Download Rotation math foundations

Computer Graphics
Mathematical Fundamentals
Matrix Math
1
2
3
4
5
6
7
8
• Matrix dimensions are specified as:
rows x cols
Matrix Addition
1 2
1 2
2
4
3 4
3 4
6
8
• The dimensions must match to add
Scalar Multiplication
1 2
5 10
3 4
15 20
• The dimensions can be any size to
perform scalar multiplication
Matrix Multiplication
1 2 3
4 5 6
1
4
7
2
5
8
3
6
9
30
36
42
66
81
96
1x2 + 2x5 + 3x8 = 2 + 10 + 24 = 36
• rows from the first matrix
• columns from the second matrix
Matrix Multiplication
• “inner” dimensions must match
• result is “outer” dimension
• Examples:
– 2x3 X 3x3 = 2x3
– 3x4 X 4x5 = 3x5
– 2x3 X 4x3 = cannot multiply
• Question: Does AB = BA?
– Hint: let A be 2x3 and B be 3x2
Matrices and Graphics
• Matrices are used to represent equations
• Used as an efficient way to represent
transformation equations in graphics
– Multiple transformations can be composed
into a single matrix
– Matrix multiplications are often optimized in
hardware
Transformations
• View a complex shape as a set of points
• To transform the shape, one transforms
each of the points that define the shape
• Major types of transformations:
– Translation
– Rotation
– Scaling
Translation
• The equation to translate a single point
(x, y) by tx in the x direction and ty in the
y direction:
x’ = x + tx
y’ = y + ty
Resulting in the new point (x’, y’)
Translation
• These equation can be represented using
matrices:
P=
x
y
P’ =
x’
y’
T=
tx
ty
• Then using matrix math we simply have:
P’ = P + T
Rotation
• The equations to rotate a point through an
angle, q (ccw), around the origin:
x’ = x cos(q) – y sin(q)
y’ = x sin(q) + y cos(q)
• In matrix form:
R = cos(q) -sin(q)
sin(q) cos(q)
• Using matrix math: P’ = R P
Scaling
• The equations to scale a point along the
coordinate axes:
x’ = x sx
y’ = y sy
• Using matrix math: P’ = S P
• Where S =
?
Scaling
• Scaling is used to grow/shrink objects
centered on the origin
• s numbers bigger than 1 grow objects
• s numbers smaller than 1 shrink objects
• uniform scaling: both s numbers the same
Transformation Sequences
• In general you need to perform a
sequence of transformation:
This is a rotation followed by a translation
P’ = R P + T
Do a dimension check to make sure the math
will work out:
– R is 2x2 and P is 2x1, thus R P is 2x1
– R P is 2x1 and T is 2x1, thus P’ is 2x1
Transformation Sequences
• However, it is often more efficient if we can
combine operations into a single matrix
calculation
• How can this be done?
– Here are the equations:
x’ = (x cos(q) – y sin(q) ) + tx
y’ = (x sin(q) + y cos(q) ) + ty
Transformation Sequences
• Put them into matrix form:
x’
cos(q) –sin(q) tx
x
y’
sin(q) cos(q)
y
ty
1
• The only problem is that our original point, P,
now need a 3x1 matrix while the resulting
point, P’, needs a 2x1 matrix
Homogeneous Coordinates
• We resolve this using homogeneous
coordinates
• Lots of theory here, but…
• Basically boils down to all points being
represented as a triple (x, y, 1)
• Thus, we need to fix our transformation
matrix so that P’ comes out as a triple
Homogeneous Coordinates
x’
y’
1
cos(q) -sin(q) tx
sin(q) cos(q) ty
0
0
1
x
y
1
• This 3x3 matrix is called a transformation matrix
– This particular matrix rotates then translates
• However, it is often easier to work with transformations
matrices that handle one type of transformation at a
time, then compose then to obtain our final
transformation
2D Transformation Matrices
• Translation: T(x, y)
1
0
0
• Rotation: R(q)
cos(q)
-sin(q)
0
sin(q)
cos(q)
0
0
0
1
• Scaling: S(x, y)
0
1
0
tx
ty
1
sx 0 0
0 sy 0
0 0 1
Transformation Composition
•
Example: How do you rotate point P
about a random “pivot point”?
