9. Problems

Problem M-1. Give homogeneous coordinates for each of the points indicated below. Points indicated with arrows are at infinity. (Remember, the point at infinity is determined by the direction of the line to it.)

book/04dir/points.eps

Problem M-2. Recall the definition of a projective transformation: To transform a point, you choose some triple representing the point in homogeneous coordinates, multiply the triple by the matrix to get a new triple, and then take the point represented by the new triple. Why doesn't it matter which triple you choose to represent the first point?

Problem M-3. (a) Is the standard square the same thing as the standard quartet? (b) If it is, explain why. If it is not, say whether the standard square is even a quartet at all, and why.

Problem M-4. (a) In P$_2$, find a matrix for the projective transformation that takes the ordinary points $(0,0), (1,0), (1,1), (0,1)$ (the ``standard square'') to the ordinary points $(0,0), (2,0), (3,3), (0,2)$, respectively. (If you wish you may list the vertices in another order, as long as you list the image vertices in the corresponding order. You may be able to take advantage of Problem .)

(b) Give all possible answers to (a).

(c) If one person took advantage of the possibility of listing the vertices in another order and another person did not, would their answers to (a) necessarily be the same? Their answers to (b)?

Problem M-5. Is there a projective transformation that takes $X,Y,O,E$ respectively to $(1,0)$, $(1,1)$ (ordinary points), $X, Y$ ? Why or why not?

Problem M-6. Check that the matrix given as the solution to Problem above does give a projective transformation that does what it is supposed to.

Problem M-7.

Show that the only projective transformation of P$_2$ that leaves each of $X,Y,O,E$ fixed is the identity transformation, i.e., the transformation that leaves all points fixed.

(Method: If $T$ is such a transformation, then $T$ comes from a $3 \times 3$ matrix $M$. Show that $M$ must be a nonzero scalar matrix; i.e., $M = rI$, by using the fact that x $(1,0,0)_h M$ is a scalar times $(1,0,0)_h$, etc. What transformation is produced by a scalar matrix?)

Problem M-8. (a) Show that if $P,Q,R,S$ is a quartet in P$_2$, then the only projective transformation that leaves each of $P,Q,R,S$ fixed is the identity transformation.

(Method: Suppose $T$ leaves $P,Q,R,S$ fixed. Choose a projective transformation $W$ taking $X,Y,O,E$ to $P,Q,R,S$ respectively. Apply the result of Problem M- to show $W^{-1} T W =$   1, the identity transformation. Here the composition $W^{-1}TW$ means to apply $W$, then $T$, then $W^{-1}$. Solve for $T$, just as you would using matrices.)

(b) Show that if $P,Q,R,S$ and $P',Q',R',S'$ are two quartets in P$_2$, there is only one projective transformation that takes $P$ to $P'$ and so on.

(Method: Suppose both $T$ and $U$ take $P$ to $P'$ and so on. Apply (a) to $T^{-1}U$.)

Problem M-9. (a) Explain why it is not possible to define the value of the determinant for a projective transformation. (Method: As in Problem M-, there is more than one possible matrix for the transformation. Do all possible matrices have the same determinant? Be careful in saying what happens to the determinant of a matrix when all entries are multiplied by the same scalar.)

(b) Explain why in P$_2$ it is not even possible to talk about the sign of the determinant. (Method: If all entries of the $3 \times 3$ matrix are multiplied by $-1$, what happens to the sign of the determinant?)

(c) Explain how it is possible to define the sign of a projective transformation in P$_3$ using determinants, even though the value of the determinant itself cannot be defined.

(For homogeneous linear transformations, the sign of the determinant tells whether the transformation preserves orientation or reverses it. The same is true for affine transformations. The facts (b) and (c) say in effect that orientation of objects cannot be defined in P$_2$ but can be defined in P$_3$.)

