Advanced Game Engineering RealTime Rendering

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Title: Advanced Game Engineering RealTime Rendering

1
Advanced Game EngineeringReal-Time Rendering
Transforms

2
Contents

3
Orthographic Projection

4
Orthographic Projection
5
Orthographic Projection

??
Clearly,p0 is noninvertible, since its
determinant po 0
In order words , The transform drops from three
to two dimensions,
And there is no way to retrieve the dropped
dimension
Problem
A problem with using this kind of orthographic
projection for viewing is that it projects both
points with positive and points with negative
z-values onto the projection plane
?? ??-Useful ways
It is usually useful to restrict the z-values(
and the x- and y- values) to a certain interval
from, say n (near plane) to f (far plane).
This is the purpose of the next transformation.

6
Orthographic Projection

A more common matrix for performing orthographic
projection is expressd in terms of the six
tuple, (l,r,b,t,n,f)
This matrix Essentially scales and translates the
AABB(axis aligned bounding boxing see the
definition in Section 13.2) formed by these
planes into an axis-aligned cube centered around
the origin
They minimum corner of the AABB is (l,b,n) and
the maximum corner (r,t,f)
The axis-aligned cube has a minimum corner
of(-1,-1,-1) and a maximum corner of (1,1,1,) .
This cube is called the canonical view volume(??
?? ??)
the coordinates in this volume are called
normalized device coordinates
The transformation procedure is shown in Figure
3.14

7
Orthographic Projection
8
Orthographic Projection

The reason for transforming into the canonical
view volume is that clipping is more efficiently
performed there, especially in the case of a
hardware implementation
After the transformation into the canonical view
volume, vertices of the geometry to be rendered
are clipped against this cube.
This orthographic transform is shown next page.

9
Orthographic Projection
10
Orthographic Projection

As suggested b the above equation, p0 can be
written as the concatenation of a translation,
T(t), followed by a scaling matrix, S(s), where
s (2/(r-l),2/(t-b),2/(f-n)), and t
(-t1)/2,-(tb)2,-(fn)/20. this matrix in
invertible
Po-1T(-t)S((r-l)/2,(t-b)/2,(f-n)/2)
In computer graphics, a left-hand coordinate
system is most often used after projection
Because the far value is less than the near value
for the way we defined our AABB. The orthographic
transform will always include a mirroring
transform.
The AABBs coordinates are (-1 ,-1, 1) for
(l,b,n) and (1,1,-1)

11
Orthographic Projection

Before page matrix is a mirroring matrix.
the right-handed viewing coordinate -gtleft handed
normalized device coordinates.
Some systems, such as DirectX, also map the
z-depths to the range 0,1
Instead of-1,1.
This can be accomplished by applying a simple
scaling and translation matrix applied after the
orthographic matrix, that is

12
Orthographic Projection

So the orthographic matrix used in DirectX is
Which is normally presented in transposed form.
As DirectX uses a row-major form for writing
matrices

13
Perspective Projection

Perspective projection is used in the majority of
computer graphics applications,
Here, parallel lines are generally not parallel
after projection rather, they may converge to a
single point at their extreme
Perspective more closely matches how we perceive
the world, i.e., object further away are smaller

14
Perspective Projection

Assume that the camera (viewpoint) is located at
the origin, and that we want to project a point
,P, onto the plane z-d, dgt0, yielding a new
point q(qz,qy,-d).
This scenario is depicted in Figure 3.15.

15
Perspective Projection

From the similar triangles shown in this figure,
the following derivation, for the x-component of
q, is obtained
The expressions for the other components of q are
qy-dpy/pz (obtained similarly to qx) , and
qz-d.
Together with the above formula, these give us
the perspective projection matrix, Pp, as shown
here

16
Perspective Projection

That this matrix yields the correct perspective
projection is confirmed by the simple
verification of Equation 3.66
The last step comes from the fact that the whole
vector is divided by the w-component (in this
case pz/d), In order to get a 1 in the last
position.
The resulting z value is always d since we are
projecting onto this plane.
Intuitively ,it is easy to understand why
homogeneous allow for coordinates for projection

17
Perspective Projection

One geometrical transformation, there is also a
perspective transform that, rather than actually
projecting onto a plane (which is noninvertible)
, transforms the view frustum into the canonical
view volume described previously.
Here the view frustum is assumed to start at zn
and end at zf , with 0gtngtf The rectangle at z
n has the minimum corner at (l,b,n) and the
maximum corner at (r,t,n) this shown in Figure
3.16

18
Perspective Projection

19
Perspective Projection

The perspective Transform matrix that transforms
the frustum into a unit cube is given by Equation
3.68
After applying this transform to a point , we
will get another point q (qx,qy,qz,qw)T .
The w-component, qw , of this point will (most
often) be nonzero and not equal to one.

20
Perspective Projection