Mumbai University > Computer Engineering > Sem 4 > Computer Graphics

Marks: 10 Marks

Year: May 2016

Shearing:

A transformation that slants the shape of an object is called the shear transformation. There are two shear transformations X-Shear and Y-Shear. One shifts X coordinates values and other shifts Y coordinate values. However; in both the cases only one coordinate changes its coordinates and other preserves its values. Shearing is also termed as Skewing.

X-Shear:

The X-Shear preserves the Y coordinate and changes are made to X coordinates, which causes the vertical lines to tilt right or left as shown in below figure.

The transformation matrix for X-Shear can be represented as:

Y-Shear:

The Y-Shear preserves the X coordinates and changes the Y coordinates which causes the horizontal lines to transform into lines which slopes up or down as shown in the following figure.

The Y-Shear can be represented in matrix from as:

Viewing Transformation:

One of the most important operations in rendering is the projection of a a three-dimensional scene onto a two-dimensional screen from an arbitrary camera position. A fundamental part of this operation is the specification of a viewing transformation, a $ 4 \times 4$ matrix that transforms a region of space into image space.

1. The camera coordinate system is a coordinate system with the camera at the origin, looking out over the positive z axis.
2. It is, essentially, the scene from the camera's point of view. The raster coordinate system is the space of the pixels on the monitor.
3. Connecting these two coordinate systems there is a special coordinate system known as the screen coordinate system.
4. The screen coordinate system is, conceptually, the same as the film plane of a camera.
5. It is usually best to consider both the screen coordinate system and the raster coordinate system to be two-dimensional, even though we know that RenderMan can output depth information.
6. The RenderMan Interface Specification has a rather complex viewing transformation. The interface has many calls which each set-up a small piece of the transformation.
7. Each of these values has a “reasonable” default, which is to say that if you don't set it, it will default to something which is probably appropriate, given the values that you have already set.
8. The viewing transformation has lots of controls, but typically they are not all used together. Rather, a couple important controls are set and the rest are let to default to their “logical” values.

9. The viewing transformation can be broken down into two pieces, the camera-to-screen projection and the screen-to-raster projection.

   i. The camera-to-screen projection flattens the 3-D world onto the 2-D screen.

   ii. The screen-to-raster projection maps every point on the screen onto some output pixel.

The Viewing Transformation Matrix:

Given the specification of the parameters $ (\alpha,n,f)$, we define a transformation that can be applied to all elements of a scene and takes the truncated viewing volume (bounded by the viewing pyramid and the planes $ z=-n$ and $ z=-f$) to the cube $ -1 \leq u,v,w \leq 1$. This transformation is given by

The transformation $ A _{\alpha, n, f}$ is commonly referred to as the viewing transformation