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Homework Help: Reflecting a circle off another circle

  1. Jul 1, 2010 #1
    1. The problem statement, all variables and given/known data

    I am working on a computer game and I need to correctly reflect a ball off a circle object. I am trying to do it as a line and circle intersection. I know the intersect point of the line (ball path) and the circle. Now I want to rotate the ending point of the ball path about the intersection point to get the correct angle of reflection. The following are known:

    ball current x
    ball current y
    ball end x
    ball end y
    ball radius

    circle center x
    circle center y
    circle radius

    intersection point of ball path and circle x and y

    2. Relevant equations

    I don't know.

    3. The attempt at a solution

    I know I need to find the angle of incidence between the tangent line and the incoming ball path which will also equal my angle of reflection. I think once I know those two angles I can subtract them from 180 to get my rotation angle then rotate my end point about the angle of intersection by that amount. I just don't know how.

    I have attached a pic. Again, I know the two end points of my line segment, the point of intersection and the radius of the circle.

    I am ultimately trying to get the point marked by the open circle, so I need to know the angle between the ball path and the norm, which is the blue line.

    Thanks.
     

    Attached Files:

  2. jcsd
  3. Jul 2, 2010 #2

    LCKurtz

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    Call the starting position of the ball (x0,y0), the bounce point on the circle (a,b), and the end point (x1,y1). The vector from the (a,b) to (x0,y0) is

    [tex]\vec V = \langle x_0 - a, y_0 - b\rangle[/tex]

    The position vector of (a,b) is

    [tex]\vec R = \langle a,b\rangle[/tex]

    and a vector of length one in that direction is

    [tex]\hat r = \frac{\vec R}{| \vec R|}[/tex]

    The orthogonal projection of V on the radius direction is

    [tex]\vec V_{perp} = \vec V - (\vec V \cdot \hat r)\hat r[/tex]

    Then the ending point is:

    [tex]\langle x_1,y_1\rangle = \vec R + \vec V - 2\vec V_{perp}[/tex]
     
  4. Jul 2, 2010 #3

    LCKurtz

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    Since this apparently isn't homework, I will save you a little more work. Here's the coordinates (x1,y1) of the reflection point in terms of the given point and the bounce point on the circle:

    [tex]x_1=\frac{x_0(a^2-b^2)+2aby_0}{a^2+b^2}[/tex]

    [tex]x_2=\frac{y_0(b^2-a^2)+2abx_0}{a^2+b^2}[/tex]
     
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