restart;

ImplicitPlot3D_withCurveProjection := proc(f, n, x_range, y_range, z_range, x_t, y_t, z_t, t_range, num_vertical_lines)
    local implicit_function, plot_result, space_curve, projection_curve, combined_plot, is_closed, projection_curve_length, space_curve_length, solve_z, t_values, t_increment, points, i, t, t_increment_vert, vertical_lines, x_val, y_val, z_val_space, z_val_projection;
    uses plots, plottools, Student[Calculus1];

    # Basic check on the correct number of arguments and their types
    if nargs <> 10 or not type(f, algebraic) or not type(n, numeric) or
       not type(x_range, range) or not type(y_range, range) or not type(z_range, range) or
       not type(x_t, procedure) or not type(y_t, procedure) or not type(z_t, procedure) or
       not type(t_range, range) or not type(num_vertical_lines, posint) then
        error "Error: Incorrect input. Ensure all parameters are correctly typed and provided in the following order: ImplicitPlot3D_withCurveProjection_withPoints(f, n, x_range, y_range, z_range, x_t, y_t, z_t, t_range, num_vertical_lines). Each parameter must match its expected type and range.";
    end if;

    # Create the implicit expression
    implicit_function := f = n;

    # Generate the 3D plot of the implicit surface
    plot_result := implicitplot3d(implicit_function, x = x_range, y = y_range, z = z_range, axes = boxed, grid = [30,30,30], style = surface, title = sprintf("Implicit plot of %a = %a with space curve and its vertical projection", f, n));

    # Define the parametric space curve
    space_curve := spacecurve([x_t(t), y_t(t), z_t(t)], t = t_range, color = "red", thickness = 2);

    # Solve the surface equation for z if possible
    solve_z := solve(f = n, z);

    # Define the projection curve on the surface, vertical projection to z solved from surface equation
    projection_curve := spacecurve([x_t(t), y_t(t), eval(solve_z, {x = x_t(t), y = y_t(t)})], t = t_range, color = "blue", thickness = 2, linestyle = 2);

    # Check if the curve is closed
    is_closed := evalb(x_t(op(1, t_range)) = x_t(op(2, t_range)) and y_t(op(1, t_range)) = y_t(op(2, t_range)));

    # Calculate the length of the projection curve
    projection_curve_length := evalf(Int(sqrt(diff(x_t(t), t)^2 + diff(y_t(t), t)^2 + (diff(eval(solve_z, {x = x_t(t), y = y_t(t)}), t))^2), t = t_range));

    # Calculate the length of the space curve
    space_curve_length := evalf(Int(sqrt(diff(x_t(t), t)^2 + diff(y_t(t), t)^2 + diff(z_t(t), t)^2), t = t_range));

    # Calculate t values for points
    t_values := [seq(op(1, t_range) + i * (op(2, t_range) - op(1, t_range)) / num_vertical_lines, i = 0 .. num_vertical_lines)];

    # Calculate the increment for t values
    t_increment := (op(2, t_range) - op(1, t_range)) / num_vertical_lines;

    # Calculate points on the space curve
    points := [seq([eval(x_t(t_values[i])), eval(y_t(t_values[i])), eval(z_t(t_values[i]))], i = 1..num_vertical_lines)];

    # Calculate the increment for vertical t values
    t_increment_vert := (op(2, t_range) - op(1, t_range)) / num_vertical_lines;

    # Plot vertical lines from space curve to projection curve
    vertical_lines := [];
    for i from 0 to num_vertical_lines-1 do
        x_val := eval(x_t(op(1, t_range) + i * t_increment_vert));
        y_val := eval(y_t(op(1, t_range) + i * t_increment_vert));
        z_val_space := eval(z_t(op(1, t_range) + i * t_increment_vert));
        z_val_projection := eval(solve_z, {x = x_val, y = y_val});
        vertical_lines := [op(vertical_lines), plottools:-line([x_val, y_val, z_val_space], [x_val, y_val, z_val_projection], color = "green")];
    end do;

    # Combine both plots, space curve, projection curve, and vertical lines
      combined_plot := display({plot_result, space_curve, projection_curve, seq(vertical_lines[i], i = 1..num_vertical_lines)}, axes = boxed);



    # Print the combined plot
    print(combined_plot);

    # Print whether the curve is closed or not
    if is_closed then
        printf("The space curve is closed.\n");
    else
        printf("The space curve is not closed.\n");
    end if;

    # Print the length of the projection curve
    printf("Length of the projection curve (vertical green lines from spacecurve): %a\n", projection_curve_length);

    # Print the length of the space curve
    printf("Length of the space curve: %a\n", space_curve_length);

    # Provide detailed information on how to use the procedure correctly after execution
    printf("Procedure successfully executed. To correctly use this procedure, provide parameters in the following order and format:\n\n");
    printf("f: Algebraic expression for the implicit equation. Example: x^2 + y^2 - 1\n");
    printf("n: Numeric value that the implicit equation equals to. Example: 0\n");
    printf("x_range, y_range, z_range: Valid ranges for x, y, and z coordinates. Example for x and y: -1.5..1.5, and for z: -2..2\n");
    printf("x_t, y_t, z_t: Parametric expressions for the space curve as functions of t. Example: x_t: t -> cos(t), y_t: t -> sin(t), z_t: t -> 0\n");
    printf("t_range: Range for the parameter t. Example: 0..2*Pi\n");
    printf("num_vertical_lines: Number of vertical lines to be plotted.\n");
    printf("Complete example call: ImplicitPlot3D_withCurveProjection_withPoints(x^2 + y^2 - 1, 0, -1.5..1.5, -1.5..1.5, -2..2, t -> cos(t), t -> sin(t), t -> 0, 0..2*Pi, 5)\n");
end proc:

