Given the relationship between two coordinate systems (x,y,z) and (s,t,v) as

x = 2 s t , y = s² - t² z = v

Solving the first two equations, we get:

s = x/2t => t² = x²/4t² - y => 4 t⁴ + 4 t² y - x² = 0

=> t² = [ - y +- √(y² + x²)] / 2 = **(√(x²+y²) - y) / 2 --- (1)**

Then, ** s² = y + t² = ****(√(x²+y²) + y) / 2 ** ------- (2)

This is how we calculate the

we differentiate them

dx = 2 t ds + 2 s dt ---- (3)

dy = 2 s ds - 2 t dt ---- (4)

dz = dv

Infinitesimally small distance dS in (x,y,z) is :

Solving the first two equations:

-dx 2t 2s -dx

-dy 2s -2t -dy

ds = (2 t dx + 2 s dy) / (4s² + 4t²) = (t dx + s dy) /2(s²+t²)

dt = (-2t dy +2s dx) / (4s² +4t²) = (s dx - t dy)/ 2(s²+t²)

Infinitely small distance in (s,t,v) system is = dS':

(dS')² = (ds)² + (dt)² + (dv)²

= dv² + [ t²dx²+s²dy²+2st dx dy + s²dx²+t² dy-2st dx dy] /4(t²+s²)²

= (dv)² + [ (dx)² + (dy)² ] / [4 (s²+t²)]

From (1) and (2) we have s² + t² = √(x²+y²)

So the scaling factors for linear distances from (x,y,z) to (s,t,v) are given by :

ds = 1/ √ √ [4√(x²+y²) ] * dx

dt = 1/ √ √ [4√(x²+y²) ] * dy

dv = 1 * dz

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*Expression for square of the arc length is = (dS)²*

= (dx)² + (dy)² + (dz)² use equations (3) and (4)

= (dv)² + 4 t² (ds)² + 4s² dt² + 8 t s ds dt + 4 s² ds² + 4 t² dt² - 8 s t ds dt

** (dS)² ** = **(dv)² + 4 [ (ds)² + (dt)² ] [s² + t²] ------ (5)**

This is the expression for arc length dS. Then Integration has to be done for obtaining the length of an arc.

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__EXAMPLE:__

Suppose in 3-dim. x, y, z space we have a complicated curve and we have mapped that to s,t,v vector space. Let us say that the curve simplifies in (s, t, v) space to a straight line segment OP, with O(0,0,0) and P(1,1,1). Its parametric form is:

s = w t = w and v = w

ds/dw = dt/dw = dv/dw = 1

Using equation 5 we get an expression the the length of the curve in (x,y,z) vector space:

(dS/dw)² = (dv/dw)² + 4 (s²+t²) [ (ds/dw)² + (dt/dw)² ]

= 1 + 4 * 2 w² * (1 + 1) = 1 + 16 w²

dS = √(1 + 16 w²) dw

Integrate wrt w from limits 0 to w and then substitute w = 1 to get the arc length in (x,y,z) space.