# Finding How to Transform a Plane to Reflect a Trajectory through a Given Coordinate

So, for context, I am trying to analyze how this basketball machine works: https://www.youtube.com/watch?v=FycDx69px8U

I have a basic understanding of how to calculate the new velocity of the ball after colliding with the backboard (assuming an elastic collision), but how would I approach finding a transform for the plane representing the backboard if I have a coordinate that the resulting trajectory should pass through?

In other words, given an incoming trajectory and a desired resulting trajectory after a collision, how would I find the plane (or normal vector representing that plane) that would create that resulting desired trajectory?

If there isn't a direct mathematical way to solve this, would I just sequence the different planes the trajectory can collide with and find one that fits a given set of parameters?

If we assume that during the collision, the plane only exerts a force on the ball that perpendicular to the plane itself ($$\vec{F} \propto \hat{n}$$), then the impulse delivered $$\int \vec{F} \, dt = \Delta \vec{p} = m \Delta \vec{v}$$ will also perpendicular to the plane. Thus, for a given $$\vec{v}_f$$ and $$\vec{v}_i$$, the normal to the plane $$\hat{n}$$ must point in the same direction as $$\vec{v}_f - \vec{v}_i$$.
• Alright, so I would solve for the plane as follows? $\vec{n} = \frac{(\vec{v}_f - \vec{v}_i)}{(\lVert(\vec{v}_f - \vec{v}_i)\rVert)}$ Where $\vec{n}$ is the normal of the plane. If so, is there an equation to transform the plane to match that normal? Apr 27 at 19:30
• @BrandonMichelsen: Yes, if you want $\hat{n}$ to be a unit normal vector. As far as how to "transform the plane", I'm not entirely sure what you mean by that, but it sounds like more of a question for Mathematics than for this forum. Apr 27 at 19:50
• Got it, I'll ask about the transform there. Also, just to clarify, the method we discussed for finding $\hat{n}$ given $\vec{v}_f$ and $\vec{v}_i$ will work for any trajectory defined by $\vec{v}_i$, correct? In other words, it will work for any angle the trajectory approaches from? Apr 27 at 21:25
• Got it. Now, one last question. If I don't know $\vec{v}_f$ but I know I want the trajectory defined by $\vec{v}_f$ to pass through a given coordinate and I have a starting coordinate, how would I find $\vec{v}_f$? Apr 30 at 6:57