Week 9 Solution
Question 1
(a) If we break up the equation of the line we find that:
Subbing into the equation given for a plane, we can easily show that:
and it’s not hard to rearrange the above equation to prove the statement.
(b) First we have to work out the parametric equation for the line. In general, the line between p0 and p1 can parametrically be represented as: p0 + (p1-p0)t. In this case, p0=(0,0,-2) and p1=(0,1,0). So the equation of the line through the point is (0,0,-2)+(0,1,2)t.
We can now substitute into the equation from part (a) and determine that it intersects with plane A when:
and the point of intersection is:
(0,0,-2) + 2(0,1,2) = (0,2,2)
You should check that this point is on Plane A.
For plane B:
and the point of intersection is:
(0,0,-2) + 3(0,1,2) = (0,3,4)
You should check that the above point is on plane B.
Hence since plane A has the lower t value, it will intersect with plane A first and not with plane B.
(c) Since the point of intersection is (0,2,2) and the light source is at (10,10,10), then the parametric equation is (0,2,2) + (10,8,8)t. The only thing that can obstruct it is plane B. If the above line intersects with plane B and its t value is between 0 and 1, we are in shadow (as it means point B lies between the point and the light source). Working this out:
And hence we are in shadow!