Mathematics 138
Incidence properties of some models of Cayley-Klein plane geometries

If you worked out all the details of all these geometries, you'd be done with the course, but the goal is much more limited.  For each model, I have specified objects that can be called "points" and objects that can be called "lines." These may be defined in terms of physics, linear algebra, or Euclidean geometry.  The model specifies a criterion for finding the "line" (if any) specified by two "points" and the "point" (if any) specified by two "lines."  However, since the objects of the model are familiar ones, these criteria often do not need to be specified explicitly.  For example, if the "lines" are ordinary straight lines, you can use a straightedge to construct the line specified by two given points.

Where it is easy to do, I have also specified how the distance between two points can be measured and how the angle between two lines can be measured, and I have specified some "motions" that preserve lengths and angles (though not necessarily all such motions).  Some of the formulas for distance and angle yield signed numbers, and you might prefer to take the absoute value. In general, you do not need this information to answer the questions below, but you will need it when we return to these geometries.
For next Wednesday, choose at least three of these and do your best to answer the questions below.  For the following Monday, do as many as you can, working in teams if possible.
In an attempt to cover every possible Cayley-Klein geometry, I have included some fairly bizarre models, which might lead to good term projects. Some require specialized knowledge of something like relativity or hyperbolic functions, and you may just need to ignore these for the moment.  Some require difficult feats of geometrical visualization. 
Here are the questions.

1. Given a line L and a point P not on L , which of the following holds?
a. Every point on L can be joined to P by a line. (code E)
b. There is precisely one point on L that cannot be joined to P by a line. (code P)
c. There are infinitely many points on L that cannot be joined to P by a line. (code H)

2. Given a line L and a point P not on L , which of the following holds?
a. Every line though P intersects L. (code E)
b. There is precisely one line through P that does not intersect L. (code P)
c. There are infinitely many lines through P that do not intersect L. (code H)

3. Given three distinct points A, B, C on a line, is one of them unambiguously "between" the other two?

4. Given three distinct lines r, s, t passing through a point P, is one of them unambiguously "between" the other two?

5. Is there an upper limit to the distance between two points?

6. Is there an upper limit to the angle between two lines?

7. Is there a subgroup of motions that is isomorphic to a two-dimensional vector space with the operation of addition?  Or more simply: can you associate a two-component vector with some of the motions in such a way that vector addition correctly predicts the effect of successively performing the two motions?

Model 1:
Points are vectors of the form (x,y) where x and y are real numbers.
Lines are sets of points satisfying an equation ax + by = c, where a and b are not both zero.
Distance is defined by d = 
Angle is defined by q = 
Motions include
Adding (p, q) to every point 
Replacing (x, y) by (x cos q - y sin q, y cos q + x sin q)
Model 2
Points are points of the Euclidean half-plane with y > 0.
Lines are either
Euclidean straight lines with x = constant
or
Euclidean semicircles with their centers on the x-axis.
Angles are measured between the lines tangent to intersecting circles, using an ordinary protractor.
For points at coordinates y1 and y2 on a line with x = constant, distance is defined as
d = 
For points on a line that is a Euclidean circle, distance is defined as follows:

M and N are the points where the circle meets the x axis.
A and B are obtained by projecting A' and B' vertically onto the x axis.
Then the distance between A' and B' is defined in terms of the lengths of Euclidean segments on the x axis by

d = 1/2 ln  

Motions include
Translations parallel to the x-axis
Similarity transformations (changes of scale) that hold fixed any point on the x-axis.
Reflections in any "line" (if necessary, defined as inversion in a circle).
Model 3:
"Points" are diameters through the center of the Earth. Each such "point" may be represented by a unique point in the northern hemisphere (latitude > 0) or by a point on the Equator whose longitude is greater than or equal to zero and less than p.
"Lines" are planes through the center of the Earth. Each such "line" may be represented by an arc of a great circle where it meets the northern hemisphere.
The distance between two "points" is the (smaller) angle between the diameters where they meet at the center of the Earth.
The angle between two "lines" is the (smaller) angle between the tangents to the great circles where they intersect on the surface of the Earth.
Motions include rotations about any axis through the center of the Earth.