Problem M-10. In the one-dimensional space R, a linear fractional function is a function of the form $f(x) = \frac {ax + c} {bx + d}$, where $ad - bc \neq 0$. Of course, the domain of $f$ may not be all of R. Explain how a linear fractional function is really the same as a projective transformation in P$_1$. (Method: Make the extended vector $[x 1]_h$, transform it by an appropriate matrix, and scale to put the answer in the form $[y 1]_h$. What is $y$ in terms of $x$? Linear fractional functions are important in the theory of complex variables, where $x$ and $y$ can be complex.)

Problem M-11. Find all $4 \times 4$ matrices that give projective transformations on P$_3$ leaving the $x,y$-plane fixed. (In other words, $T(Q) = Q$ for every point $Q$ in the $x,y$-plane.)

Problem M-12. Find an example to show that a projective transformation in P$_2$ does not preserve ratios of line segments on the same line. In other words, if $PR$ is a line segment and $Q$ is between $P$ and $R$ on the segment, the ratio of the length of $PQ$ to the length of $PR$ may not be preserved. (In contrast, affine transformations do preserve such ratios; see the exercises of the first handout on affine transformations.)

Problem M-13. Let $T$ be the projective transformation on P$_2$ that takes $X,Y,O,E$ to the standard square e$_1,$   e$_2, (0,0), (1,1)$, in that order. A matrix for $T$ was found as part of the solution to Problem above, namely, $\left[\begin{array}{rrr} 1&0&1\\ 0&1&1\\ 0&0&-1 \end{array}\right]$. (a) Sketch the images under $T$ of the edges of the standard square (between vertices in the usual order). (Suggestion: Represent each edge parametrically and transform. Indicate just ordinary points.) (b) On your sketch, indicate the image under $T$ of the whole standard square region, with interior. (c) What is the image of the diagonal line $x + y = 1$? (d) Sketch the image of the line $x + y = 2$.

Problem M-14. Describe the image of the hyperbola $y = {{\frac{\displaystyle 1}{\displaystyle x}}}$ under the projective transformation with matrix $\left[\begin{array}{ccc} 0&0&1\\ 0&1&0\\ 1&0&0 \end{array}\right]$.

(Method: Express the hyperbola parametrically as $(t, {\frac{1} t})$ for $t \neq 0$. Rewrite in homogeneous coordinates, transform, and find the Cartesian coordinates again in terms of $t$. Finally, try to re-express the answer as a curve $y = \dots$.)

Note. Actually, projective transformations can map any kind of conic to any other, for instance, a circle to a parabola.

Problem M-15. Verify Rules 1 and 2 of Section 6 above, by discussing the different cases that can occur. (For example, in considering two lines, one might be an ordinary line and the other the line at infinity.)

Problem M-16. Prove Fact . In other words, prove that the method of Problem works for any quartet with vertices $P,Q,R,S$.

(Method: You may assume this useful fact: If $P$, $Q$, $R$ are points of P$_2$ whose homogeneous coordinates are linearly dependent as three vectors in R$^3$, then $P$, $Q$, $R$ are collinear. Using this fact, explain why, in the method of Problem above, the coefficient matrix is nonsingular and none of $r,s,t$ can be zero.)

Problem M-17. Explain how the extended matrix of an affine transformation on R$^2 \rightarrow$   R$^ 2$ can be regarded as the matrix of a projective transformation on P$_2 \rightarrow$   P$_2$. What does each row of the matrix mean, in terms of $X,Y,O$?

Problem M-18. As you now know, the extended vectors used for affine transformations were really homogeneous coordinates. Further, the $3 \times 3$ extended matrices used for an affine transformation in R$^ 2$ can be applied even to points at infinity, so that the affine transformation becomes a projective transformation.

Show that nonsingular affine transformations, if applied in P$_2$, have the special property that they take points at infinity only to points at infinity. Do this two ways:

(a) algebraically, by transforming points with coordinates $(a,b,0)_h$;

(b) geometrically, by using the fact that affine transformations on R$^ 2$ take parallel lines to parallel lines.

Problem M-19. Let $T:$   P$_2 \rightarrow$   P$_2$ be a projective transformation. This problem shows how to tell if $T$ is really an affine transformation.

