(50g) 3D curvature and torsion

[peacecalc]

Hello 50g guys,

after programming for 2D curvature, I extend a program for 3D vectors. A curve in 3D has one parameter, this parameter is named "X", so the CAS command "DERVX" can be used. The program get a vector as input on stack 1 which describe a curve, let say "[ 'X' 'X^2' 'X^3']"

The output on stack 2 is the curvature and for stack 1 the torsion.

Code:


%%HP: T(3)A(R)F(,);
\« DUP DERVX           
   DUPDUP ABS DUP 
   ROT SWAP / 
   EVAL SIMPLIFY 
   DUP DERVX 
   DUPDUP ABS DUP 
   ROT SWAP / 
   EVAL SIMPLIFY \-> V VA VAA T TA TAA N
                          \« TAA VAA / 
                              EVAL SIMPLIFY 
                              T N CROSS 
                              EVAL SIMPLIFY 
                              DERVX NEG N DOT VAA / 
                              EVAL SIMPLIFY
  \»
\»

Don't worry, the program is slow, with input mentioned above you obtain a result after 65 sec...
The results are:

stack 2: \[(\kappa=)\frac{2\cdot\sqrt{81X^8+117X^6+54X^4+13x^2+1}}{81X^8+72X^6+34X^4+8x^2+1​}\]

stack 1: \[(\tau=)\frac{3}{9X^4+9X^2+1}\]

The result for the curvature can simplified to with the help of the CAS command "FACTOR" (used extra for denominator and numerator) to the expression:

\[(\kappa=)\frac{2\cdot\sqrt{9X^4+9X^2+1}}{\left(\sqrt{9X^4+4X^2+1}\right)^3}\]


Feel free and enjoy the little program!
Every constructive critics or suggestions for improvement are welcome.

Greetings peacecalc