8. Problems

Problem O-1. Give the main classification of the projections used to make

(a) the picture of the earth in Figure .

(b) our usual drawing of $x,y,z$ axes (with $z$ vertical, $y$ horizontal, and $x$ slanted);

(c) the 3-d engineering graph paper shown at the end of this handout;

(d) the map of Math Sciences and Boelter Hall displayed by the elevator in the MS 5th floor lobby off the breezeway.

Problem O-2. In the article by Carlbom and Paciorek [1] , two picture captions have accidentally been switched. Which ones?

Problem O-3. Imagine a large cubical storage shed, 10 feet in each dimension and open in the front. It sits on horizontal ground. Sketch views of the shed as seen from these locations, and say what kind of projection each is. (No computations expected. Classify by major kind (orthographic, oblique, perspective) and by subclassification if relevant. Imagine a viewplane perpendicular to the line of sight described.)

(a) with your eye at ground level, 30 feet in front of the shed, looking directly toward it (in other words, along a line on the ground going to the middle of the front of the shed).

(b) Corner-on from 40 feet away, along a line passing directly through the top front left corner and the bottom back right corner. (Here ``left'' and ``right'' are interpreted with respect to you.)

(c) From very far away, along the same line as in (b). (``Very far'' = ``from infinity''. Imagine using a telescope, so the shed still does not appear tiny.)

(d) From very far away, along a vertical line of sight.

Problem O-4. Find a right-handed orthonormal coordinate frame v$,$   w$,$   n for a viewplane with normal N$= (1,2,3)$ and up-vector U$=$   k$= (0,0,1)$.

Problem O-5. Find a right-handed orthonormal coordinate frame v$,$   w$,$   n for a viewplane with normal N$= (a,b,c)$, using k$= (0,0,1)$ as an up-vector. Assume $a ^ 2 + b^2 \neq 0$. (Be sure to use the unit normal n where needed. Of course, your answer will be in terms of letters. It is easiest to normalize lengths last.)

Problem O-6. (a) For a perspective projection from an arbitrary viewpoint $(a,b,c)$ onto the $x,y$-plane, by what $4 \times 4$ matrix should the homogeneous coordinates of the points of the object be transformed?

(b) Compute and sketch a picture of the standard tetrahedron, with vertices e$^{(1)},$   e$^{(2)},$   e$^{(3)}$, and the origin, as projected on the $x,y$-plane from $(1,2,2)$.

Problem O-7. Consider the cube $(\pm 1, \pm 1, \pm 1)$, viewplane $x + 2y + 2z = 0$, and up-vector k$= (0,0,1)$.

(a) Find the coordinate frame and rotation matrix that go with this setup.

(b) Find expressions for the images of the vertices in the $x,y$-plane, after rotation, using an orthographic projection. Your answer may be left as a matrix product (with the first matrix being a row vector). The matrix on the right can be $3 \times 2$, since the missing third column has the effect of discarding the third entry.

(c) Repeat (b) using a perspective projection with viewpoint $(4,8,8)$ (before rotation). You may leave the answers in homogeneous coordinates. (In other words, after doing the same rotation as for the orthographic problem, multiply by an additional viewing transformation to take care of the perspective, and finally project orthographically to two dimensions. The last two steps can be combined by taking the $4 \times 4$ matrix for the viewing transformation and then deleting the third column and use the resulting $4 \times 3$ matrix.)

(In all parts, you may combine cases by using $\pm$.)

Problem O-8. Consider the ``standard unit cube'' in R$^ 3$, in other words, the cube whose vertices have coordinates with entries 0 and/or $1$ only. Give the main and sub-classification of each of the following projections of the cube. You are not asked to compute images.

(a) viewplane $x+y+z=0$, viewpoint $pt(5,5,5,0)_h$;

(b) viewplane $x + y + 2z = 0$, viewpoint $pt(1,1,2,0)_h$;

(c) viewplane $x+y+z=0$, viewpoint $pt(5,5,5,1)_h$;

(d) viewplane $x + 2z = 0$, viewpoint $pt(5,0,10,1)_h$;

(e) viewplane $z = 0$, viewpoint $pt(1, 1, 2 \sqrt 2 ,0)_h$;

(f) viewplane $z = 0$, viewpoint $pt(0,0,5,1)_h$.

Problem O-9. The three-dimensional analogue of a window in user (world) coordinates is a box-shaped viewing volume. Parts of the object that are in front of the viewing volume or in back of it would not be shown. Suppose your viewing volume is $-4 \leq x \leq 4$, $-3 \leq y \leq 3$, $0 \leq z \leq 1$, and you are making a perspective projection on the $x,y$-plane with viewpoint $(0,0,5)$. After you apply the projective transformation in P$_ 3$ to move the viewpoint to infinity on the $z$-axis, where has each of the corners of the viewing volume moved to?

Problem O-10. Consider a perspective projection on the $x,y$-plane from the viewpoint $(0,0,H)$, and the projective transformation on P$_ 3$ that moves the viewpoint to infinity on the $z$-axis (the viewing transformation).

