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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.

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### CS 455 – Computer Graphics

Viewing Transformations I

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

Object Space

Model Transformations

World Space

Viewing Transformation

Eye/Camera

Space

Projection & Window

Transformation

Screen Space

• 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

• 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

• 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

• Looking down the y axis:

(x, y, z)

x

z

• So z’ = 0, x’ = x

Parallel Projection

• Looking down the x axis:

y

(x, y, z)

z

So y’ = y

• 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

• 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

• 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

• 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

• 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