1. Computer Graphics
One of the central components of three-dimensional graphics has been a basic system that renders objects represented by a set of polygons
One approach to rendering three-dimensional objects is to build a basic renderer then add on enhancements
The basic renderer may be one which incorporates a local reflection model, such as the Phong model, into a Phong incremental shader

2. Computer Graphics Advantages gained by this approach include:
Modeling objects using polygons is straighforward
Piecewise linearities are rendered invisible by the shading technique
Geometric information is only stored at the polygonal vertices - information required for the reflection model that evaluates a shade at each pixel is interpolated from this information
This allows fast, hardware-based shading

3. Computer Graphics
More accurate representations made up of a set of bicubic patches can be converted to a polygon representation and fed to such a renderer
One drawback of using polygons to model objects is that a large number of polygons are required to achieve much detail for complex objects

4. Computer Graphics The main steps in rendering a polygonal object are:
1. Polygons representing an object are extracted from the database and transformed into the world coordinate system using linear transformations such as translation and scaling
2. A scene constructed in this way is transformed into a coordinate system based on a view point or view direction.
3. The polygons are then subjected to a visibility test. This is called ‘backface elimination’ or ‘culling’ and removes those polygons that face away from the viewer.

5. Computer Graphics The main steps in rendering a polygonal object are:
4. Unculled polygons are clipped against a three-dimensional view volume.
5. Clipped polygons are then projected onto a view plane or image plane
6. Projected polygons are then shaded by an incremental shading algorithm. First, the polygon is rasterized, or those pixels that the edges of the polygon contain are determined. Second, a depth for each pixel is evaluated and a hidden surface calculation is performed. Third, the polygon is shaded.

6. Polygonal representation of three-dimensional objects
Objects that possess curved surfaces have these surfaces approximated by polygonal facets
The error introduced by this representation may be visually diminished by using interpolative shading algorithms

7. Polygonal Representation of Curved Objects

8. Polygonal representation of three-dimensional objects
For complex objects, a number of polygons in excess of 100,000 is not uncommon
Another problem occurs when objects are scaled up
An object adequately represented at one size may degrade when the object is enlarged
This has been called ‘geometric aliasing’

9. Polygonal representation of three-dimensional objects
Polygonal representations can be generated manually from a designer’s abstraction, or automatically from real objects by using devices such as laser rangers in conjunction with the appropriate software
Complete information necessary to shade a polygon is usually stored in a hierarchical data structure - objects; surfaces; vertices and normals

10. Coordinate systems and rendering
One view of the geometric part of the rendering process is that it consists of the application of a series of coordinate transformations that takes an object database through a series of coordinate systems
For ease of modeling and application of local transformations, it makes sense to store the vertices of an object with respect to some point conveniently located in or near the the object
This is called the local coordinate system

11. Coordinate systems and rendering
Once an object has been modeled, the next stage is to place it in the scene that we wish to render
The global coordinate system of the scene is known as the world coordinate system
All the objects have to be placed into this common space in order to have their relative spatial relationships defined
The act of placing an object in a scene defines the transformation required to take the object from local space to global space
If this object is being animated, then the animation provides a time-varying transformation that takes the object into world space on a frame-by-frame basis

12. Coordinate systems and rendering
The scene is lit in world space
Light sources are specified
The eye or camera coordinate system is a a space used to establish viewing parameters and view volume
A virtual camera can be positioned anywhere in the world space and can point in any direction
The scene is projected on a view plane

13. Backface elimination and culling
This operation removes entire polygons that face away from the viewer
When dealing with a single convex object, culling completely solves the hidden surface problem
If an object contains a concavity, or if we have multiple objects in a scene, a general hidden surface removal algorithm is needed as well as culling
We can determine whether a polygon is visible from a view point by a simple geometric test
The geometric normal to the polygon is calculated and the angle between this and the line-of-sight vector is determined. (The line-of-sight vector is the vector from the polygon to the view point)
If this angle is greater than 90 degrees, then the polygon is invisible.

