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Collision detection & bouncing part 3: bouncing balls Print E-mail
Sunday, 10 September 2006
Article Index
Collision detection & bouncing part 3: bouncing balls
More about the dot product
demo 1: basic bouncing
demo 2: finding the closest wall
demo 3: recursive collision detection
demo4: rethinking the algorithm
demo5: bouncing off sloped walls
demo6: the last bug
Page 2 of 8

More about the dot product

In this article I'll be using the dot product of two vectors a lot. Here's a quick introduction.
The dot product of vector v and vector u is:
v.dot(u) = v.x * u.x + v.y * u.y;

This is not very useful at first sight, but the dot product can also be interpreted as the product of the magnitude of Vector v, the magnitude of Vector u and the cosine of the angle between the 2 vectors: |v| * |u| * cos(a)

Especially the cosine part is interesting: if the angle is > 0° and < 90° the cosine and therefore the dot product will be positive. If the angle > 90 the cosine and the dot product are negative. If the angle is 90° the cosine and the dot product are 0.

You get even more interesting values if you normalize one or both the vectors. Normalizing means that a vector's magnitude is set to 1, but it's direction remains unchanged. If you normalize both vectors, their dot product is the cosine of the angle between them: |v| * |u| * cos(a) = 1 * 1 * cos(a) = cos(a)
So the angle between 2 vectors is the arc-cosine of their dot product.

ImageWhen you normalize only one of the vectors, you get the projection of the unnormalized vector onto the normalized vector.
u.dot(v) = |v| * |u| * cos(a) = |v| * 1 * cos(a) = |v| * cos(a)

For more info on the dot product:
http://en.wikipedia.org/wiki/Dot_product#Geometric_interpretation
http://mathworld.wolfram.com/DotProduct.html

Comments
Its a gr8 tutorial for how can we use vectors instead of trigonometry for motions and collisions
  Posted by poonam sheth, Whose homepage is http://smspoonam.blogspot.com on Wednesday, 11 October 2006 at 10:17

There's a typo on step 3bis:
...the Y-component of Y...
instead of
...the Y-component of v...
but great work ;).
  Posted by Fugus, on Thursday, 19 October 2006 at 12:47

I fixed the typo. Thanx.
  Posted by Johan van Mol, on Thursday, 19 October 2006 at 3:06

Thanks for the great set of tutorials, very helpful!

One issue I've been trying to solve is dealing with walls that the ball can hit from either side. The direction of the normal throws things off. What I did was added an else after the 'if (this.walls[i].n.dot(speedVec) < 0) { ' block in move().

Instead of var vec:Vector = this.walls[i].n.getInverted().cross(this.radius); in the else I have it as var vec:Vector = this.walls[i].n.cross(this.radius);

This seems to work, but I'm worried I've recreated the bugs you addressed in demo4, and I'm guessing there might be a better solution out there.
  Posted by peter, Whose homepage is http://www.thup.com/ on Thursday, 07 December 2006 at 11:17

do you got a tool to convert a drawed wallshape to this wall array that needed?
  Posted by Peter, Whose homepage is http://index.hu on Wednesday, 30 May 2007 at 4:03

First I draw the shape in Illustrator and save the file in Illustrator 8 format. Open the file in a text editor, scroll to the bottom and you'll find something like
(Layer 1) Ln
... more stuff here ...
24 803 m
69 828 l
108 775 l
31 757 l
24 803 l
The numbers in front are the X and Y coordinates. 'm' means moveTo, 'l' means lineTo. I copy and paste these codes, save them as a string in actionscript and parse the string.
  Posted by Johan van Mol, on Wednesday, 30 May 2007 at 5:20

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