My project depends on previous homework because of that I posted all the questions to be clear, but the main project that I need help with is the homework3

Homework 1: Done and working great in the attachments
Programming problem: Interactive Natural Cubic Spline
In mathematics, a spline is a numeric function that is piecewise-defined by polynomial functions, and which possesses a high degree of smoothness at the places where the polynomial pieces connect (which are known as knots).
Cubic splines are the most popular spline functions. They are smooth functions with which to fit data, and when used for interpolation, they do not have the oscillatory behavior that is characteristic of high-degree polynomial interpolation.

The idea behind computing a cubic spline is as follows:

• Let s(x) = a(i) + b(i)*x + c(i)*x^2 + d(i)*x*3 for x(i-1)
• This gives 4*n unknown coefficients. So, we need 4*n constraints to solve the problem.
• First we require (i) s(x(i)) = y(i) for i = 0,1,...,n
• Next we require (ii) s(x(i)+0) = s(x(i)-0); (iii) s'(x(i)+0) = s'(x(i)-0); (iv) s''(x(i)+0) = s''(x(i)-0) for i = 1,...,n-1
• This gives us n+1 constraints from (i) and 3(n-1) constraints from (ii)-(iv). Thus, we have 4n-2 constraints leaving 2 degrees of freedom for choosing the coefficients.

Detail algorithm and pseudo code can be found on http://blog.ivank.net/interpolation-with-cubic-splines.html .

Your programming tasks include:

Interactively define the initial control spline, i.e. end-users will use the mouse to click points on the screen to define the vertices of the cubic spline, and the program should draw the current natural cubic spline every time a vertex is defined. You are also required to use the motion callback function to generate "picking-and-moving" of control vertices Receiving a character from keyboard as a keyboard event. The character invokes the following actions :
a. ‘q' or ‘Q': exit the program

b. ‘+' or ‘-': increase or decrease the interpolation steps between control points

Generate a menu for all of the above keyboard functionalities
a. Keyboard event ‘d' and mouse-picking to delete an existing control vertex
b. Keyboard ‘ ' (space bar) to close the spline as a closed natural cubic spline
Homework 2: not complete and don't works well with me in the attachments also

The programming problem is based on Homework #1. Our goal is now to extend the 2-dimensional spline into full 3-dimensional space, with user interface elements to rotate and scale the 3D graphic scene. Other tasks include using cylinders to replace the original 2D line segments, with colors and materials.

Your programming tasks include:

Extend the original cubic spline algorithms into 3D. Your mouse will be able to pick and move the control points, moving in x-y plane with mouse pointer, and moving in z-direction with the support of a modifier key.
Implement a "rolling-ball" interface to rotate the scene in 3D with mouse. Implement a "zoom-in/out" interface with mouse motion and a modifier key.
Replacing the line segment with cylinders, adding appropriate colors and materials.
When the rolling ball interface receives mouse event with momentum, the rotation will be kept until another mouse click happens.
Homework 3: I need help to achieve this part for now

The programming problem is based on Homework #2. Our goal is now to apply force-directed relaxation algorithm to refine the curve topology.

Two forces are used, an attractive "mechanical" force ( ) applied between adjacent beads on the same component and a repulsive "electrical" force ( ) applied between all other pairs of beads.
Your programming tasks include:

Implement the force model so that your sketched curves can be relaxed automatically to a refined shape.
Adding collision avoidance mechanism: during each iteration of shape refinement, check to determine if moving the bead to its desired location will cause the knot to move into an unsafe position. If so, bring the unsafe components apart to avoid collisions.

Adding a shadow to the 3D scene.

Implement view finding function, so that after each relaxation is done the system will rotate the curves into a position that users see the fewest intersections.
I post all the homework because it is related to each others, but I need help with the assignment 3 for now

Specific requirements: 1. Don't use external libraries just opengl glut

the first homework is a correct code use the same math for hw 3
I upload some of professor demo use it to help you