# Voorbeeldaanroep
ImplicitPlot3D_withCurveProjection(x + y + z, 0, -10.5..10.5, -10.5..10.5, -20..20, t -> t, t -> 3*cos(t), t -> 3*sin(t), 0..1*Pi, 5);

The space curve is not closed.
Length of the projection curve (vertical green lines from spacecurve): 7.907090108
Length of the space curve: 9.934588266
Procedure successfully executed. To correctly use this procedure, provide parameters in the following order and format:

f: Algebraic expression for the implicit equation. Example: x^2 + y^2 - 1
n: Numeric value that the implicit equation equals to. Example: 0
x_range, y_range, z_range: Valid ranges for x, y, and z coordinates. Example for x and y: -1.5..1.5, and for z: -2..2
x_t, y_t, z_t: Parametric expressions for the space curve as functions of t. Example: x_t: t -> cos(t), y_t: t -> sin(t), z_t: t -> 0
t_range: Range for the parameter t. Example: 0..2*Pi
num_vertical_lines: Number of vertical lines to be plotted.
Complete example call: ImplicitPlot3D_withCurveProjection_withPoints(x^2 + y^2 - 1, 0, -1.5..1.5, -1.5..1.5, -2..2, t -> cos(t), t -> sin(t), t -> 0, 0..2*Pi, 5)

ImplicitPlot3D_withCurveProjection(x + y + z, 0, -10.5..10.5, -10.5..10.5, -20..20, t -> cos(t), t -> sin(t), t -> 0, 0..2*Pi,8);

The space curve is closed.
Length of the projection curve (vertical green lines from spacecurve): 8.737752571
Length of the space curve: 6.283185307
Procedure successfully executed. To correctly use this procedure, provide parameters in the following order and format:

f: Algebraic expression for the implicit equation. Example: x^2 + y^2 - 1
n: Numeric value that the implicit equation equals to. Example: 0
x_range, y_range, z_range: Valid ranges for x, y, and z coordinates. Example for x and y: -1.5..1.5, and for z: -2..2
x_t, y_t, z_t: Parametric expressions for the space curve as functions of t. Example: x_t: t -> cos(t), y_t: t -> sin(t), z_t: t -> 0
t_range: Range for the parameter t. Example: 0..2*Pi
num_vertical_lines: Number of vertical lines to be plotted.
Complete example call: ImplicitPlot3D_withCurveProjection_withPoints(x^2 + y^2 - 1, 0, -1.5..1.5, -1.5..1.5, -2..2, t -> cos(t), t -> sin(t), t -> 0, 0..2*Pi, 5)

# 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;
 

# Define functions and variables
f := (x, y) -> 2*x + y^2;  # Function f(x, y)
x := t -> 1 + t;           # Function x(t)
y := t -> 2*t;             # Function y(t)
 

# Compute derivatives
dx_dt := D(x);  # Derivative of x with respect to t
dy_dt := D(y);  # Derivative of y with respect to t
 

# Definitie van de functies x(t) en y(t)
x(t) := t + 1;
y(t) := 2*t;

# Definitie van de functie f(x(t), y(t))
f(x, y) := 2*x + y^2;
f_t := unapply(f(x(t), y(t)), t);  # Omzetten van f in termen van t

# Berekenen van de afgeleiden
dx_dt := diff(x(t), t);
dy_dt := diff(y(t), t);

# Definitie van de uitdrukking voor booglengte (ds)
ds := sqrt(dx_dt^2 + dy_dt^2);

# Bereken de booglengte van de kromme van t = 0 tot t = 1
curve_length := Int(f_t(t)*ds, t = 0 .. 1);
evaluated_curve_length := value(Int(f_t(t)*ds, t = 0 .. 1));

# Output van de inerte en geëvalueerde vorm van de booglengte
curve_length = evaluated_curve_length;

# Plotting
with(plots):
    surface := plot3d(f(x, y), x = 0..3, y = 0..4, axes=boxed, labels=['x', 'y', 'z'], style=surfacecontour):  # 3D surface plot of the function f
    curve := spacecurve([x(t), y(t), f(x(t), y(t))], t = 0..1, color=red, thickness=2):  # Parametric space curve
    line_xy := spacecurve([[1, 0, 0], [2, 2, 0]], color=black, thickness=2, linestyle=2):  # Line in the xy-plane from (1,0) to (2,2)
    vertical_line1 := spacecurve([[1, 0, 0], [1, 0, f(1,0)]], color=blue, thickness=2):  # Vertical line from the xy-plane to the spacecurve at x=(1,0)
    vertical_line2 := spacecurve([[x(0.5), y(0.5), 0], [x(0.5), y(0.5), f(x(0.5), y(0.5))]], color=blue, thickness=2):  # Vertical line at t=0.5
    vertical_line3 := spacecurve([[x(1), y(1), 0], [x(1), y(1), f(x(1), y(1))]], color=blue, thickness=2):  # Vertical line at t=1

   

 # Display all plots together in a single visualization
    display(surface, curve, line_xy, vertical_line1, vertical_line2, vertical_line3, title="3D Plot of f(x,y) = 2x + y^2 with Curve and Vertical Lines");