(a) For $X, Y$ as usual, show that if $T(X)$ and $T(Y)$ both are points at infinity, then $T$ is affine. (Method: Narrow down possibilities for the matrix of $T$. See if you can get the desired entries to be zero. Scale the whole matrix by a nonzero scalar to get the desired entry to be a 1.)

(b) Explain why if $T$ takes points at infinity to points at infinity only (in other words, the line at infinity stays at infinity), then $T$ must be affine.

(c) Explain why if $T$ takes even two points at infinity to points at infinity, then $T$ must be affine. (Part (a) showed one example of this fact. Method: $T$ takes lines to lines. Where does the line at infinity go?)

(d) Explain why, if $T$ takes even one parallelogram to a parallelogram, then $T$ must be affine. (Method: Use (c).)

This last part shows that even with a little information you can tell that $T$ is affine. An equivalent statement is this: If $T$ is not affine, then $T$ distorts every parallelogram into a non-parallelogram.

Problem M-20. Consider all projective transformations on P$_2$ that leave $X,Y,O$ fixed. These are the same as some transformations you knew about before ever hearing of P$_2$. Which ones, exactly?

Problem M-21. (a) Explain why each real eigenvector of a nonsingular real $3 \times 3$ matrix gives a fixed point of the corresponding projective transformation on P$_2$.

(b) Show that every projective transformation on P$_2 \rightarrow$   P$_2$ has at least one fixed point. (Notice that the characteristic polynomial is cubic, and recall that every cubic polynomial has at least one real root, since its graph goes from the third quadrant to the first quadrant and so crosses the $x$ axis. Mention why the eigenvalue 0 can't occur.)

(c) For a translation, regarded as a projective transformation, describe one fixed point. Are there any others?

Problem M-22. In P$_3$, (a) define the standard points $X,Y,Z,O,E$; (b) find a projective transformation taking $X,Y,Z,O,E$ to $(1,0,0)$, $(0,1,0)$, $(0,0,1)$, $(0,0,0)$, $(1,1,1)$ respectively.

Problem M-23. Surprisingly, the intersection of two lines in P$_2$ can be found by using a cross product. Consider the two lines whose equations in ordinary coordinates are $x + 3y + 4 = 0$ and $4x + y + 5 = 0$.

(a) Find equations for these two lines in homogeneous coordinates. (Method: Given a line $ax + b y + c = 0$ in ordinary coordinates, put $x/z$ for $x$ and $y/z$ for $y$ and then clear the denominator; you get simply $ax + b y + cz = 0$.)

(b) Use a cross product to find the homogeneous coordinates of the point where the two lines intersect. (Method: Solving the two equations simultaneously will give the desired homogeneous coordinates. If you regard the triples as being in R$^3$ instead, you are finding the line of intersection of two planes, and you know how to do this using a cross product.)

(c) Express the answer in ordinary coordinates.

(d) Does this method work even if one line is the line at infinity? Give an example.

Problem M-24. Projective geometry is concerned with theorems that mention only which lines meet at which points, and not with angles and distances. There are actually some good theorems of this type that could have been understood in high school geometry but were probably not mentioned. Here is one [with problems following the diagram]:

Desargues' Theorem. Choose a point $P$ in the ordinary plane. Draw three lines from $P$. Choose two triangles, each with a vertex on each line (but not using $P$), as in the left diagram. If corresponding sides are not parallel, extend the corresponding sides until they meet. Then the three points of intersection are collinear (i.e., they all lie on one line).

book/04dir/desargues.eps

There are also versions of the theorem covering cases where one pair of corresponding sides is parallel but the other two pairs are not; where at least two pairs of corresponding sides are parallel; and similar cases for a drawing where the original three lines are chosen to be parallel instead of going through a point $P$.

(a) Write down statements for five of these cases. (Choose interesting ones.)

(b) Explain how in the projective plane there is only one case, of which all your cases are really instances. (This is an example of how using the projective plane can actually make some kinds of geometry simpler and yet more powerful at the same time.)

