3D Graphics and Virtual Reality (Example answers 2)

Section A ( 3 marks )

Original point location is (1, 1).

    1. translation by (2, -1) gives the new point at (3, 0)
    2. scaling by (2.0, 0.5) gives the new point at (2.0, 0.5)
    3. rotation by 90 clockwise gives the new point at (1, -1)

Section B ( 3 marks )

To rotate an object about a point (Px, Py), You would perform the following sequence of transformations:

This operation could be concatenated into a single transformation.

Section C ( 3 marks )

In order to concatenate these two transformations, we simply multiply the two transformation matrices to gain a single transformation matrix.

Section D ( 5 marks )

Inserting the values for q , Sx and Sy, we get the following transformation matrix:

By multiplying each of the vertices by our transformation matrix we get the following co-ordinates:

Vertex Num.

Original position

New position

1

( 0, 0 )

( 0, 0 )

2

( 0, 3 )

( -3, 0 )

3

( -1, 3 )

( -3, -0.5 )

4

( -1, 4 )

( -4, -0.5 )

5

( 2, 4 )

( -4, 1.0 )

6

( 2, 3 )

( -3, 1.0 )

7

( 1, 3 )

( -3, 0.5 )

8

( 1, 0 )

( 0, 0.5 )

The image below shows the final position of the object:

Section E ( 3 marks )

The second condition is that all faces must be convex, i.e. they should not contain any concave sections.

A face must also be planar, which means it must be completely flat in one dimension.

Section F ( 3 marks )

No, the object can not be represented as a single face because it contains two concave sections created by vertices 2 and 7.

You can split this shape into 2 faces, one defined by points (1, 8, 7 and 2) and the other defined by points ( 3, 6, 5 and 4 ).

NB: there are many more perfectly correct answers to this last question - this is just a single example.

( Total 20 )