Hi Mark,

(Side/2)^2 = Side^2/4

The rule applied here is
   (A/B) ^ C = (A^C) / (B^C)

You can see this by replacing exponentiation by repeated multiplication,
for example with C=5: (use an non-proportional font to read this)

   (A/B)^5

     A   A   A   A   A
   = - * - * - * - * -
     B   B   B   B   B

     A*A*A*A*A
   = ---------
     B*B*B*B*B

     A^5
   = ---
     B^5

   = (A^5) / (B^5)

   = A^5/B^5

The last equation is true because "^" has higher priority than "/" (and
"*"). (This is a convention to save some parantheses, not a mathematical
necessity!)

In the trefoil Height formula,
   (Side/2)^2 = Side^2 / 2^2 = Side^2 / 4

so the Height is calculated as follows:
      [law of pythagoras]
   Height^2 + (Side/2)^2 = Side^2
      [subtract (Side/2)^2 on both sides of equation]
   Height^2 = Side^2 - (Side/2)^2
      [power (here: sqare) of quotient as above]
   Height^2 = Side^2 - Side^2 / 4
      ["*1 doesn't change anything; "/ 4" = "* 1/4"]
   Height^2 = Side^2 * 1 - Side^2 * 1/4
      [Side^2 is common factor]
   Height^2 = Side^2 * (1-1/4) = Side^2 * 3/4
      [square root on both sides of equation; sqrt(A*B) = sqrt(A)*sqrt(B)]
   sqrt(Height^2) = sqrt( Side^2 * 3/4 ) = sqrt(Side^2) * sqrt(3) / sqrt(4)
      [sqrt(A^2) = A (when A>=0)]
   Height = Side * sqrt(3)/2


Probably you have noticed that at some places in my explanation post I have
removed square roots from the denominator, because generally root-less
denominators are (aesthetically) preferred. The following examples show
how this can be done. Of course this is not required for POV-Ray's
calculations, I've done it only for humans :)
A, B, ... are arbitrary expressions; if the denominator contains more than
one root, these methods will have to be repeatedly applied.

[case 1: nothing added to the root]

      A            A     * sqrt(C)   A * sqrt(C)    A
   --------- = ------------------- = ----------- = --- * sqrt(C)
   B*sqrt(C)   B*sqrt(C) * sqrt(C)       B*C       B*C

[case 2: denominator is a sum]

         A               A       * (B*sqrt(C)-D)
   ------------- = -----------------------------
   B*sqrt(C) + D   (B*sqrt(C)+D) * (B*sqrt(C)-D)

      [in denominator: (X+Y)*(X-Y) = X*X-X*Y+Y*X-Y*Y = X^2 - Y^2]

      A*B*sqrt(C) - A*D    A*B*sqrt(C) - A*D
   = ------------------- = -----------------
     (B*sqrt(C))^2 - D^2      B^2*C - D^2


How far can you proceed now in understanding the trefoil calculations?


   Sputnik

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