal size.
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towr
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Re: split a rectangle.
« Reply #10 on: Nov 15th, 2007, 11:39am » |
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I'm sorry, but I really don't understand what you've just said..
But any straight line through the triangle's center of gravity should split it exactly in two.
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Grimbal
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Re: split a rectangle.
« Reply #11 on: Nov 15th, 2007, 2:12pm » |
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That's what I also assumed, but that intuition turned out to be wrong.
/\
/__\
/\ /\
/__\/__\
/\ /\ /\
/__\/__\/__\
If you cut horizontally through the center of gravity, the top part has 4 triangles, the bottom part 5.
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| « Last Edit: Nov 15th, 2007, 2:14pm by Grimbal » |
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towr
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Re: split a rectangle.
« Reply #12 on: Nov 15th, 2007, 2:16pm » |
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Ah yes, the center of gravity means equal 'moment', not equal 'weight'.. doh!
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| « Last Edit: Nov 15th, 2007, 2:17pm by towr » |
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bbskill
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Re: split a rectangle.
« Reply #13 on: Nov 16th, 2007, 3:02am » |
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on Nov 15th, 2007, 11:39am, towr wrote:I'm sorry, but I really don't understand what you've just said..
But any straight line through the triangle's center of gravity should split it exactly in two. |
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I meaned if the line through the orignal rectangle's center could split the triangle exactly in two, then the line is what we want.
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mikedagr8
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Re: split a rectangle.
« Reply #14 on: Nov 16th, 2007, 3:17am » |
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Does the line always have to be straight, or can it be curved? If it were curved it would be a lot easier, but I feel that it will be deemed as cheating.
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bbskill
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Re: split a rectangle.
« Reply #15 on: Nov 16th, 2007, 9:13am » |
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Of course should be straight. 
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Grimbal
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To finish the remark I did earlier, To cut a triangle in half means to cut it along a line tangent to the red line below. So, the method is to find a line that goes through the center of the rectangle and is tangent to the red line.
I am not sure you can do it geometrically, so it is not really a solution to the problem.
But I mention it because of the interesting fact that the red curve takes the role of a point. Which made me wonder about some exotic geometry, where points are replaced with curves.
If you consider multiple copies of such a curve (all with the same size, shape and orientation), you have the property that for 2 such curves there is a single line that is tangent to both. And for 2 non-parallel lines, there is a single such curve that is tangent to both lines.
After writing all this, I am a bit suspicious that it is really different from standard geometry, and not just a transformation of it. It was just a thought.
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Hippo
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Re: split a rectangle.
« Reply #17 on: Nov 19th, 2007, 11:47pm » |
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What is wrong with towr's solution:
Find rectangle center (diagonal cross), find triangle center (cross of lines going from vertices to edge midpoints). Cut the shape by any line going through both centers.
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towr
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Re: split a rectangle.
« Reply #18 on: Nov 20th, 2007, 12:19am » |
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on Nov 19th, 2007, 11:47pm, Hippo wrote:What is wrong with towr's solution:
Find rectangle center (diagonal cross), find triangle center (cross of lines going from vertices to edge midpoints). Cut the shape by any line going through both centers. |
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Consider reply #11 by Grimbal; a line through the center doesn't necessarily divide the triangle in two parts of equal area.
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Grimbal
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Re: split a rectangle.
« Reply #19 on: Nov 20th, 2007, 12:30am » |
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The gray lines in the picture of my post #16 each cut the triangle in 2 equal areas. As you can see, they don't generally go through the center of the triangle.
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Hippo
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Re: split a rectangle.
« Reply #20 on: Nov 20th, 2007, 12:35am » |
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Oh yes,  thanks ... I should read what is the problem 
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