14. Screen space
The fundamental transformation that takes us into screen space is the perspective transformation which takes a point in the scene and projects it onto a view plane positioned at disance D away from the view point and oriented normal to the viewing direction

15. Perspective Transformation

16. Screen Space
Screen space is defined to act within a closed volume called the viewing frustum that delineates the volume of space which is to be rendered
Objects that lie outside the viewing frustum are not rendered

17. View Frustum

18. Homogeneous Coordinates
A point in three dimensional space requires 3 coordinates - x, y and z - to describe its location
In homoegeneous coordinates, an extra coordinate (which can be thought of as a scaling term) is added
By using homogeneous coordinates, we are able to represent transformations as matrix multiplications
Then in order to compute the transformation of some object, we need only perform matrix multiplications for the vertices of the polygons making up the object’s representation

19. Homogeneous Coordinates
The transformations we are most interested in are scaling, translation and rotation
These can be defined as follows

20. Scaling Transformation

21. Scaling Transformation

22. Translation Transformation

23. Translation Transformation

24. Rotation About the X-Axis

25. Rotation About the X-Axis

26. Homogeneous Coordinates
Rotations about the y and z axis are similar
A series of operations can be combined by performing a series of multiplications
If the scaling factor is greater than one, the resulting coordinates must be divided by that number at the end of the operation

27. Perspective Transformation
The near plane is the view plane at distance D. The far plane is at distance F
If the height of the view plane is 2h, then the perspective transformation can be given by the following matrix:
0 0 hF/(D(F-D)) h/D
0 0 -hF/(F-D) 0

28. Perspective Transformation
This transformation can simply be concatenated with all of the other transformations in the rendering pipeline
Clipping is performed against the viewing frustum
Objects lying completely outside the frustum are discarded
Objects inside the frustum are transformed into screen space and then rendered
Objects intersecting the frustum are clipped and then transformed into screen space

29. Pixel-Level Processes
Once the objects in a scene have been translated into screen space, the processing is oriented towards pixels rather than polygons
Rasterization, hidden-surface removal and shading are all pixel-level processes

30. Rasterization
Rasterization is the process of finding out which pixels a polygon projects onto in screen space
Calculate, for each scan line within a polygon, by linear interpolation, the x values of the edge pixels, xstart and xend
This results in a span of pixels between two edges that have to be rendered
Care must be taken in these calculations, otherwise we will have aliasing problems (e.g. a straight line may appear to be a series of steps)
Most of these problems are caused by rounding errors in the interpolation of polygon edges

31. Hidden Surface Removal
We now know which pixels contain which objects, however since some pixels may contain two or more objects we must calculate which of these objects is visible and which are hidden
Hidden surface removal is generally accomplished using the Z-buffer algorithm

32. Hidden Surface Removal
In this algorithm, we set aside a two-dimensional array of memory (the Z-buffer) of the same size as the screen (#rows x #columns)
This is in addition to the buffer we will use to store the values of pixels which will be displayed (color values)
The Z-buffer will hold values which are depths (or z-values)
The buffer is initialized so that each element has the value of the far clipping plane (the largest possible z-value after clipping has been performed)
The other buffer is initialized so that each element contains a value which is the background color

33. Hidden Surface Removal
Now for each polygon we have a set of pixel values which that polygon covers
For each one of these pixels, we compare its interpolated depth (z-value) with the value of the corresponding element already stored in the Z-buffer
If this value is less than the previously stored value, the pixel is nearer the viewer than previously encountered pixels
Replace the old value of the Z- buffer with the new, interpolated value and replace the old value of the other buffer with the value (color) of the pixel
Repeat for all polygons in the image

34. Interpolative or Incremental Shading

35. Interpolative or Incremental Shading
A major reason for the growth of popularity of polygon-based renderers is the existence of two interpolative or incremental shading algorithms usually known as Gouraud shading and Phong shading
Gouraud shading is faster than the other method, but cannot produce accurate highlights. It is usually used in applications where speed is important

36. Interpolative or Incremental Shading
Phong shading gives higher quality images, but is more expensive
Gouraud shading calculates intensities at polygon vertices only, using a local reflection model, and interpolates these for pixels within the polygon
Phong shading interpolates vertex normals and applies a local reflection model at each pixel

37. Interpolative or Incremental Shading
The motivation of both schemes is efficiency and the ‘convincing’ rendering of polygonal objects
This means that as well as giving an impression of solidity or three-dimensionality to the model, the shared edges in adjacent polygons that approximate a curved surface, are made invisible

38. Gouraud Shading
Generally, the intensity of light at a polygon vertex is described as a function of the orientation of vertex normals with respect to both the light source and the eye
This function is called a reflection model
Gouraud shading applies a local reflection model at each vertex of a polygon to calculate a set of vertex intensities
These intensities are linearly interpolated across the polygon interior on a scan line basis

39. Phong Shading
In Phong shading, the vertex normals (calculated exactly as in Gouraud shading) are linearly interpolated
The local reflection model is then applied at each pixel using the interpolated normal
This solves the highlight problem and Mach banding, but is more computationally expensive