(a) Under the projection, what ultimately happens to the viewpoint itself? (Apply the projective transformation and then the orthographic projection to the viewpoint and interpret what your answer means.)

(b) In (a), what happens to other points $(a,b,H)$ where $a,b$ are not both zero?

(c) Under just the viewing transformation, as the viewpoint is being taken to $Z$ at infinity, the point $Z$ is being taken to some other point. What point?

Problem O-11. In our usual setup for a perspective projection, the viewplane is the $z = 0$ plane and the viewpoint is at $(0,0,H)$. As in formula , the result of the projecting is $(x,y,z)\rightarrow (x/(1-\frac zH),y/(1-\frac zH))$.

What formula is obtained instead if the viewplane is the $z=H$ plane and the viewpoint is the origin?

Suggestion: Instead of trying to do this from the beginning, make the transformation $(x,y,z)\rightarrow (x,y,H-z)$ that trades the $z = 0$ and $z=H$ planes and then use formula (so you put $H-z$ for $z$ in that formula). Simplify algebraically.

Problem O-12. Suppose that a car is represented in three dimensions in a coordinate system for which the positive $z$-axis is horizontal and points forward on the car, the positive $x$-axis is horizontal and points to the left, and the positive $y$-axis is up. For the origin, take some point that is roughly in the center of the car, and for the units use meters.

(a) Is this coordinate system right-handed or left-handed?

(b) Suppose you want to make an a view of the car from the outside, looking toward the right front corner of the car (right as seen by the driver). Specifically, suppose that you use an orthographic projection along rays slanted at $30 ^\circ$ to the ground and at $45 ^\circ$ sideways from forward. Give an appropriate normal vector for the viewplane and an appropriate up-vector. (You may leave your answer as a matrix product.)

(c) Suppose you want to make successive views as if you are walking around the car in a circle five meters in radius and centered on the ground under the origin you chose. Use a perspective projection with a viewpoint at your eyes, one meter higher than the origin. For a viewplane normal, use the line from your eyes to the origin. Give the viewplane normal, the distance of the viewpoint from the viewplane, and the up-vector to use. (You may leave your answer in the form of a matrix product. The normal will be in terms of some angle or time parameter.)

Problem O-13. A puzzle: Figure , similar to one in Carlbom and Paciorek [1], shows two oblique projections of the same I-beam, but with different faces being undistorted and with possibly different foreshortening ratios. Find these foreshortening ratios and (if relevant) give the name(s) of the projection(s). (Measure the pictures with a ruler in millimeters and use what logic you can. A good idea is to find a distance that is shown undistorted in both pictures and use that as a ``unit'', in terms of which to measure all distances on the picture and on the object. One catch is that the pictures are not to the same scale.)

Figure: Two oblique projections of an I-beam

.8book/05dir/I-beam.eps

Problem O-14. Reconstruction of a cube from its image: the orthographic case. Consider a cube with one vertex at the origin in R$^ 3$ but otherwise in no special position or orientation and of no special size. Suppose that this cube is projected orthographically on the $x,y$-plane, and consider the problem of reconstructing the cube from just knowing its image. Specifically, let the vertices adjacent to the vertex at the origin be $(x _ 1, y_1, z_{1})$, $(x_2, y_2, z_{2})$, and $(x _ 3, y_3, z_3)$. Looking at the image, you can see the pairs $(x_i, y_i)$ for each $i$, but the values of the $z_i$ are not obvious. This problem shows how the $z_i$ can be found.

It is handy to define the three vectors x $=$ $(x_1, x_2, x_3)$, y $=$ $(y_1, y_2, y_3)$, and z $=$ $(z_1, z_2, z_3)$, even though these have no obvious pictorial interpretation in terms of the original cube. (If the cube is a unit cube, they are triples consisting of the cosines of the angles that the axes of the cube make with the $x$-, $y$-, and $z$-axes.)

(a) Show that if the cube is a unit cube then x and y are orthogonal to each other and of length 1, and that z is plus or minus x$\times$   y.

(Method: Write the original vertices as rows of a $3 \times 3$ matrix. What is special about this matrix? x and y are two of its columns.)

(b) Explain why if the cube is not necessarily a unit cube, x and y are still orthogonal to each other and have equal lengths $S$, where $S$ is the length of the sides of the cube. (Method: x$/S$, y$/S$, z$/S$ are as in (a).)

(c) Invent a way of finding $z_1, z_2, z_3$ (except for a factor of $\pm 1$) from a knowledge of the three image points $(x_i, y_i)$. (The factor of $\pm 1$ is needed because if you have one solution then its reflection in the $x,y$-plane is another solution. The method is the same idea as in (b).)

Problem O-15. Reconstruction of a cube from its image: the general parallel case. Suppose a picture of a cube with sides of length $S$ is using with a parallel projection on the $x,y$-plane from the viewpoint $pt(a,b,1,0)_h$. If $a$ and $b$ are 0, the projection is orthographic; otherwise, it is oblique. In either case, allow the cube to be tilted with respect to the viewplane. (In the oblique case this is contrary to our usual guarantee that we won't use oblique projections unless one face is parallel to the viewplane.)

