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Computer Graphics I > Perspective Transformation > Flashcards

Perspective Transformation Flashcards

1
Q

Planar Projections

A

Reduces a 3D scene to a 2D by interscting rays with a projection plane.
* Types:
*Parallel: Rays are paralle (e.g., orthographic, oblique)
*Perspective: Rays converge at a point for realistic depth.
* Comparison:
*Parallel: Maintains dimensions; eye at infinity.
*Perspective: Distorts for depth; eye at a finite point.
* Uses:
*Parallel: Technical drawings
*Perspective: Realistic 3D visuals

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2
Q

What is the simplest parallel projection that can be made.

A

Primary Views. Projects a 3D scene onto a 2D plane using rays orthogonal to the projection plane along a canonical axis. So on axis vanishes.A

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3
Q

What are the limitations of the primary views?

A
  • No distinction between points in front of or behind the projection plane
  • Depth information is lost.
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4
Q

What are the applications for the primary view?

A

Technical drawings and basic visualizations.

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5
Q

How to project a 3D scene from a direction other than along a canonical axis?

A
  1. Transform the camera space:
    * apply an affine transformation K to align the view direction with a canonical axis.
    * Change of basis matrix T handles rotation and alignment.
  2. Apply primary view projection:
    * Once in camera space, use a standart orthographic projection.
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6
Q

What are isometric projections?

A

A type of parallel projections where the projection plane’s normal makes equal angles with all three coordinate axes, resuling in equal scaling along all axes.
Key Features:
* Equal Angles: Axes are 120 degrees apart in the projection
* Equal Scaling: No distortion; dimensions along x, y and z are preseved proportionally.
* Parallel Lines: Stay parallel; no vanishing points.

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7
Q

What elements are required to completely define a isometric projection?

A
  • View direction p: Specifies where the camera is looking. Determines the projection planes orientation
  • Up vector u: Defines “what is up” in the image. Prevents the camera from tilting unpredictably around p.
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8
Q

What is the transformation Pipeline for an Orthographic Projection

A

Object Space (T) -> World Space (K) -> Camera Space (P) -> Screen Space

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9
Q

When is a projection isometric?

A

When the view direction p = (x, y, z)^T has |x| = |y| = |z|

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10
Q

How do we compute the Matrix K to transfrom from world space into camera space?

A
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11
Q

With which parameters can we describe any orthographic projection?

A

With two angles alpha and beta

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12
Q

What are orthographic projections?

A

Projections where the projection direction is orthogonal to the projection plane.

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13
Q

What are Dimetry Projections?

A

Orthogonal projections where exactly two axes have the same ratio

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14
Q

What are Trimetry Projections?

A

Projections where all three axes have different ratios.

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15
Q

What are oblique Projections?

A

A type of parallel projection where the projection direction is not orthogonal to the projection plane

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16
Q

What is the simple approach to a perspective transformation in 2D?

A
17
Q

What is the problem with the simple approach to a perspective transformation in 2D?

A

When we solve for y_0 and write it as a matrix multiplication we can see that this matrix multiplication is not a linear or affine matrix transformation so we can not tranform from object space to screen space using a single matrix multiplication.

18
Q

What is a projective Space?

A

Given a Vector Space V, the projective space of V called P(V) is the set of lines in V that go through the origin. The dim P(V) = dim V -1.

19
Q

How do i get from an affine space into a projective space?

A
20
Q

How do i get from a projective space into an affine space?

A

Given that w != 0 there a point in affine space for every point in projective space. A consistent choice is:

21
Q

why is this matrix transformation not linear and affine?

A

because the matrix is dependent on x

22
Q

How can we define points at inifinity with homogeneous coordinates and affine/projective spaces?

A
23
Q

How can we use points at infinty to represent vectors?

A

points at infinty respresent vectors

24
Q

point/vector arithmetic rules

A
  • point + point = undefined
  • vector + vector = vector
  • point + vector = point
  • point - point = vector
  • scalar * point = undefined
  • Sum(scalar * vector) = vector
  • Sum(scalar * point) = undefined, or point if Sum(scalar) = 1
25
Q

How can we describe projective transformations?

A

A transformation from P(R^4) to P(R^4) is described by a homogeneous 4x4 matrix. (With the convention that the last entry is a 1)

26
Q

What does it mean that the hyper plane at infinity is not invariant und der projective transformation?

A

That finite points can become infinit and infinit points can become finite.

27
Q

When is a projective transformation affine?

A

If and only if A(H) = H so:

28
Q

In what parts can we decomposite a simple projective transformation now?

A

We can decompose it into a parallel projection and a perspective transformation.

29
Q

properties of projective transfromation: fixed points

A
30
Q

properties of projective transformation: fixed points and parallel lines

A
31
Q

properties of projective transformation: view plane maps to infinity

A
32
Q

properties of projective transformation: staight lines

A

straight lines stay staight lines

33
Q

properties of projective transformation: more parallel lines

A
34
Q

properties of projective transformation: points at infinity map to the vanishing line.

A
35
Q

properties of projective transformation: vanishing point

A
36
Q

general perspective transformation

A

directions parallel to the coordinate axis are mapped to the vanishing points

Computer Graphics I (9 decks)

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