# Deflection formula doesn't work when angle is 0

I am trying to make objects bounce off of a circle in a realistic direction. The equation is $new = \theta - 2N * \theta N$
Where $\theta$ is the object angle in degrees,
$new$ is the new object angle in degrees,
and $N$ is a vector pointing from the circle origin to the object.

But if $\theta$ is 0 then as you can see here it doesn't work:

The object direction and normal are 0:
new direction = 0 - (2*0) * (0*0) = 0 - 360 * 0 = 0 (the direction did not change)

My question is, how can I modify this formula without making a special case when the object direction is 0$^{\circ}$?

• The normal is not $180^o$, it's $0^o$. – K. Rmth Dec 14 '13 at 19:00
• Thank you for your response. I have fixed my error. – person27 Dec 15 '13 at 17:58
• Could you label a figure of an example that does work, clearly labeling the angle $\theta$ so we can see what is meant by "the object angle"? Also, if the formula is $\theta - 2 N * \theta N$, and N is a vector, why do you write 0 - (2*0) * (0*0) while evaluating a new direction? It looks like you are substituting the scalar value 0 for the vector N, which makes no sense. – David K Apr 4 '14 at 1:37
• @DavidK I knew a lot less back then than I do now regarding trigonometry (I didn't even know what radians were). I answered my question below; the formula was wrong. I apologize for not revisiting this question earlier, I suppose I had forgotten about it. – person27 Apr 5 '14 at 23:05
• I should pay more attention to the dates on postings. But I'm glad it worked out for you. Your diagram is crystal-clear. – David K Apr 6 '14 at 2:36

Nevermind this question; I had the formula wrong. The equation is $V_{fd} = N_d + (N_d - V_{id})$
Where $V_{id}$ is the initial vector's direction in radians,
$V_{fd}$ is the final vector's direction in radians,
and $N_d$ is the normal vector's direction $\perp$ to the tangent and pointing outwards.

At the point of impact, the initial vector that describes the object's movement changes direction by a certain number of radians. As you can see, the difference in radians between $V_{id}$ and $V_{fd}$ is $2\theta$ because it is the sum of angles formed by $\angle NF_{id}$ and $\angle NF_{fd}$, which are always equal.

The equation now works when $\theta$ is $0$:

If $V_{id} = \pi$, then $0 + (0 - \pi) = -\pi$.