Our Friend the Dot Product

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Our Friend the Dot Product

• Jim Van Verth (jim_at_essentialmath.com)

2
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

3
Background

• Assume you know something about vectors
• Will be skipping some steps for time
• Just follow along

4
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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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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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 gt 0 if angle lt 90
• a b 0 if angle 90 (orthogonal)
• a b lt 0 if angle gt 90

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Angles View Testing

• Simple view culling
• Suppose have view vector v and vector t to object
  in scene (t o - e)
• If v t lt 0, object behind us, cull

e
v
t
o
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Angles View Testing

• Note doesnt work well for
• large objects
• objects close to view plane
• Best for AI, not rendering

e
v
t
o
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Angles Collision Response

• Have normal n (from object A to object B),
  relative velocity va-vb
• Three cases of contact
• Separating
• (va-vb) n lt 0
• Colliding
• (va-vb) n gt 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 lt cos ?v, cull
• Nice for spotlights
• Must normalize

L
?v

d
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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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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
b?v
a?v
c?v
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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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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 w2 w2

Q
w?
w
w

P
v
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Projection Line-Point Distance

• Final formula
• If v isn't normalized

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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
• axbyczd 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
P
P0
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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
axbyczd gt 0
axbyczd 0
axbyczd lt 0
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Planes

• Project point to plane
• Assuming n normalized,

n
P
v
P0
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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
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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 its 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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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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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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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.