Review on Graphics Basics

Outline

• Polygon rendering pipeline• Affine transformations• Projective transformations• Lighting and shading• From vertices to fragments

Outline

• Polygon rendering pipeline• Affine transformations• Projective transformations• Lighting and shading• From vertices to fragments

Polygon Rendering Pipeline

• Process objects one at a time in the order they are generated by the application

• Pipeline architecture

• All steps can be implemented in hardware on the graphics card

application program display

Vertex Processing

• Much of the work in the pipeline is in converting object representations from one coordinate system to another– Object coordinates– Camera (eye) coordinates– Screen coordinates

• Every change of coordinates is equivalent to a matrix transformation

• Vertex processor also computes vertex colors

Projection

• Projection is the process that combines the 3D viewer with the 3D objects to produce the 2D image– Perspective projections: all projectors meet at the

center of projection– Parallel projection: projectors are parallel, center

of projection is replaced by a direction of projection

Primitive Assembly

Vertices must be collected into geometric objects before clipping and rasterization can take place– Line segments– Polygons– Curves and surfaces

Clipping

Just as a real camera cannot “see” the whole world, the virtual camera can only see part of the world or object space– Objects that are not within this volume are said to

be clipped out of the scene

Rasterization• If an object is not clipped out, the appropriate pixels

in the frame buffer must be assigned colors• Rasterizer produces a set of fragments for each

object• Fragments are “potential pixels”

– Have a location in frame buffer– Color and depth attributes

• Vertex attributes are interpolated over objects by the rasterizer

Fragment Processing

• Fragments are processed to determine the color of the corresponding pixel in the frame buffer

• Colors can be determined by texture mapping or interpolation of vertex colors

• Fragments may be blocked by other fragments closer to the camera – Hidden-surface removal

Outline

• Polygon rendering pipeline• Affine transformations• Projective transformations• Lighting and shading• From vertices to fragments

Affine Transformations

• Line and parallelism preserving• Affine transformations used in Graphics

– Rigid body transformations: rotation, translation– Scaling, shear

• Importance in graphics is that we need only transform endpoints of line segments and let implementation draw line segment between the transformed endpoints

Homogeneous CoordinatesThe homogeneous coordinates form for a three dimensional

point [x y z] is given as

p =[x’ y’ z’ w] T =[wx wy wz w] T

We return to a three dimensional point (for w0) byxx’/wyy’/wzz’/wIf w=0, the representation is that of a vectorNote that homogeneous coordinates replaces points in three

dimensions by lines through the origin in four dimensionsFor w=1, the representation of a point is [x y z 1]

Homogeneous Coordinates and Computer Graphics

• Homogeneous coordinates are key to all computer graphics systems– All standard transformations (rotation, translation,

scaling) can be implemented with matrix multiplications using 4 x 4 matrices

– Hardware pipeline works with 4 dimensional representations

Translation MatrixWe can also express translation using a 4 x 4 matrix T in homogeneous coordinatesp’=Tp where

T = T(dx, dy, dz) =

This form is better for implementation because all affine transformations can be expressed this way and multiple transformations can be concatenated together

1000

d100

d010

d001

z

y

x

Rotation (2D)

Consider rotation about the origin by q degrees– radius stays the same, angle increases by q

x’=x cos q –y sin qy’ = x sin q + y cos q

x = r cos fy = r sin f

x’ = r cos ( + )f qy’ = r sin ( + )f q

Rotation about the z axis

• Rotation about z axis in three dimensions leaves all points with the same z– Equivalent to rotation in two dimensions in planes

of constant z

– or in homogeneous coordinates

p’=Rz(q)p

x’=x cos q –y sin qy’ = x sin q + y cos qz’ =z

Rotation Matrix

1000

0100

00 cossin

00sin cos

R = Rz(q) =

Scaling

1000

000

000

000

z

y

x

s

s

s

S = S(sx, sy, sz) =

x’=sxxy’=syxz’=szx

p’=Sp

Expand or contract along each axis (fixed point of origin)

Reflection

corresponds to negative scale factors

originalsx = -1 sy = 1

sx = -1 sy = -1 sx = 1 sy = -1

Shear

• Helpful to add one more basic transformation• Equivalent to pulling faces in opposite directions

Shear Matrix

Consider simple shear along x axis

x’ = x + y cot qy’ = yz’ = z

1000

0100

0010

00cot 1

H(q) =

Outline

• Polygon rendering pipeline• Affine transformations• Projective transformations• Lighting and shading• From vertices to fragments

Affine vs. Projective Transformation

• Difference is in the last line of the transformation matrix. – Projective transformation preserves collinearity but not

parallelism, length, and angle. – Affine transformation is a special case of the projective

transformation. However, it preserves parallelism.

1000

lkji

hgfe

dcba

Ma = Mp =

1onm

lkji

hgfe

dcba

Projections and Normalization

• The default projection in the eye (camera) frame is orthogonal

• For points within the default view volume

• Most graphics systems use view normalization– All other views are converted to the default view by

transformations that determine the projection matrix– Allows use of the same pipeline for all views

xp = xyp = yzp = 0

Homogeneous Coordinate Representation

xp = xyp = yzp = 0wp = 1

pp = Mp

M =

1000

0000

0010

0001

In practice, we can let M = I and set the z term to zero later

default orthographic projection

Simple Perspective

• Center of projection at the origin• Projection plane z = d, d < 0

Perspective Equations

Consider top and side views

xp =

dz

x

/

dz

x

/yp =

dz

y

/zp = d

Homogeneous Coordinate Form

M =

0/100

0100

0010

0001

d

consider q = Mp where

1

z

y

x

dz

z

y

x

/

q = p =

Perspective Division

• However w 1, so we must divide by w to return from homogeneous coordinates

• This perspective division yields

the desired perspective equations • We will consider the corresponding clipping

volume with the OpenGL functions

xp =

dz

x

/yp =

dz

y

/zp = d

Outline

• Polygon rendering pipeline• Affine transformations• Projective transformations• Lighting and shading• From vertices to fragments

Phong Model

For each light source and each color component, the Phong model can be written as

I = (a+bd+cd2)-1[ kd Id l · n+ks Is (v · r ) a +kaIa]

For each color componentwe add contributions fromall sources

Example

Only differences in these teapots are the parametersin the modifiedPhong model

Outline

• Polygon rendering pipeline• Affine transformations• Projective transformations• Lighting and shading• From vertices to fragments

Pipeline Clipping of Polygons

• Three dimensions: add front and back clippers• Strategy used in SGI Geometry Engine• Small increase in latency

Scan Conversion and Interpolation

span

C1

C3

C2

C5

C4

scan line

C1 C2 C3 specified by glColor or by vertex shadingC4 determined by interpolating between C1 and C2

C5 determined by interpolating between C2 and C3

interpolate between C4 and C5 along span

Hidden Surface Removal

• z-Buffer Algorithm– Use a buffer called the z or depth buffer to store the depth of

the closest object at each pixel found so far– As we render each polygon, compare the depth of each pixel

to depth in z buffer– If less, place shade of pixel in color buffer and update z buffer