3D Graphics and Virtual Reality (Example answers 2)
Section A ( 3 marks )
Original point location is (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 )