1. Our Friend the Dot Product
Jim Van Verth

2. Essential Math for Games
So… Why The Dot Product?
Probably the single most important vector operation
The Ginzu™ knife of graphics
It slices, it dices, it has 1001 uses
Understanding is a 3-D sword… So
Define
Utilize
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Background
Assume you know something about vectors
Will be skipping some steps for time
Just follow along
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What is the Dot Product?
One of a class of vector functions known as inner products
Means that it satisfies various axioms
Useful for dot product math
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5. Definition #1: Coordinant-Free
In Euclidean space, define dot product of vectors a & b as
where
|| || == length
 == angle between a & b a  b
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6. Definition #2: Coordinant-Dependant
By using Law of Cosines, can compute coordinate-dependant definition in 3-space:
or 2-space:
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Use #1: Measure Angles
Look at def #1:
Can rearrange to compute angle between vectors:
A little expensive – cheaper tests available
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Use #1: Measure Angles
Rather than determining exact angle, can use sign to classify angle
||a|| and ||b|| are always non-negative, so sign depends on cos , therefore
a • b > 0 if angle < 90°
a • b = 0 if angle = 90° (orthogonal)
a • b < 0 if angle > 90°
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9. Essential Math for Games
Angles: View Testing
Simple view culling
Suppose have view vector v and vector t to object in scene (t = o - e)
If v • t < 0, object behind us, cull e v t o
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Angles: View Testing
Note: doesn’t work well for
large objects
objects close to view plane
Best for AI, not rendering e v t o
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11. Angles: Collision Response
Have normal n (from object A to object B), relative velocity va-vb
Three cases of contact:
Separating
(va-vb) • n < 0
Colliding
(va-vb) • n > 0
Resting
(va-vb) • n = 0
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Angles: View Cone
Test against view cone – store cosine of view angle and compare
If d • L < cos v, cull
Nice for spotlights
Must normalize ^ ^ ^ L v ^ d
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13. Angles: Collinear Test
Test for parallel vectors
If v and w parallel & pointing same direction, v • w ~ 1
If v and w parallel & pointing opposite direction, v • w ~ -1
Has problems w/floating point, though
And you have to normalize ^ ^ ^ ^ ^ ^ ^ ^
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Use #2: Projection
Suppose want to project a onto b
Is part of a pointing along b
Represented as a  b
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Use #2: Projection a
From trig
Now multiply by normalized b, so  b
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Use #2: Projection
So have
If b is already normalized (often the case), then becomes
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17. Projection: OBB Collision
Idea: determine if separating plane between boxes exists
Project box extent onto plane normal, test against projection btwn centers c a b
av
bv
cv
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18. Projection: OBB Collision
To ensure maximum extents, take dot product using only absolute values
Check against axes for both boxes, plus cross products of all axes
See Gottschalk for more details
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Use #2: Projection
Can use this to break a into components parallel and perpendicular to b a b
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20. Projection: Line-Point Distance^
Line defined by point P and vector v
Break vector w = Q – P into w and w||
w|| = (w  v) v
||w||2 = ||w||2 – ||w||||2 ^ ^ Q w w
w|| P ^ v
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21. Projection: Line-Point Distance
Final formula:
If v isn't normalized:
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22. Use #3: Hidden Dot Products
There are dot products everywhere… just need to know to look for them
Examples:
Plane equation
Matrix multiplication
Luminance calculation
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Planes
Defined by
normal n = (a, b, c)
point on plane P0=(x0, y0, z0)
Plane equation
ax+by+cz+d = 0
d=-(ax0 + by0 + cz0)
Dot products!
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Planes
Why a dot product?
Plane is all points P = (x, y, z) passing through P0 and orthogonal to normal
Can express as n P0 P
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Planes
Plane is all points P = (x, y, z) such that
Can rewrite as or
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Planes
Can use plane equation to test locality of point
If n is normalized, gives distance to plane n
ax+by+cz+d > 0
ax+by+cz+d = 0
ax+by+cz+d < 0
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Planes
Project point to plane
Assuming n normalized, n P
v|| v P0 P’
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Planes
Want to mirror point across a plane
Take dot product of vector with normal
Subtract twice from vector n
v|| v
-v||
v - 2v||
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Matrix Product or
In general, element ABij is dot product of row i from A and column j from B
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Matrix Product
So what does this mean?
Beats me… but it’s cool
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Luminance
Can convert RGB color to single ‘luminance’ value by computing
In this case, a projection of sorts
“Luminance” vector not normalized
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32. Use #4: Perpendicular Dot Product
2D cross product, sort of
Perpendicular takes 2D vector and rotates it 90 ccw
Take another vector and dot it, get
Looks like z term from cross product
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33. Perpendicular Dot Product
Result is
So if you want to compute signed angle from v0 to v1, could do
(works for 3D, too, just need to get length of cross product)
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34. Perpendicular Dot Product
Another use: sign of perp dot product indicates turning direction
Have velocity v, direction to target d
If v• d is positive, turn left
If negative, turn right v v d d v v
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Conclusion
Dot product is cool
Dot product has many uses
Use it heavily and often
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References
Van Verth, James M. and Lars Bishop, Essential Mathematics for Games and Interactive Applications, Morgan Kaufmann, 2004.
Anton, Howard and Chris Rorres, Elementary Linear Algebra, 7th Ed, Wiley & Sons, 1994.
Axler, Sheldon, Linear Algebra Done Right, Springer Verlag, 1997.
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