•
Note that normal rotation has the origin as a pivot
1. Translate P so pivot point lines up with
the origin
2. Rotate P around origin
3. Translate P back so pivot point is back
on original position
Rotation Around Pivot Point
Rotation Around Pivot Point
• Assume pivot point is at (x, y)
• Assume P is the point you wish to rotate
P’ = T(x, y) R(q) T(-x, -y) P
– Recall that order make a huge difference!
Other Transformations
• Reflection
– about x axis (left)
– about y axis (right)
• Shear
– along x axis (left)
– along y axis (right)
• Identity
1 0 0
0 -1 0
0 0 1
-1 0
0 1
0 0
1 shx 0
1 0 0
shy 1 0
0 0 1
0
1
0
0
0
1
1
0
0
0
1
0
0
0
1
0
0
1
Transformation Between
Coordinate Systems
• Point P is represented in the original coordinate
system. To transform it to the new coordinate
system one needs to know how this system is
oriented and positioned relative to the original
coordinate system.
3D Transformations
• I’ll be using a right-handed coordinate
system
– Positive z-axis is sticking out of the screen
– Dashed lines will be used
for lines that stick into
the screen
3D Transformations
• Points now have 3 coordinates (x, y, z)
• Written in homogeneous coordinates as
(x, y, z, 1)
• Since the points are now 4x1 matrices, the
transformation matrices need to be 4x4
3D Translation
T(x, y, z):
1
0
0
tx
0
1
0
ty
0
0
1
tz
0
0
0
1
3D Rotation
• 3D rotation is a bit harder than in 2D
– In 2D we always rotated about the z-axis
– Now we have a choice of 3 axes, Rx, Ry, Rz
– Rotation is still ccw about the axis
• Looking from the positive half of the axis back
towards the origin (same as it was in 2D)
Rz(q)
• Extending our 2D rotation equations to 3D:
x’ = x cos(q) – y sin(q)
y’ = x sin(q) + y cos(q)
same as in 2D
z’ = z
the height stays the same
Rz(q):
cos(q) -sin(q)
sin(q) cos(q)
0
0
0
0
0
0
1
0
0
0
0
1
Rx(q) and Ry(q)
Rx(q):
Ry(q):
1
0
0
0
0
0
0
cos(q)
sin(q)
0
-sin(q)
cos(q)
0
0
0
1
cos(q)
0
sin(q)
0
0
-sin(q)
0
1
0
0
0
cos(q)
0
0
0
1
Rotation Around a General Axis
1. Translate P so that the rotation axis
passes through the coordinate system
origin
2. Rotate so that the axis of rotation
coincides with one of the coordinate axes
•
In general, requires 2 rotations (about the other 2 axes)
3. Rotate around the axis
4. Perform the reverse of the rotations (2)
5. Perform the reverse of the translation (1)
Rotation Around a General Axis
P’ = T(-a,-b,-c) Rx(-y) Ry(-r) Rz(q) Ry(r) Rx(y) T(a, b, c) P
• Requires 7 matrix multiplications
– Each one requiring 64 multiplies and 48 adds
• For a total of 448 multiplies and 336 adds
– Per point!
