Distance formula without using sqrt?

Question

the distance of line basically uses pythagorean theorm. is there a way to solve this without using the sqrt function?

d = sqrt (dX^2 + dY^2)

Answer 1

Sure, you could find the length of the hypotenuse of a right triangle without using a square root... if you're up on your trig functions.

d = (Δy) / { sin [ Atan (Δy / Δx) ] } is one way to do it.

Using a square root with the Pythagorean Theorem seems easier to me, though.

Answer 2

Essentially, you can't get around the square root.

You can see that as follows. Suppose dX and dY are rational numbers (fractions of integers). Any number you can make by adding, subtraction, multiplying and dividing using dX and dY will be fractions of integers again. In order to "escape" the rational numbers, you will need more advanced functions such as sqrt.

Now consider a few simple situations:
dX = 1, dY = 1 --> d = sqrt 2
dX = 1, dY = 2 --> d = sqrt 5
The numbers sqrt 2, sqrt 5, etc. are not rational numbers. They can never be formed by adding, subtraction, multiplying or dividing integers.

Answer 3

It seems your question is not properly formated.

If you need to know the length of shortest distance between two points the, the formula you have is correct.

Also to know it, you can use any trig ratio, and one dx oy dy

Answer 4

yes for instance,
transform the coordinates so that the line is horizontal or vertical

translate you line so that starting point is in origin,. gives an endpoint (Px,Py)

if Px==0 done Py is the distance
if Py = 0 done Px is the distance
else phi = arctg(Py/Px)
rotate the line around origin over phi degrees, this can be done by multiplying the vector (Px,Py) with a rotation-matrix of phi degree, this gives you the point (0,Q)
Q is the length .

Answer 5

you need the sqrt if you want the euclidean distance in R^2.
if your distance metric was say the manhatten metric, then the distance would be dX + dY

Answer 6

Sure, just rotate your coordinates so that dY is always zero.

then d = dX, no square root.

Answer 7

YOu can use different coordinates.
Fro instance: distance between two points A(r1, p1), B(r2, p2) on polar coordinates:

d = sqrt { r1^2 + r2^2 - r1 * r2 * cos(p1 - p2) }

...sorry, again there's a root.

Answer 8

Depends on what line you're talking about. If you're talking about the longest line of a right triangle, then I don't think there is. Couldn't you measure the line?

Answer 9

d^2 = dx^2 + dy^2.

Bada bing!

Answer 10

dX^2=dX^2 + dY^2