Model 4:
"Points" are events in the motion of a particle along the x axis in Newtonian mechanics. An event is specified by a time coordinate and a space coordinate in some frame of reference.
"Lines" are motions along the x axis that satisfy Newton's First Law of motion (constant velocity with respect to some inertial frame of reference).
The distance between two "points" is the difference in time between the two events.
The angle between two "lines" is the relative velocity of the two motions (equivalently, the velocity of one moving particle in a frame of reference where the other is at rest)
Motions include
a change in the origin on the x axis
a change in the origin of time (like a switch to Daylight Savings time)
a change to a new inertial frame of reference, moving with constant velocity with respect to the original frame of reference.
Model 5:
(This will be familiar if you have studied motion with constant acceleration or conservation of momentum in a physics course.)
"Points" are states of motion (velocity vectors) for a particle that moves in a plane according to Newtonian mechanics, where there is no upper limit to velocity.
A "line" is the set of "points" associated with a particle undergoing motion with constant acceleration (a vector) in the plane.
The distance between two "points" is their relative speed (the magnitude of the velocity of one motion relative to a frame of reference where the other has zero velocity).
The angle between two "lines" is the Euclidean angle between their acceleration vectors.
Motions include
change of frame of reference (equivalently, adding a fixed vector to all velocities)
rotation of all velocity and acceleration vectors through a specified angle.

Model 6:
"Points" are states of motion of a particle that moves in a plane according to relativistic mechanics, where there is an upper limit to velocity given by the speed of light c. These "points" therefore correspond to the interior of a disc of radius c.
"Lines" are Euclidean straight lines within this disc.
Motions include Lorentz transformations that change the frame of reference.  Given two "points" A and B there is always a Lorentz transformation that transforms to a frame of reference "moving with point A." In that frame of reference the velocity of point A is zero, so point A is shown at the center of the disc.  Since Lorentz transformations also include rotations around the center of the disc, point B can also be moved onto the horizontal axis. This means that there exists a Lorentz transformation by which an arbitrary line MABN gets transformed into the diameter M'A'B'N'

Now the point A' is at (0,0) while B' is at (v,0), where v is the speed of point B' in a frame of reference where point A' is at rest. The distance between A' and B' is defined as

ln(), where the quantity whose logarithm is taken equals the "Doppler shift factor"
if observers at A' and B' are sending light signals to one another.
An equivalent purely geometric way to calculate the distance between A and B is as
1/2 ln ()
The angle between two intersecting lines is defined by making a Lorentz transformation that carries their point of intersection into the center of the disc, then measuring the angle with an ordinary protractor.  But unless the point of intersection of the lines is at the center of the circle, the Euclidean angle between the lines does not give the correct value.

Model 7
"Points" are all the points of the Euclidean plane.
"Lines" are Euclidean lines whose slope is positive.  This excludes horizontal and vertical lines as well as those of negative slope.
The distance between the points (x1, y1) and (x2, y2) is 
The angle between two intersecting lines s and t is defined as follows:

Through the point of intersection O draw horizontal and vertical lines h and v.  Then draw any line that cuts v, s, t, and n.  The "angle" between A and B is defined in terms of Euclidean lengths of segments on MABN as 1/2 ln()
Motions include
any translation in the Euclidean plane
stretching the plane horizontally by a factor K > 0 while shrinking it vertically by the same factor.

