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CS 455 – Computer Graphics PowerPoint Presentation

CS 455 – Computer Graphics

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### CS 455 – Computer Graphics

Viewing Transformations I

Motivation

Want to see our “virtual” 3-D world on a 2-D screen

Graphics Pipeline

Object Space

Model Transformations

World Space

Viewing Transformation

Eye/Camera

Space

Projection & Window

Transformation

Screen Space

Viewing Transformations

- Projection: take a point from m dimensions to n dimensions where n < m
- There are essentially two types of viewing transforms:
- Orthographic: parallel projection
- Points project directly onto the view plane
- In eye/camera space (after viewing
tranformation): drop z

- Perspective: convergent projection
- Points project through the origin
onto the view plane

- In eye/camera space (after viewing
tranformation): divide by z

- Points project through the origin

- Orthographic: parallel projection

Parallel Projections

- We will first deal with orthographic projection
- Get the concept and models down
- Projection direction is parallel to projection plane normal
- Center of projection (COP) is at infinity
- Parallel lines remain parallel
- All angles are preserved for faces parallel to the projection plane

p1

p1’

Center of

projection

at infinity

p2

p2’

Projectors

Orthographic Projection

- Points project orthogonally onto (i.e., normal to) the view plane:
- Projection lines are parallel

y

z

x

d

x

z

Projection Environment- We will use a right-handed view system

- The eyepoint or camera position is on the +z axis, a distance d from the origin

- The view direction is parallel to the z axis

- The view plane is in the xy plane and passes through the origin

(x, y, z)

(x’, y’, z’)

x

z

Parallel Projection- A point in 3-space projects onto the viewplane via a projector which is parallel to the z axis

- What is (x’, y’, z’)?

Parallel Projection

- Thus, for parallel, orthographic projections,
- x’ = x, y’ = y, z’ = 0
- So, to perform a parallel projection on an object, we need to multiply it by some matrix that has this effect

What is M?

i.e., we simply

drop the z

coordinate

Perspective Projection

- In the real-world, we see things in perspective:
- Parallel lines do not look parallel
- They converge at some point

x

z

Perspective Projection- Points project through the focal point (e.g., eyepoint) onto the view plane:
- Projection lines are convergent

Perspective Projection

- Center of projection (COP) is no longer at infinity
- Projectors form a view frustum that is a pyramid with the tip at the COP

eye

view plane

Perspective Projection

- We will start with the projection plane parallel to the XY plane and perpendicular to the Z axis
- Lines parallel to the X or Y axis remain parallel
- X and Y distances become shorter as Z becomes more negative, e.g. a cube viewed in perspective:

y

x

Perspective Projection Computation

- Assume the projection plane is normal to the Z axis, located at Z = 0.
- Assume the center of projection (COP, eyepoint) is located at Z = d
- What is P’(x’, y’, z’)?

P(x, y, z)

projection plane

y

P’(x’, y’, z’)

x

Center of Projection

z

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