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Taxicab Geometry investigation
By Jack on Monday, January 24, 2000 - 07:05 pm:

hi, im doing an investigation entitled 'taxicab geometry', ive attempted it and have done most of the practical part of it. But i cant seem to see the equation behind it all. please guide me in the right direction.

they provide a regular grid with regular horizontal and vertical lines. the lines are roads and the actual squares are blocks of hoouses/ buildings.

in taxicab geometry the distance between place A & B is worked out by adding together the horizontal and vertical distances.

each square is counted as 1 unit.

Part 1 (ive done this)
a taxicab firm is based at A. the position B is 7 units away from A.
investigate all the positions of B at 7 units. investigate for different distances.

i found that all the points when plotted form a diamond shape. the same shape is formed for different amounts of units.

what formula can u get for the positions of the points? i have already found a formula for the number of points there are.

part2
the bus station is X. the railway station is Y. the distance XY=6 units.
investigate to find where the taxicab firm could be sited, so that the same distance from the bus station and the railway station. What does happen if the locations of the bus and the railway station are varied???

part 3
the taxi cab firm decides to site its office so that the distance from the office to the bus station is twice the distance to the railway station. Where could the office be sited? ALL the possible positions and formulas????

Please help, i havent left it to the last minute, but i know already i dont understand!!!

Desperatley Jack

(by the way im Yr 10 UK, (1 year before GCSE)

By Richard Dwight (Rpd25) on Wednesday, January 26, 2000 - 03:13 pm:

Hi Jack,

I think you'll find that once you have the basic idea of how to construct and use the formula things will become clear, so we'll start with the first part where we already know the answer.

If you're going to have a formula that tells you which points on the grid B can be at, you're going to need to label the points on your grid by numbers, as you are doing when you draw a graph. Now you could start your grid with the (0,0) point anywhere and have the point A at (4,5) for example, but thinking ahead, the problem might be simplified if we choose A to be in a "nice" simple place, and the nicest place on the grid is (0,0).

So the points a distance 1 from A are (1,0), (-1,0), (0,1) and (0,-1). I hope you can see how the points on the grid are being labelled by this example. Now we are ready to start considering a formula. The points we are interested in are those a distance 7 away from the (0,0) point, but how can we express this mathematically? Well how about the point (4,5)? You already know that this is a distance 9 from (0,0) since to get to it you must go 4 across and 5 up, 4+5=9. But how about the point (-4,5)? Again you must move 4 then 5 points, so the distance is again 9. With a bit of thought you can see that the sign of the co-ordinates is irrelevant, the minus signs are just ignored when we are calculating the distance. There is a function that is used widely in maths that allows us to write down this ignoring of the minus sign, I'll write it as square brackets, it goes something like [-3]=3, [3]=3, [-5]=5 etc. You get the idea.

Take now, a general point on the grid, call it (x,y) where x and y stand for numbers. The distance of this point from (0,0) is [x]+[y], and our problem requires that this distance be equal to 7, giving the formula [x]+[y]=7.

It may look a bit scary at first with the square brackets but all you have to do to find which points are a distance 7 from A is put in a value of x and solve for y. So to find the points with x co-ordinate 4 that are a distance 7 from x we say:

[4]+[y]=7
so [y]=7-[4]=3
so y=+3 and -3

and our points are (4,3) and (4,-3). By using different x's you can build up a picture of the shape the points form. By the way, what will happen if you try to find points with an x-value of more than 7, and why?

I know that you have done this part already and found the shape to be a diamond, but the method is very powerful, and using the same ideas you can get an equation that tells you the possible positions of the cab company in all the questions in your project. For part 2, there are two points that you start with, so you can only put one of them at (0,0) when you make your grid, so what will be the distance of the general point (x,y) from the other point, if it is (1,5) for example? Well I'll tell you the answer and see if you can see where it came from. The distance from (x,y) to (1,5) is [x-1]+[y-5].

You should get (roughly) straight lines of points for your answer to part 2, but the answers to part 3 are the most interesting of all.

I hope this has helped, as always, please tell me if you have any queries about any of the above and I'll do my best to answer them.
Best wishes,
Richard Dwight.

By The Editor:

If this is a GCSE coursework, you will need to give details of the help you have had. This will not necessarily affect your mark, especially as Richard has given you hints and suggestions rather than answers.