04 - Transformations and Homogeneous Coordinates Flashcards by Deleted Deleted

What are the “Transformation Groups” (5)

Euclidean Maps
Similarity Transformations
Linear Maps
Affine Transformations
Projective Transformations

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What are “Euclidean Maps” + examples (3)

preserve distances
preserve volume
preserve angles

identity
rotation
translation

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What are “Similarity Transformations” + examples (4)

preserve angles

identity
rotation
translation
isotropic scaling

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What are “Linear Maps” + examples (6)

can be achieved by matrix multiplication
linear map of sum of two elements -> distributable ( T(p + q) = T(p) + T(q) )
scalar can be outside ( T(ap) = a * T(p) )

identity
rotation
isotropic scaling
scaling
mirroring
shearing

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What are “Affine Transformations” + examples (7)

parallel lines are preserved

identity
rotation
translation
isotropic scaling
scaling
mirroring
shearing

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What are “Projective Transformations” + examples (7)

straight lines are mapped to straight lines

identity
rotation
translation
isotropic scaling
scaling
mirroring
shearing

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What does a “Scaling” matrix look like

sx 0
0 sy

if sx = sy => isotropic

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What does a “Shearing” matrix look like

1 s
0 1
=> horizontal shear

1 0
s 1
=> vertical shear

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What does a “Rotation” matrix look like

cos𝜙 -sin𝜙
sin𝜙 cos𝜙
=> 𝜙 counterclockwise angle

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What does a “Mirroring” matrix look like

-1 0
0 1
=> x-axis mirroring

1 0
0 -1
=> y-axis mirroring

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What are “Compound Transformations”

multiple transformations
matrix multiplication
p’ = CBAp => right to left transformations

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How can one “Change Between Coordinate Systems”

INTO coordinate system u, v
ux uy
vx vy
Ruv * pxy = puv
FROM coordinate system u, v
ux vx
uy vy
=> transpose of before
Ruv^T * puv = pxy

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What are the “Rotation Matrices for 3D”

Rx(𝜙)
1 0 0
0 cos𝜙 -sin𝜙
0 sin𝜙 cos𝜙

Ry(𝜙)
cos𝜙 0 sin𝜙
0 1 0
-sin𝜙 0 cos𝜙

Rz(𝜙)
cos𝜙 -sin𝜙 0
sin𝜙 cos𝜙 0
0 0 1

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What is the principle of “Euler Rotations”

rotation represented by 3 rotations about principle axis
z, x, z
z, y, z
x, y, z

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What are “Affine Mappings”? What are the properties (4)?

combination of linear transformation and translation
+> x -> Ax + b
straight lines mapped to straight lines
parallel lines stay parallel
preservation of division ratio
NO preservation of angles

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What are advantages of “Homogeneous Coordinates”

include points at infinity
affine mappings as matrix

How are “Homogeneous Coordinates” build

include one more dimension z
other values / z must result in original value
=> (x, y) -> (xz, yz, z)
if z = 0 => direction
else => point

How do “Affine Mappings” work in “Homogeneous Coordinates” compared to “Euclidian Coordinates”

Ax + b

a11 a12 b1
a21 a22 b2
0 0 1

What does a “Translation Matrix” look like in “Homogeneous Coordinates”

1 0 Δx
0 1 Δy
0 0 1

How does one rotate around a point with “Homogeneous Coordinates”

move point to origin = A
1 0 -cx
0 1 -cy
0 0 1
rotate = B
cos𝜙 -sin𝜙 0
sin𝜙 cos𝜙 0
0 0 1
move point back from origin = C
1 0 cx
0 1 cy
0 0 1

=> p’ = CBAp

How to calculate the coordinates of a point in different coordinate systems with “Homogeneous Coordinates”

pu = ux uy 0 * 1 0 -ex * px
pv = vx vy 0 0 1 -ey py
1 = 0 0 1 0 0 1 1

px = 1 0 ex * ux uy 0 * pu
py = 0 1 ey vx vy 0 pv
1 = 0 0 1 0 0 1 1

How to scale with “Homogeneous Coordinates”

sx 0 0 0
0 sy 0 0
0 0 sz 0
0 0 0 1

How to shear with “Homogeneous Coordinates” (z unchanged)

1 0 dx 0
0 1 dy 0
0 0 1 0
0 0 0 1

How to rotate with “Homogeneous Coordinates”

Like normal but with 4ht dimension at 1

What are the "Coordinate Systems in Computer Graphics" and what are their translations called

- Object Coordinates => model transformation - World Coordinates => camera transformation - Camera Coordinates => projection transformation - Clip Coordinates => normalization transformation - Normalized Device Coordinates => viewport transformation - Window Coordinates

What is "Hierarchical Modeling"

- generate copies to create scene - create scene graph (directed and acyclic) - utilize matrix stack

What's the problem with "Transformation of Normals"? How do you fix that?

- some transformations don't preserve angles => instead transform tangent-plane => normal transformation by multiplication with "Inverse Transpose"

How do you perform an "Intersection Test" with a "Transformed Object"? What problem occurs here?

- problem: implicit representation of transformed object - transform ray into object coordinates -> eos = M^-1 * ews -> dos = M^-1 * dws - problem: directional vector dos might not be normalized (if M contains scaling) - two options: 1. normalize dos -> dos / ||dos|| -> tos \= tws 2. don't normalize -> tos = tws -> tos not true distance (because in different coordinate system)

How to perform "Projective Mappings" with "Homogeneous Coordinates"? (map onto z = 1 plane)

example: map onto z = 1 plane 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0

What is a "View Frustum" / "View Volume"

- cuboid [l; r] x [b; t] x [n; f] -> n>f (because view in negative direction)

How does "Orthographic Projection" work

- map "View Frustum" to unit cube [-1; 1]^3 - general matrix: 2/(r-l) 0 0 -(r+l)/(r-l) 0 2/(t-b) 0 -(t+b)/(t-b) 0 0 2/(n-f) -(n+f)/(n+f) 0 0 0 1

How does "Perspective Projection" work? What is a problem of this projection and how can it be solved?

- includes distance / orientation of camera to object - includes zoom - projection onto image plane with z=0 - because of triangle similarity: x'/D = x/(z+D) => x' = x/(z/D + 1) = (xh wh)^T same for y z is 0 => no need for projection BUT we need to know order of objects solution: map z to projection plane Overall: (1 0 0) (x) (0 1 0) (z) (0 1/D 1) (1) => shear in w-direction - de-homogenize = "Normalization Transformation" => unit cube

How does the z value of the "Perspective Projection" effect z'

- large z result in a small resolution between z-values -> low precision - small z result in negative z' (until z behind camera) => solution: "View Frustum" with near an far plane => use full resolution in this frustum, clip everything else

What is the "Normalization Transformation"? What's special about this?

- de-homogenization after projection transformation - camera coordinates: looking along -z - after de-homogenization: z now pointing in -z direction

Why do we need 3 x 4 entries to unambiguously define an "Affine 3D Transformation"

- linear part (3 x 3) - translation part (3 x 1)

Why do we need 3 x 3 - 1 entries to define an "2D Projection"? How do you define the "Projection Matrix"

- homogeneous coordinates are scale invariant (bottom right value is free) 1. camera position mapped to -∞ M (0, 0, 1) = k1 (0 -1 0) 2. x-direction preserved M (1, 0, 0) = k2 (1, 0, 0) 3. 4. Two additional points (not 3 on a line) M (0, -n, 1) = k3 (0, -1, 1) M (f, -f, 1) = k4 (1, 1, 1) => 12 equations (3 equations for each point) => 4 + 8 unknowns