Model 8:
"Points" are pairs of points in the Euclidean plane that are related by inversion in the unit circle:
if one such point has polar coordinates r and q the other has polar coordinates 1/r and (q+p).
It is usually sufficient to consider the point of the pair that lies inside the unit circle.
"Lines" are Euclidean circles that intersect the unit circle at opposite ends of a diameter. A diameter itself is also a "line."
The angle between two "lines" is measured between tangents to the Euclidean circles, using a protractor.
The distance between two "points" A and B is determined by constructing the "line" AB (a circle), then constructing perpendiculars (Euclidean circles) to this line through points A and B and using a protractor to measure the Euclidean angle between the tangents to these circles at their point of intersection.
Motions include
rotation about the center of the unit circle.
Reflection in any "line." Where the "line" is a diameter this is ordinary reflection as in a flat mirror. Where the "line" is a circle, a point and its mirror image lie on the same radius of that circle, and the product of their distances from the center of the circle equals the square of the radius of the circle.
The unit circle is centered at O.  The pairs AA' and BB' are two "points" on a "line"

The diagram above illustrates the definition of the angle between lines.
f is the "line" on which "points" A and B lie.
The diameter g through a is one "line" perpendicular to f
The "line" j is perpendicular both to f and to g.
The "point" P where f and g intersect is the "pole" of line f.
The "line" h through P is perpendicular to "line" f at B
The angle a between the tangents to g and h at P defines the distance AB.
Model 9:
"Points" are ordered pairs (x,y) of real numbers.
"Lines" are equations of the form y = mx+b (this excludes vertical lines!).
The distance between two points is the difference in their x-coordinates.
The angle between two lines is the difference in their "slope" m.
Motions include
adding a fixed vector (a,b) to every point.
adding a constant p to the slope of every line.

Model 10:
Imagine a set of railroad tracks, all parallel to the x axis, along which relativistic trains operate, each moving at a constant velocity less in magnitude than the speed of light c..
"Points" are train schedules, each completely specified by the velocity v of the train and the "initial position" x0 that it occupies at time t = 0.
"Lines" are sets of trains that are all in the same place at the same time, though moving with different velocities. A "line" can therefore be specified by one time coordinate and one space coordinate.
The distance between two "points" is the velocity of one train relative to another.
The angle between two "lines" is the time that elapses on a clock riding on a train that belongs to both the sets of trains specified by the two "lines."  But such a train cannot move faster than light!
Motions include
changing the "zero milestone" on all the train tracks.
changing the origin of time for all trains (switching to Daylight Savings Time).
changing the frame of reference, which affects both the velocity and the initial position associated with each train.

Model 11:
"Points" are all the points in a Euclidean plane, which can be viewed as being tangent to the Earth at the North Pole.  This plane is supplemented by a set of "ideal points," one for each angle 0 <= q < p. Each point represents the intersection of a diameter of the Earth with the plane .  The ideal points represent diameters that cannot intersect a plane tangent to the Earth at the North Pole because they lie in the equatorial plane.
"Lines" are all the lines in the Euclidean plane. Each corresponds to a plane through the center of the Earth, which cuts the Earth in a great circle and cuts the plane tangent to the Earth at the North Pole in a straight line.  There is an additional "ideal line" that contains all the "ideal points;" this corresponds to the equatorial plane.  Parallel lines in the Euclidean plane correspond to planes that all have in common a single line in the equatorial plane; they are deemed to intersect in the ideal "point" that corresponds to this "line."
The distance between two points is measured by the angle that the corresponding diameters make at the center of the Earth.
The angle between two lines is measured by the dihedral angle between the corresponding planes.
Each motion corresponds to a rotation of the Earth through some angle about some axis.

Model 12:
"Points" are all the points in a Euclidean plane that lie outside the unit circle.  This plane is supplemented by a set of "ideal points," one for each angle 0 <= q < p. 
"Lines" are all the lines in the Euclidean plane, outside the unit circle, that do not touch the unit circle. Parallel lines with inclination q to the x axis are deemed to meet at the ideal "point" specified by q.  There is an additional "ideal line" that contains all the "ideal points." 
For points very far from the unit circle, the distance between two points is measured by the angle that the segment joining the points subtends at the center of the unit circle. A more precise rule is illustrated by the diagram below: for each point, construct two tangents to the unit circle and join the points of tangency to form a segment inside the unit circle.  Then measure the angle between these segments as in model 6.

