>  # Herstart de sessie om alle oude variabelen en instellingen te wissen restart; # Definieer de parametrische vergelijkingen van de trefoil knoop x := t -> sin(t) + 2*sin(2*t); y := t -> cos(t) - 2*cos(2*t); z := t -> -sin(3*t); # Plot de kromme plots:-spacecurve([x(t), y(t), z(t), t = 0..2*Pi], color = "Blue", thickness = 2, axes = boxed); >  # Herstart de sessie om alle oude variabelen en instellingen te wissen restart; # Definieer de kromme als een benoemde functie 'TrefoilKnot' TrefoilKnot := t -> [sin(t) + 2*sin(2*t), cos(t) - 2*cos(2*t), -sin(3*t)]; # Gebruik de functie 'TrefoilKnot' om de kromme te plotten plots:-spacecurve(TrefoilKnot(t), t = 0..2*Pi, color = "Blue", thickness = 2, axes = boxed); restart; with(plots): with(CurveFitting): # Define the Trefoil Knot function TrefoilKnot := t -> [sin(t) + 2*sin(2*t), cos(t) - 2*cos(2*t), -sin(3*t)]; # Generate interpolation points interpolatiepunten := [seq(TrefoilKnot(t), t = evalf(0)..evalf(2*Pi), evalf(Pi/16))]; print("Interpolatiepunten:", interpolatiepunten); # Check the interpolation points # Extract x, y, z coordinates from the points xPoints := [seq(p[1], p = interpolatiepunten)]; yPoints := [seq(p[2], p = interpolatiepunten)]; zPoints := [seq(p[3], p = interpolatiepunten)]; print("x Points:", xPoints); # Check x points print("y Points:", yPoints); # Check y points print("z Points:", zPoints); # Check z points # Generate t-values for the spline interpolation tValues := [seq(i, i = 0..nops(interpolatiepunten)-1)]: print("t Values:", tValues); # Check t values # Create splines for each coordinate xSpline := Spline(tValues, xPoints, t, degree=3); ySpline := Spline(tValues, yPoints, t, degree=3); zSpline := Spline(tValues, zPoints, t, degree=3); print("x Spline:", xSpline); # Check x Spline print("y Spline:", ySpline); # Check y Spline print("z Spline:", zSpline); # Check z Spline # Define the interpolated Trefoil Knot function InterpolatedTrefoilKnot := t -> [xSpline(t), ySpline(t), zSpline(t)]; # Create plots for the original and interpolated knots originalPlot := spacecurve(TrefoilKnot(t), t = 0..2*Pi, color = "Blue", thickness = 2, title = "Original Trefoil Knot"); interpolatedPlot := spacecurve(InterpolatedTrefoilKnot(t), t = 0..nops(interpolatiepunten)-1, color = "Red", thickness = 2, title = "Interpolated Trefoil Knot"); # Display the plots display(Array([originalPlot, interpolatedPlot]), axes = boxed, orientation = [45, 45]); >      >    original curve- interpolation curve- derative curve : take the integral to get the equation ?

Download interpolatiekromme_uitzoeken.mw

I am not quite finished with the procedure yet
In the plot caption :
- line integral notation with /symbolic numerical value 
- parameter curve equation entered
-equation of the projection curve 
 
On how many ways i can program this for storing data ? 

@Carl Love 
Thanks, very instructive this procedure. 

Thanks, looks good.
Try to add some more text to get even a clearer picture.
This procedure is a nice basis to extend to more complicated surfaces f(x,y,z) with a spatial domain curve.
This example was already very instructive.

Thanks, Admit it may be a bit unclear, will try to word it more clearly.

Integral expression = Elliptic expression = float expression 

That can be shown in the plot itself this above output.
Also outside the plot would be useful to show the outcome of the procedure.  

Thanks , It seems that in the procedure there is a provision if a special function occurs in the arc length ( red curve) calculation then a numeric value of this is shown?
Can the special function symbolic expression  just remain with the numeric value behind it in the plot?
 

VisualizeFunctionAndCurve(2*x+y^2, sin(t)+1, 2*cos(t), 0, [0, 1/2*Pi]);

Thanks, yes this definitely became an overhaul of the procedure ( I didn't expect )
I definitely need to study this : how some things have been changed and why.
What is the objection to using a function operator instead of an expression notation ?
This procedure is still for simple surfaces it appears
For more complicated surfaces it becomes f(x,y,z) and the domain becomes a space curve
I still want to try to extend the existing procedure... 

If this should work the VisualizeFunctionAndCurve () procedure, then a further modification for the domain curve  

A plane curve can be represented in two ways: by an equation H(x,y)=0 (the implicit description) and parametrically by a pair of equations with one parameter (the parameter description).

@sija 
Thanks, i will look at it again

Interesting, but from the looks of it, none of the animations work in Maple 2024, or am I mistaken? 

Have revisited the task and have the idea that this elaboration could be better in Maple ? 

Download lijnintegraal_2e_poging_maple_primes.mw

@Carl Love 
I asked the AI for an alternative to the is () command and it didn't come up with a thoughtful answer anyway.
Fine to is ( ) command and apparently there are no alternatives to this?
I find it tricky with that evaluation in Maple with the cauchy-riemann equations.

@janhardo 
Can I also do that check on the cauchy riemann equations via : evalb ?

if evalb(u_x = v_y) and evalb(u_y = -v_x) then
    printf("The function is analytic (holomorphic) at this point.\n");
    printf("The derivative f'(z) is %a + I*%a\n", u_x, v_y);
else
    printf("The function does not satisfy the Cauchy-Riemann equations and is not analytic.\n");
end if

@Carl Love 
Thanks, Yes, that "is" command is essential to establish that the cauchy-riemannt equations are true for all variables

@Axel Vogt 
Thanks, forms a nice basis to rewrite the procedure and the hereby use the D operator instead of diff command.

@acer 
Yes, when to use a branch from a topic?
There are several ways to derive that zeta(2) value. 
I thought this was 1 branch to explore this further and another branch for a different approach to derive zeta( 2).
Otherwise it will be one long thread of proof methods topics , fine by me too and forget the branches.