Coordinates and Positions

Question 1

Orthographic

Answer 1

Orthographic projection is a form of parallel projection, where every projection line runs orthogonal to the projection plane.

Question 2

Name two vector interpretations of the numeric vector representation (2,3).

Answer 2

A position vector and a direction vector. The numeric representation is the same, but the two interpretations are different objects.

Question 3

Describe the difference between position vectors and direction vectors in their 4-number representations in terms of xyzw.

Answer 3

Position vectors have W=1, direction vectors have W=0.

Question 4

GLKMatrix4MakePerspective

Answer 4

Parameters:
Field of View (Y) in Radians
Aspect (height/width)
Near Z
Far Z
Creates a perspective projection matrix (perspective, not orthographic)

Question 5

Given XYZW, how do you get normalized coordinates?

Answer 5

Divide by W.

Question 6

Describe the projection matrix mathematically.

Answer 6

It is a 4x4 matrix that performs a linear transformation in homogeneous space. It doesn’t actually perform the transformation, but it sets vector positions up for the next step.

Question 7

What are the advantages of homogeneous coordinates?

Answer 7

They allow points at infinity to be represented with finite coordinates (w=0), and they allow all projective transformations (including affine transformations, such as translation) to be performed using matrix multiplications.

Question 8

What is the effect of multiplying a homogeneous coordinate by a scalar?

Answer 8

The coordinate is not changed.

Question 9

Give a 4x4 translation matrix.

Answer 9

1 0 0 0
0 1 0 0
0 0 1 0
tx ty tz 1

Question 10

Give a 4x4 rotation matrix in terms of the original orthonormal basis x, y, z, and x’, y’, z’.

Answer 10

xx' yx' zx' 0
xy' yy' zy' 0
xz' yz' zz'  0
  0   0   0   1
(dot products)

Question 11

What is the dot product of two normal vectors?

Answer 11

Since the normal vectors have magnitude of one, the dot product is simply the cosine of the angle between them.

Question 12

Describe skew or shear in terms of x,y, and z.

Answer 12

Skew adds an offset to one coordinate based on the magnitude of another coordinate. For example, an image could be shifted right at the top, shifted left at the bottom, and not shifted at all in the middle.

Question 13

Give a 4x4 shear matrix skewing x and y in z.

Answer 13

1 0 0 0
0 1 0 0
kx ky 1 0
0 0 0 1

Question 14

Give a 4x4 scale matrix.

Answer 14

a 0 0 0
0 a 0 0
0 0 a 0
0 0 0 1

Question 15

Give a 4x4 projection matrix.

Answer 15

1  0  0  0
0  1  0  0
0  0  0  -1/E
0  0  0  1
where
E = EyeZ, or the distance from the eye to the screen, or near clipping plane. This matrix transforms a vector into :
x  y  0  1-z/EyeZ
So it scales them by that ratio. For example, with an EyeZ of 1 and a z of -10, the transformation of (x,y,z) would be (x/11, y/11, 1). So it brings far-away things into the center of the screen.

Question 16

What direction is x, y and z?

Answer 16

X goes right, Y goes up. The cross product leaves us with Z coming out of the screen.

Question 17

Why are homogeneous coordinates called “homogeneous”?

Answer 17

Because (x,y,z,w) = a(x,y,z,w).

Question 18

Why are triangles used in rendering engines?

Answer 18

Because they are always planar. Triangle strips and fans are also very efficient to render.

Question 19

What are the stages of the rasterization pipeline?

Answer 19

Transformations: Model Space -> Clip Space -> NDC -> Scan Conversion -> Fragment Processing
NDC = Normalized Device Coordinates

Question 20

How are triangles clipped?

Answer 20

Triangles are clipped at the clipping plane by turning them into one or more new triangles depending on how many points are inside of the viewing area.

Question 21

Where does the eye look?

Answer 21

The eye looks in the negative z direction.

Question 22

What is perspective division?

Answer 22

Converting 4D homogeneous coordinates to 3D coordinates by dividing x,y,z by w. This is the final step to convert clip coordinates to NDC.

Question 23

What is a frustum?

Answer 23

A pyramid with the tip chopped off.

Question 24

What are the dimensions of the viewing volume in clip coordinates?

Answer 24

x=[-1,1], y=[-1,1], z=[-1,1], w=[-inf,inf]. Vertices outside of the xyz ranges can be discarded, and their triangles reconfigured.

Question 25

Describe the viewport transformation.

Answer 25

Takes NDC, scales xy to screen height/width in pixels, scales z to [0,1] and calls it depth. xyz becomes xyd.

Question 26

Describe the ModelView matrix in terms of Model and View.

Answer 26

Model refers to a transformation from object coordinates to world coordinates. This can account for the fact that objects are facing different directions, for instance. These are entirely application constructs, OpenGL does not support them directly.

View transforms from world coordinates to camera coordinates.

The two matrices are usually combined.

Question 27

What is the range and direction of Z as a depth buffer?

Answer 27

Z = [0,1], with 0 being near and 1 being far (opposite direction of Z coordinate!).

Question 28

What is scan conversion?

Answer 28

Changing a triangle into fragments using the pixel sample (center-of-pixel coordinate) and texture information.

Question 29

What is the invariance guarantee?

Answer 29

OpenGL guarantees that two polygons with shared edge vertex positions will have no gaps, and also guarantees that the same vertex data passed through the same vertex processor will have identical output.