Often when exploring a new math problem it is beneficial to start out with the theory and then prove it. Not only is this a good place to start, but it gives the reader an understanding of what the math is intended to mean. Often when creating a new math application you start with the theory and build from there testing the hypothesis.
The field of dynamics deals with movement, velocities, and force. As shown in the video_game_curves problem, when objects move in straight lines we can use a sine curve to figure out angles, velocities, distance traveled, and force. We start with creating a sine curve for each member (member meaning each length of straight line). We are interested in determining how one straight line effects the next straight line and so forth until the movement has been described. Then we graph the lengths with it max and min values as if it would revolve in a circle. Once a sine curve has been established for all lengths, the sine curve is then summed to give one sine curve. It is also relevant to find the phase angles between these sine curves.
The part of the theory that needs to be verified is how exactly the length of the lines in rotation effects the over all velocity. And since we know velocity we can find force needed to move a certain distance. See Chord_vs_Circular_Function . This theory is based upon relative velocity (a member such as the given force + its length as it rotates around that given force). It also helps graph and maybe even put into a useable form for the computer a complex relationship between members. That is the result from the many vectors of the members.
So if we know the angular velocity or instantaneous velocity, there is the question, “What is the relationship between a member as it rotates around the reference member + the velocity of the reference member, both, compared to second members position with the combined members?” In other words if the sine curve of the members are known and the instantaneous velocity at a given point is found, “Does the value of the new found instantaneous velocity relate proportionally to the sine curve of members length as they rotate around a circular path?” Simply put is there a relationship between velocity and members position? If there is can we use Chord_vs_Circular_Function to find the unknown sine curve values?
** It is also important to not that the integral of the balls combined, member sine curves equals the total distance the ball travels.
Also it might be interesting to ask, “Is there a coordinate system here?” Specialized coordinate system can be fabricated to fit the needs of the individual math problem. This is nothing new. Imagine a rectangle (pool table), triangle, or any other measurable shape with balls bouncing inside the shape. After many collisions or reflections inside the shape, the path of the ball is changed accordingly. Angles return and give the angles of the shape. That is where the name sonar_coordinates comes from. Just a “hunch.”
Hopefully it is clear what is being attempted to be solved here. I will post updates to better explain and hopefully solve this problem. This is a good group project. If you have read this and want to work on a problem email: firstname.lastname@example.org . Also more math can be found in the math_hunches section of Constructor’s Corner.