babtrices, babogeneous, betc.

quaternions

• who introduced them to graphics?

• what are its components? how many dimensions?

• why are they not susceptible to gimbal lock?

1. ken shoemake introduced to graphics, but they were discovered by william hamilton (important)

2. 4d entities, a single real value and 3 imaginary axes

3. they represent a single, direct rotation in 3d space, not 3 independent axes in a sequence

where is rotation controlled?

it is passed as a uniform into the vertex shader

e.g. uniform mat4 modelview contains all the rotation data

wwhy larry roberts famous

' the interpretation of a matrix as a geometrical operator is the foundation of mathematical transformatiosn in computer graphcis"

what is a primitive? what is a vertex?

a triangle or quadilateral

a 2d or 3d location and any associated attributes (position attribute mandatory)

vector dot product info (5)

dot product code

1. a dot b = axbx+ayby + azbz

2. results in ascalar

3. if a and b are unit length, a dot b = cos theta. if not, cos theta = a dot b / magnitude of a * magnitude of b

4. a dot b = 0 if they are perpendicular

5. for unit vectors, a dot b = 1 if same vector and -1 if opposite direction

float cosTheta = dot(a,b) / length(a) * length(b)

left handed system vs right handed system

in left handed system, positive z goes away from you, in RHS it goes towards you

alternatively, for flat xy plane. z goes down for LHS, up for RHS

in math its usually RHS

vector cross product

cross product code

1. result of 2 3d vectors is another 3d vector

2. computers the surface normal

3. the direction of the resulting vector depends on the handedness of the coordinate system

vec3 N = normalize(cross(p2-p1, p3-p2))

where the 3 poitns of the triangle are p1, p2, p3

gemoetric transformations and their operations (4)

euclidean (rigid body): rotation, translation

linear: rotation, uniform/differential scale

affine: rotation, translation, uniform/differential scale

projective: rotation, translation, scale, perspective

what kind of system do the rotation as complex multiplication equations form? why is this relevant?

a linear system, which means they can be represented by a matrix

what does the 2d scale matrix look like

can a 2x2 matrix represent an affine transformation? (translation)

what matrix can?

no, because a 2x2 matrix can only do multiplication/scale, cannot do addition to input (affine, not linear, transformation)

a 2x3 matrix, with an additional homogeneous coordinate on the input

why are square matrices preferred?

what does this do to the transformation matrices?

what will the transformation matrix now yield?

they are invertible and any vector times a square matrix yields a vector of the same size

since an extra column was added to support translation, an extra row is also added to make it a square matrix

the result will be the same x' and y' plus an additional homogenous coordinate

what do the 2d transformation matrices look like? what by what size?

if you multiply a 3x3 matrix by a 3x1 input matrix, how many dot products is it equivalent to?

3 dot products, one for each row

what do the 3d geometric transformation matrixes look like? what by what size?

4x4 matrices, 3 rotation matrices, one for each axis

• multiple rotations are order dependent

how do you do multiple transformations at once? what is this called?

concatenation

yields a single matrix that is a continuous, linear transformation with a geometric interpolation

e.g. mat4 m = Rotate(xAng, yAng, zAng) * Scale(sx, sy, sz) * Translate(dx, dy, dz);

• rightmost one is the first transformation

process of rotating a point/object about a fixed center

code example

move c to origin, rotate, then move back

vec4 pTransform = Translate(c)RotateZ(θ)Translate(-c)*vec4(p, 1);

• same process for scaling a point about a fixed center; move, scale, move back

what is the general order for matrices? (e.g. row major or column-major)

row major for most, but open gl uses column major

• to convert between the two, transpose the matrix

also random fact most graphics and opengl use right handed coord system but pixar uses left

what is graphical projection

creates a 2d image from a 3d object.

if the image is flat (e.g. your computer screen vs imax curved screen), the projection is planar and straight lines remain straight

perspective projection vs parallel projection

1. one point, finite projector, has perspective and parallel lines may not be parallel

2. orthographic, infinite projector so parallel lines remain parallel

how are points in scene translated to the image plane to achieve perspective?

what is the key axis here?

what is the division by pz known as and how can it be implemented?

1. screen points (sx, sy) are computed via similar triangles

2. the z axis, depth

3. the perspective divide

4. with homogeneous

in computer graphics, how are ordinary 3d points represented?

as 4d homogeneous points

how do you convert an ordinary point into a 4d homogeneous one

how to convert 4d homogeneous point to an ordinary 3d one? why is this significant?

shows that homogeneous points inherently represent division

what is a homogeneous vector? how does it differ from homogeneous point?

(x,y,z,0), a point at infinity

w is 0 isntead of 1

• as w approaches 0, the ordinary 2d point moves in the x,y direction towards infinity

does a transformed homogeneous point/vector remain a homogeneous point/vector?

yes

• for the vectors, they have no translation component

what is the modelview matrix?

it is combination of model transformation (moves object from object space into world space) and view transformation (shifts the world so that camera is at 0,0)

basically takes points from object into eye/camera space, where lighting is calculated

it is for affine transformations, basically combines rotation, scale, and translation into one matrix

• separate from perspective transformation/perspective matrix

what is the view frustum

the volume of 3d space that represents everything be camera can see, sandwiched between the near plane and the far plane and bounded by the top left bottom right planes