– Recall objects sometimes have >100k points
Pre-multiplication of
Transformations
• Recall that matrix multiplication is not
commutative: AB != BA
• However, it is associative: (AB)C = A(BC)
– This implies that you don’t have to perform the matrix
multiplies right-to-left
• Thus, we can pre-multiply the 7 matrices that
don’t change between points to obtain:
A = T(-a,-b,-c) Rx(-y) Ry(-r) Rz(q) Ry(r) Rx(y) T(a, b, c)
• Then we repeat the single multiplication P’= A P
for each of the 100k points
Pre-multiplication of
Transformations
• The old method produced (for 100k points)
44.8M multiplies and 33.6M adds
• The pre-multiplication method produces
approx 6.4M multiplies and 4.8M adds
• Plus, not all the multiplies/adds need to be done
since we are assuming 0 0 0 1 in the last row
• Plus, these multiplies/adds are often done on
optimized graphics hardware
Vectors
• Another important mathematical concept
used in graphics is the Vector
– If P1 = (x1, y1, z1) is the starting point and
P2 = (x2, y2, z2) is the ending point, then
the vector V = (x2 – x1, y2 – y1, z2 – z1)
– This just defines length and direction, but
not position
Vector Projections
• Projection of v onto the x-axis
Vector Projections
• Projection of v onto the xz plane
2D Magnitude and Direction
• The magnitude (length) of a vector:
|V| = sqrt(
2
Vx
+
2
Vy )
• Derived from the Pythagorean theorem
• The direction of a vector:
a = tan-1 (Vy / Vx)
• a.k.a. arctan or atan
• atan vs. atan2
• a is angular displacement
from the x-axis
3D Magnitude and Direction
• 3D magnitude is a simple extension of 2D
2
2
2
|V| = sqrt( Vx + Vy + Vz )
• 3D direction is a bit harder than in 2D
– Need 2 angles to fully describe direction
• Latitude/longitude is a real-world example
– Direction Cosines are often used:
• a, b, and g are the positive angles that the vector makes with
each positive coordinate axes x, y, and z, respectively
cos a = Vx / |V|
cos b = Vy / |V|
cos g = Vz / |V|
Vector Normalization
• “Normalizing” a vector means shrinking or stretching it so
its magnitude is 1
– a.k.a. creating unit vectors
– Note that this does not change the direction
• Normalize by dividing by its magnitude
V = (1, 2, 3)
|V| = sqrt( 12 + 22 + 32) = sqrt(14) = 3.74
Vnorm = V / |V| = (1, 2, 3) / 3.74 =
(1 / 3.74, 2 / 3.74, 3 / 3.74) = (.27, .53, .80)
|Vnorm| = sqrt( .272 + .532 + .802) = sqrt( .9 ) = .95
– Note that the last calculation doesn’t come out to exactly 1. This is because of the
error introduced by using only 2 decimal places in the calculations above.
Vector Addition
• Equation:
V3 = V1 + V2 = (V1x + V2x , V1y + V2y , V1z + V2z)
• Visually:
Vector Subtraction
• Equation:
V3 = V1 - V2 = (V1x - V2x , V1y - V2y , V1z - V2z)
• Visually:
Dot Product
• The dot product of 2 vectors is a scalar
V1 . V2 = (V1x V2x) + (V1y V2y ) + (V1z V2z )
• Or, perhaps more importantly for graphics:
V1 . V2 = |V1| |V2| cos(q)
where q is the angle between the 2 vectors
and q is in the range 0  q  
Dot Product
• Why is dot product important for graphics?
– It is zero iff the 2 vectors are perpendicular
• cos(90) = 0
Dot Product
• The Dot Product computation can be
simplified when it is known that the vectors
are unit vectors
V1 . V2 = cos(q)
because |V1| and |V2| are both 1
– Saves 6 squares, 4 additions, and 2 sqrts
Cross Product
• The cross product of 2 vectors is a vector
V1 x V2 = ( V1y V2z - V1z V2y ,
V1z V2x - V1x V2z ,
V1x V2y - V1y V2x )
– Note that if you are big into linear algebra
there is also a way to do the cross product
calculation using matrices and determinants
Cross Product
• Again, just as with the dot product, there is
a more graphical definition:
V1 x V2 = u |V1| |V2| sin(q)
where q is the angle between the 2 vectors
and q is in the range 0  q  
and u is the unit vector that is perpendicular
to both vectors
– Why u?
• |V1| |V2| sin(q) produces a scalar and the result
needs to be a vector
Cross Product
• The direction of u is determined by the right
hand rule
– The perpendicular definition leaves an ambiguity in
terms of the direction of u
• Note that you can’t take the cross product of 2
vectors that are parallel to each other
– sin(0) = sin(180) = 0  produces the vector (0, 0, 0)
Forming Coordinate Systems
• Cross products are great for forming
coordinate system frames (3 vectors that
are perpendicular to each other) from 2
random vectors
– Cross V1 and V2 to form V3
– V3 is now perpendicular to both V1 and V2
– Cross V2 and V3 to form V4
– V4 is now perpendicular to both V2 and V3
– Then V2, V4, and V3 form your new frame
Forming Coordinate Systems
• V1 and V2 are in the new xy plane