Problem M-25. The diagram of Desargues' Theorem looks somewhat three-dimensional, even though it isn't. However, you can invent a three-dimensional version of Desargues' Theorem in which the lines through $P$ may not be not coplanar.

(a) Give such a three-dimensional statement.

(b) Actually prove your statement in the case where the lines through $P$ are not coplanar. To keep things simple, assume that no two lines in the whole diagram are parallel in R$^3$.

Note. One good proof in the two-dimensional case is this: Given the diagram, imagine constructing a three-dimensional diagram whose perpendicular projection in two dimensions is the given diagram. The line in three dimensions containing the three intersection points of the sides of the triangles projects to a line in two dimensions with the same property.

Problem M-26. An affine transformation in extended coordinates had a ``hidden'' geometrical interpretation as a map on R$^3 \rightarrow$   R$^3$, in which the transformation acted on the $z=1$ plane. The extended matrix of an affine transformation always takes the $z=1$ plane to itself. Develop a similar ``hidden'' geometrical interpretation for how you use projective transformations when you take $(x,y) \rightarrow (x,y,1)_h \rightarrow (x,y,1)_h A$ and then normalize the result to the form $(u,v,1)_h$. (This time the matrix will usually take the $z=1$ plane to another plane. However, the final step of normalizing projects points back to the $z=1$ plane. Make a sketch.)

Problem M-27. In the survey article by Carlbom and Paciorek it is stated that various entries of $4 \times 4$ matrix represent a homogeneous linear transformation, a translation, a projection, and a scaling. This would seem to suggest that an arbitrary nonsingular $4 \times 4$ matrix $\left[\begin{array}{cc} A&\mbox{\bf p} ^ t\\ \mbox{\bf b}&s \end{array}\right]$ is the product of the matrices $\left[\begin{array}{cc} A&\mbox{\bf0} ^ t\\ \mbox{\bf0}&1 \end{array}\right]$, $\left[\begin{array}{cc} I&\mbox{\bf0} ^ t\\ \mbox{\bf b}&1 \end{array}\right]$, $\left[\begin{array}{cc} I&\mbox{\bf p} ^ t\\ \mbox{\bf0}&1 \end{array}\right]$, $\left[\begin{array}{cc} I&\mbox{\bf0} ^ t\\ \mbox{\bf0}&s \end{array}\right]$, in some order. Is this true?

Problem M-28. Suppose we take a $4 \times 3$ matrix $A$ and attempt to define a transformation on P$_3 \rightarrow$   P$_2$ by setting $T($x$_h)=$   x$_h A$. (a) Why doesn't this define a transformation on all of P$_3$, even if $A$ has rank 3 (the largest possible)? (Method: Is x$_h A$ always a nonzero vector?) (b) Even so, this does define a transformation whose domain is contained in P$_3$. Describe how you could find $A$ taking each of $X, Y, Z, O$ to given points of P$_2$.

(Essentially, you have made a two-dimensional picture of [most of] P$_3$ in which the images of $X, Y, Z$ are ``vanishing points'' for the corresponding families of parallel lines in R$^3$.)

Problem M-29. Suppose you want to make a perspective picture of a box-shaped building whose families of parallel lines are lined up with the coordinate axes. Suppose you want a 3-point perspective projection, with vanishing points on the viewplane being $Q = (4,-2)$, $R = (-4,-2)$, $S = (0,6)$, with $X \rightarrow Q, Y\rightarrow R, Z\rightarrow S, O\rightarrow O$ (where $O$ means an origin in each dimension).

(a) By a direct method as in Problem M-, find a 4-by-3 matrix $M$ that accomplishes this projection, when homogeneous coordinates are used in P$_3$ and P$_2$.

(b) Observe that $M$ makes a transformation that is not defined on all of P$_3$, because some legitimate points are mapped to a triple $(0,0,0) _ h$. Find such a point. Geometrically, why should there be such undefined points for a perspective projection?

(c) Is there more than one transformation possible, depending on the choice of $M$? (In other words, are there choices of (a) that are not just scalar multiples of each other?)