As in Problem O-, let one vertex of the cube be the origin and let the adjacent vertices be $(x_i, y_i, z_i)$ for $i = 1,2,3$. This time, since the projection rays might be slanted, the images in the viewplane are not just the first two coordinates of the points, so let the images be called $(u_i, w_i)$ for $i = 1,2,3$. As in Problem O-, define vectors u$= (u _ 1, u_2, u_3)$ and w$= (w_1, w_2, w_3)$.

(a) It is a fact that u$=$   x$- a$   z and w$=$   y$- b$   z. These sound similar to something you know for oblique projections, but notice that these are vector equations, not scalar equations. Why are they true? (Method: Think coordinatewise.)

(b) For short, let $U =$   u$\cdot$   u, $W =$   w$\cdot$   w, $D =$   u$\cdot$   w. Explain why, if the projection is orthographic, then $U = W$ and $D = 0$. (You may quote what you need from (b) of Problem O-, which makes this very easy.)

(c) Show that, conversely, if $U = W$ and $D = 0$, then the projection is orthographic. (Method: You may quote what you need from Problem O-. Start by finding $U,W,D$ in terms of x$,$   y$,$   z, carefully using rules for dot products that are like rules of high-school algebra. Then use what you know about x$,$   y$,$   z.)

Note. By (b) and (c), you can tell just from the image of a cube whether a parallel projection is orthographic or oblique, if you're willing to measure the coordinates of the images of the vertices. (In fact, since it's also easy to see if a projection is perspective, all three of the possibilities orthographic, oblique, and perspective can be determined from the image alone.)

Problem O-16. This problem carries Problem O- further for the case of a unit cube, $S = 1$. Show how to compute the points $(x_i, y_i, z_i)$ and the oblique projection direction $(a,b,1)$ just from the image points $(u_i, w_i)$. In other words, from an oblique image of a tilted unit cube show how to reconstruct the whole projection setup precisely.

(It is not required to find a formula for everything in terms of the image points; rather, you can solve for unknowns in stages, with a recipe for a solution at each stage in terms of quantities already computed at preceding stages. Outline: In the notation of Problem O-, (c) of that problem gives $M = 1 + a^2$, $N = 1 + b^2$, $D = ab$, so you can solve for $a$ and $b$ up to a factor of $\pm 1$. Next concentrate on finding z, the list of $z$ coordinates, as follows. Using the equations of (a) of Problem O- and facts about x, y, z, you get z$\cdot$   u$= -a$, z$\cdot$   w$= -b$. Notice that these are equations of planes in R$^ 3$; since z is of length 1, it lies where the line of intersection of these planes cuts the unit sphere $z_1^2 + z_2^2 + z_3^2 = 1$. [There could be two possibilities. To find them, you'd need to represent the line parametrically, then substitute coordinates in the equation for the sphere and simplify to get a quadratic equation in $t$; solve.] Then, knowing z, how can you get x, y?)

Problem O-17. This problem generalizes Problem O- by allowing a cube of any side $S$. Show how to compute the points $(x_i, y_i, z_i)$ and the oblique projection direction $(a,b,1)$ just from the image points $(u_i, w_i)$. In other words, from an oblique image of a tilted cube show how to reconstruct the whole projection setup precisely, even without knowing the length of the side of the cube.

(Method: With notation as in Problem O- and Problem O- you get $M = S^2 (1 + a^{2})$, $N = S^2 (1 + b^{2})$, $D = S^2 a b$. Here $M$, $N$, and $D$ are constants known from the coordinates of the images of the vertices, and $a,b,S$ are unknown. For convenience let $A = S a$ and $B = S b$, so that $M = S^2 + A^2$, $N = S^2 + B^2$, $D = AB$. Then $M - N = \dots$ and $D = AB$ are two equations in just the unknowns $A,B$. Solve the second for $B$ and substitute in the first to get an equation in $A$ alone, which clears to a quadratic equation in $A^2$. Solve for $A^2$ using the quadratic formula. Then use $M = \dots$ to solve for $S^2$. You should get $S^2 = \frac{1}{2} (M+N - ( (M-N)^2 + 4 D^2 )^{{\frac 1 2}} )$. Now, knowing $S$, try to reduce everything to Problem O-.)

Problem O-18. You wish to view the earth from above Los Angeles, which is approximately at latitude $34 ^\circ$, longitude $-118 ^\circ$. What rotation should you use? (You may leave your answer as a product of standard rotations, without explicit entries. Don't bother to convert to radians.)

Problem O-19. If you use the latitude-longitude method to make a picture of the earth from above a point in the southern hemisphere, does the north pole come out at the top of your picture (i.e., after being rotated does it project to the positive $y$-axis), or at the bottom (negative $y$-axis)?

Problem O-20. (a) Find a rotation formula, similar to the one in formula , for a viewing direction with altitude $\alpha$ and azimuth $\beta$. (b) Find a rotation formula for a viewing direction with co-latitude $\lambda$ and longitude $\phi$.