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* writeup.txt   *
* HingOn Miu    *
* hmiu          *
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Implementation decisions:
1.) In detecting collisions between sphere and triangle, we need to check if
    the sphere is within the triangle range before checking the distance from
    the sphere to the triangle. To check whether the projected sphere is
    within the triangle, instead of changing everything to barycentric
    coordinates (which is relatively slow), I simply compare the cross product
    of an edge and the displacement of an interior vertex and the projected
    point. Therefore, if the dot product of the two cross products is positive,
    it means that the projected point is on the same side as an interior
    vertex, which implies that the projected point must be within the triangle.
    Two cross products and one dot product is far more efficient than the
    whole barycentric coordinates calculation. This same side calculation
    method is referenced from "http://www.blackpawn.com/texts/pointinpoly/".
    
2.) Instead of returning the change in position and change in orientation from
    step_position and step_orientation, I decided to change them to return
    the acceleration and angular acceleration. This greatly simplies the
    implmentation of RK4 because after all the forces are applied, I can simply
    call them to get the acceleration and angular acceleration of the newest
    state.
    
3.) Since we can assume that at most one collision happens for each sphere at
    a time (clarified on Piazza), I have a boolean flag named collided to
    keep a record of whether the sphere have collided already. Therefore,
    this optimizes the collision detection process because spheres that
    collided would not be checked for another collision.
    
4.) I have a exert_force function to apply gravity and spring force to a
    sphere. This implmentation helps me to calculate RK4 integration because
    I can use exert_force to apply all forces to calculate the current state
    and clear the force and torque accumulators to prepare calculating the
    next state.
    
5.) The RK4 calculation code is placed in physics.cpp because the step function
    loops through every sphere to update the changes in position, velocity,
    orientation and angular velocity. All the interections among geometries
    are also processed in the step function (collisions lead to velocity
    changes). Most importantly, the spring force needs to be applied for
    each sphere when calculating RK4 integration. Therefore, it is more
    convinient and logical that the RK4 calculation code is placed in
    physics.cpp instead of spherebody.cpp.