The equation for the point incident on two lines is the same.
Problem 4
Prove that the definition of incidence is independent of the choice of
the representatives of $p$ and $L$.
That is, if $p_{1}$, $p_{2}$, $p_{3}$, and $q_{1}$, $q_{2}$, $q_{3}$ are two triples of
homogeneous coordinates for $p$, and
$L_{1}$, $L_{2}$, $L_{3}$, and $M_{1}$, $M_{2}$, $M_{3}$ are two triples of
homogeneous coordinates for $L$, prove that
$p_{1}L_{1}+p_{2}L_{2}+p_{3}L_{3}=0$ if and only if
$q_{1}M_{1}+q_{2}M_{2}+q_{3}M_{3}=0$.
Answer
If $p_{1}$, $p_{2}$, $p_{3}$, and $q_{1}$, $q_{2}$, $q_{3}$ are two triples of
homogeneous coordinates for $p$ then the two column vectors
are in proportion, that is, lie on the same line through the
origin.
Similarly, the two row vectors are in proportion.
Then multiplying gives the answer $(km)\cdot (p_{1}L_{1}+p_{2}L_{2}+p_{3}L_{3})=q_{1}M_{1}+q_{2}M_{2}+q_{3}M_{3}=0$.
Problem 5
Give a drawing to show that central projection does not preserve
circles, that a circle may project to an ellipse.
Can a (non-circular) ellipse project to a circle?
Answer
The picture of the solar eclipse — unless
the image plane is exactly perpendicular
to the line from the sun through the pinhole — shows the circle
of the sun projecting to an image that is an ellipse.
(Another example is that in many pictures in this
Topic, the circle that is the sphere's equator is drawn as an ellipse,
that is, is seen by a viewer of the drawing as an ellipse.)
The solar eclipse picture also shows the converse. If we picture the projection as going from left to right through the pinhole then the ellipse $I$ projects through $P$ to a circle $S$.
Problem 6
Give the formula for the correspondence between the
non-equatorial part of the antipodal modal
of the projective plane, and the plane $z=1$.
since that is intersection of the line containing the vector and the plane.
Problem 7
(Pappus's Theorem)
Assume that $T_{0}$, $U_{0}$, and $V_{0}$ are collinear and that
$T_{1}$, $U_{1}$, and $V_{1}$ are collinear.
Consider these three points:
(i) the intersection $V_{2}$ of the lines $T_{0}U_{1}$ and $T_{1}U_{0}$,
(ii) the intersection $U_{2}$ of the lines $T_{0}V_{1}$ and $T_{1}V_{0}$, and
(iii) the intersection $T_{2}$ of $U_{0}V_{1}$ and $U_{1}V_{0}$.
Draw a (Euclidean) picture.
Apply the lemma used in Desargue's Theorem
to get simple homogeneous coordinate vectors for the
$T$'s and $V_{0}$.
Find the resulting homogeneous coordinate vectors
for $U$'s (these must each involve a parameter as, e.g., $U_{0}$ could
be anywhere on the $T_{0}V_{0}$ line).
Find the resulting homogeneous coordinate vectors for
$V_{1}$.
(Hint: it involves two parameters.)
Find the resulting homogeneous coordinate vectors for
$V_{2}$.
(It also involves two parameters.)
Show that the product of the three parameters is $1$.
Verify that $V_{2}$ is on the $T_{2}U_{2}$ line..
Answer
Other pictures are possible, but this is one.
The intersections
$T_{0}U_{1}\,\cap T_{1}U_{0}=V_{2}$, $T_{0}V_{1}\,\cap T_{1}V_{0}=U_{2}$, and $U_{0}V_{1}\,\cap U_{1}V_{0}=T_{2}$
are labeled so that on each line is a $T$, a $U$, and a $V$.
The lemma used in Desargue's Theorem gives a
basis $B$ with respect to which the points have these
homogeneous coordinate vectors.
($u_{0}$ is a parameter; it depends on where on the $T_{0}V_{0}$ line
the point $U_{0}$ is, but any point on that line has
a homogeneous coordinate vector of this form for some $u_{0}\in \mathbb {R}$).
Similarly, $U_{2}$ is on $T_{1}V_{0}$