3D Graphics and Virtual Reality (Example question 2)

a) A point in 2D space can be transformed using translation, scaling and rotation. By default these transformations are performed centred on the origin. Imagine a point positioned at X and Y co-ordinates (1, 1). For each of the following transformations give the new position of the point. Note - do not apply these transformations in succession. In each case begin with the original point.

( 3 marks )

b) A more useful transformation allows us to define the centre point for the transformation, as well as its normal parameters. For example, we could rotate a point by 90 but centre this rotation around the point (1, 1). Briefly describe how you would actually perform this using only the three standard transformations of translation, scaling and rotation.

( 3 marks )

c) A certain operation requires that you both rotate an object by an angle q and scale it by a factor of (Sx, Sy). The transformation matrices for rotation and scaling are shown below, but this operation can be simplified by compounding these two matrices into a single transformation matrix. Calculate the single matrix you would need.

Scaling transformation

Rotation transformation

( 3 marks )

d) Using your new transformation matrix, apply it to the object shown in figure 1 below. The rotation angle q is 90 (positive angle = anti-clockwise) and the scaling factor is (1.0, 0.5). Calculate the new co-ordinates for each vertex of the object (1 to 8) using your transformation matrix, then sketch the new position of the object.

( 5 marks )

Figure 1.

e) When defining a face set to create an object, there are two limiting conditions for each of the faces you define. The first condition is that each face must be planar, mention what the second condition is and briefly describe what each of these means.

( 3 marks )

f) Can the object shown in figure 1 above be represented as a single face? If not, mention why and then list the faces necessary to construct this 2 dimensional object using the vertex numbers in the figure. Bear in mind the two conditions mentioned above.

( 3 marks )

( Total 20 )