point deflecting off of a circle

Question

I know that this is a very simple question, but I am stuck at the very last part of this process and can't find the solution elsewhere (I figured I'd find it on this site, but I didn't see it).

I have an object that is colliding with a circle and I need it to deflect properly, like this:
I know the coordinates of the center of the circle and the object when it is on the circle's perimeter. I know the direction that the object is traveling on contact and can calculate the direction to the center (pointing inwards).

From similar questions, I know that the tangent line is perpendicular to the radius line I calculate. But, I'm not sure where to go after that. I need to calculate the new direction of the object in degrees, but my idea $\theta = \theta + 2(radiusline - \theta)$ , where $radiusline$ is the vector pointing towards the center, is inaccurate.

What is the proper formula for this deflection?

Answer 1

$$\begin{align} \vec N & = \text{normal at point of incidence}=-a \hat{\mathbf{i}}-b \hat{\mathbf{j}} \\ \vec V & = \text{incident vector}=u \hat{\mathbf{i}}+v \hat{\mathbf{j}} \\ \vec R & = \text{reflected vector}=c \hat{\mathbf{i}}+d \hat{\mathbf{j}} \\ \end{align}$$ $$\begin{align} \text{using}\ \vec R =\vec V -2\vec N(\vec V \cdot \vec N)&={u \choose v}-2{-a \choose -b}\left[{u \choose v}\cdot {-a \choose -b}\right]\\ &\\ &={u \choose v}-2(au-bv){-a \choose -b}\\ &={u+2a^2u-2abv \choose v+2a^2u-2b^2v} \equiv {c \choose d}\\ \end{align}$$ Hence $\vec R=(u+2a^2u-2abv)\hat{\mathbf{i}}+(v+2a^2u-2b^2v)\hat{\mathbf{j}}$

Answer 2

Another way of doing this would be to get the tangent that runs through the point of collision and reflect off of this. The tangent is perpendicular to the line OP where O is the center of the circle and P is the point of collision.