6. Suppose you want to use ray-tracing with Phong shading to make a picture of the surface $z = x^2 - y^2$ on the viewplane $z = 0$. You choose the viewpoint $(0,0,4)$ and the lighting direction s$= (0,0,1)$ (in the direction of the viewpoint). How bright should the pixel at $P = (1,1,0)$ be? (Notice that the surface is $F(x,y,z) = 0$ for $F(x,y,z) = x^2 - y^2 - z$. Be sure to choose the surface normal direction on the side of the surface towards the viewpoint, so you get a positive brightness.)

Brief-answer questions.

(a) Sketch the points that are convex combinations of the vertices of the standard triangle.

(b) List all $3 \times 3$ diagonal matrices that are rotation matrices.

(c) For the relaxed uniform B-spline basis function $f_3(t)$ with $n=4$, find $f_3(2)$.

(d) For the relaxed uniform interpolating spline basis function $f_2(t)$ with $n=4$, find $f _ 2 ^ {\prime \prime}(0)$.

(e) For the projection on the viewplane $x+y+2z = 0$ with viewpoint $pt(0,0,1,0) _ h$, identify the main classification.

(f) Find the area (in absolute value) of the triangle in R$^ 2$ whose vertices are $(8,64)$, $(9,81)$, and $(10,100)$.

(g) Among the line segments $\overline{AB}, \overline{CD}, \overline{EF}$, which pairs cross, if values of affine functions have signs as follows?

function	$A$	$B$	$C$	$D$	$E$	$F$
$f _ {AB}()$	0	0	$-$	$+$	$-$	$+$
$f _ {CD}()$	$+$	$-$	0	0	$-$	$+$
$f _ {EF}()$	$+$	$-$	$+$	$-$	0	0

(h) A certain projective transformation takes the standard unit square to a parallelogram. What is the most you can say about the image of the line at infinity, under this transformation?

(i) Pick four Bézier control points in the answer box at random and indicate how to use the graphical method to calculate $P(\frac 14)$.

(j) For the cubic Bézier curve $P(t)$ in R$^ 2$ with control points $(0,0)$, $(1,10)$, $(2,20)$, $(0,0)$, find $P''(0)$ explicitly.

(k) Write down the matrix needed to turn the perspective projection from $(0,0,5)$ into an orthographic projection, with the viewplane being the $x,y$-plane in both cases.

(l) Write down the matrix needed to turn the oblique projection from the direction v$= (1,2,3)$ into an orthographic projection, with the viewplane being the $x,y$-plane in both cases.

(m) For a relaxed, uniform B-spline curve $B(t)$ with $n=4$, which of the control points $B _ 0,\dots, B _ 4$ affect the value of $B(.23)$?

(n) For the points $A = (2,0)$ and $B = (0,3)$, find $f _ {AB}(x,y)$ in the form $ax + by + c$ with explicit $a,b,c$.

(o) (4 points) Invent an example of a ruled parametric surface (a surface that can be swept out by a moving straight line). Your answer should be some kind of explicit formula for a function $P(t,u)$.

(p) (4 points) For the parametric bilinear patch with data points $P _ {ij}$, $i=0,1$, $j = 0,1$, find the normal vector for the point where $t=0$, $u=0$. A unit normal is not required.