Question 1A.(June 1997)
Consider the drawing of a scene in three-dimensions involving a number of coloured bricks against a white background. The first is a red cube whose vertices are at the positions ( (10,20,0) , (20,20,0), (20,10,0), (10,10,0), (10,20,10), (20,20,10), (20,10,10) and (10,10,10)). The second is a green pyramid whose vertices are at positions ( (15,15,15), (15,35,15), (35,35,15), (35,15,15) and (25, 25,35)). The third is a yellow sphere whose centre is at the position (20,20,-20) and whose radius is of length 10.

(a) Considering a parallel projection from z=100 on to the plane z=0, what output would you expect for this scene if z=0 is an opaque plane? (15%)

Now consider the stereo projection to give the view seen on the plane z=-40 by a viewer at the point (0,0,50) whose eyes are 5 units apart horizontally. What would such a viewer see with each eye? (20%)

(This should take about 50 mins.)

Question 2A.
Three rods (i.e. straight lines) AB, CD, and EF are suspended in a room in such a way that their ends are at the following points. A(-10,10,20), B(10,20,20), C(10,20,20), D(0,10,0), E(-10,10,0) and F(20,10,20). Do they intersect to form a triangle in three-dimensions? (12%)

At one particular time of day, the sun shines into the room parallel to the x-axis and throws a shadow of the rods onto the wall corresponding to the plane x=-50. Calculate the coordinates of the shadows of the rods and determine whether they intersect to form a triangle. (12%)

After dark, a lamp at the point (30,30,30) also throws a shadow onto the same wall. What are the coordinates of the shadow in this case and do they intersect to form a triangle? (11%)

(This should take about 40 mins.)

Question 3A.
Imagine a country with a flat expanse of grey sand reaching as far as the eye can see (e.g. the plane z=0). On this flat plane are built three pyramids, painted in bright colours. The first one (pyramid A) rises from a square base (corners at (20,20,0), (20,40,0), (40,40,0) and (40,20,0)) to a height of 20 units and is deep pink. The second one (pyramid B) rises from a triangular base (corners at (50,40,0),(55,60,0) and (65,42,0)) to a height of 15 units and is emerald green. The third, which is a bright buttercup yellow, rises from a square base (corners at (32,48,0), (32,60,0), (44,60,0) and (44,48,0)) to a height of 12 units.

One day an artist visits the country and decides to paint the pyramids. He positions himself at the point (35,-20,0) and is 6 units tall. He views the projection of the scene on to the plane y=0 in a window just big enough to contain all the pyramids. It is a fine day, with a clear blue sky and no clouds. At midday, the sun is vertically above the scene and so no shadows are cast.

Calculate and draw the scene painted by the artist. (35%)

(This should take about 50 mins.)

Question 4A (January 1998). Are you being observed?

The layout of part of a clothes shop is as shown below. Staff are standing at positions A and B, C, D and E represent racks of clothes and there is a mirror along the wall as shown. The counter is low enough to allow a clear view of the clothes over it. The shop wishes to install security cameras to cover those parts of the clothes racks which cannot be seen by the staff serving at the counter.

Identify those parts which are visible to one or other of the assistants. (20%)

Hence identify the areas which are hidden from them. (5%)

(c) Suggest how many security cameras will be needed and where they should be placed. (10%)

(This should take about 50 mins).

Question 5A
Assume you are implementing a graphical package.

(i) Describe the most usual technique for specifying plane surfaces. State why this is especially important for representing 3D objects in computer graphics.

(ii) You are required to shade a triangular plane surface defined by the vertices A(6,10,-3), B(-2,6,5) and C(8,-2,8). State the equation of the vector N normal to the triangle. Then determine the equation of the plane.

(iii) Define the terms `absolute coordinates' and `relative' or `local coordinates'. What are the advantages of using local coordinates to define graphical objects?

(This should take about 30 minutes.)

Question 6A.
A tetrahedron in three-dimensional space has the vertices at A(0,0,3), B(1,3,2), C(-3,1,2) and D(1,-2,2). Calculate the coordinates of the vertices after a perspective projection on to the plane z=0 from the viewpoint V1(0,0,-10) and sketch the projected view of the tetrahedron.

Now consider a stereoscopic projection, with the view for the left eye drawn in red and that for the right eye drawn in blue. If the observer is at viewpoint V1 and his (or her) eyes are 3 units apart, calculate the new vertices of the tetrahedron for each of the stereo views and draw these in the appropriate colours.

(This should take about 40 minutes.)

Question 7A
A triangle ABC is defined in three-dimensions by the three vertices A(3,4,18), B (1,0,10) and C (-3,6,14).

(i) If a parallel projection is applied to this scene (parallel to the Z-axis and using the XYplane as the view plane) find the projected coordinates and determine the lengths of the sides of the triangle after projection.

(ii) Perform the same calculations for a perspective projection (use the XY plane as the view plane and assume a single vanishing-point at (0, 0, -6) ).

(iii) Describe how the lengths of the sides after projection will vary as the vanishing-point recedes successively further away from the XY-plane towards infinity.

(This should take about 30 minutes.)