The distance AB can be related to the size of the angle a, which however cannot be measured with a Euclidean protractor. 

The angle between two lines can be measured by a similar sort of construction.

Suppose lines a and b intersect at P.  Draw tangents from P to the unit circle and join them by the chord MN,  Then drop perpendiculars from the center of the circle to lines a and b: these intersect lines M and N in points A and B.  Define the angle between a and b as

1/2 log ()
Motions include
Euclidean rotations about the center of the unit circle.
Lorentz transformations as in Model 6.

Model 13:
"Points" are all points of the Euclidean plane that lie outside the unit circle, supplemented by a "point at infinity" for each angle 0 <= q < p.
"Lines" are Euclidean lines that cut the unit circle in two points, with those points and the segment between them excluded.  The two disconnected Euclidean segments of each line
are joined at a "point at infinity."  Lines that are parallel (in the Euclidean sense) are deemed to meet at a "point at infinity."
In this geometry one can associate each "line" with a conjugate "point." If a "line" cuts the unit circle at M and N, the intersection of the tangents to the circle at M and N is the
conjugate point P.

To define the angle between two lines a and b that meet at P, construct the tangents from P to the unit circle at M and N, then determine where a and b intersect the segment M and N. The
angle between A and B is 1/2 ln(). 
To define the distance between two points, simply construct the conjugate lines and determine the angle at which they intersect.
Motions include Euclidean rotations about the center of the unit circle and the Lorentz transformations mentioned in Model 6.  There are also "reflections" in an arbitrary "line."
In the diagram above, the points C and C' are "mirror images" in the "line" MN, as are the points D and D'.  Products of such reflections give all possible motions.

Model 14:
"Points" are lines through the origin in R3, excluding the z axis. Each such line intersects the unit cylinder x2 + y2 = 1 in a pair of ordinary points with coordinates (cos q, sin q, z) and (-cos q, -sin q, -z).
One ordinary point from each pair is chosen to represent the "point" of the geometry by requiring that .
"Lines" are planes through the origin that intersect the unit cylinder in ellipses. Each such plane is normal to a vector of the form (x, y, 1). Planes containing the z axis, which intersect the unit cylinder in two parallel lines, are not included!
The distance between two "points" is defined by projecting the associated lines in R3 onto the plane z = 0 and taking the Euclidean angle between them.
The angle between two "lines" is defined by choosing normal vectors of the form (a, b, 1) for each "line" and taking the Euclidean length of their difference vector.
Motions of the geometry are linear transformations of R3 that carry the unit cylinder into itself: notably
a. rotations about the z axis.
b. "shears" of the form (x, y, z) --> (x, y, z + ex + fy). Such a shear carries the normal vector (a, b, 1) into (a - e, b - f, 1).
Model 15:
"Points" are points of the plane with coordinates ((cos q)/z, (sin q)/z) for  and all real values of z  When z = 0, the point is called an "ideal point" and is regarded as lying between points with positive z and those with negative z for the given value of q.
"Lines" are straight lines in this plane that do not pass through the origin.
The distance between the points ((cos q1)/z, (sin q1)/z) and ((cos q2)/z, (sin q2)/z) is q2 - q1.
Since the "lines" are straight lines, each has an equation of the form
a (cos q)/z + b (sin q)/z = -1, or a cos q + b sin q + z = 0  The angle between two lines is defined as  
Motions include
a. rotations (adding a constant to q)
b. transformations of the form (x, y) --> . These preserve distance as defined above, since they do not change q, and they preserve angle since they add the same constants -e and -f to the coefficients a and b in the linear equation for all lines.
Model 16:
"Points" are vectors in R2.
A "line" is defined by a "point" p and a unit "direction vector" v  . It is the set of "points" of the form p  + v t  for all real values of t.
Distance between points on a line is defined by parameterizing the line as p  + v t  
and taking the difference t2 - t1 for the two points.
The angle between two intersecting lines is defined as the angle between their direction vectors where they intersect.
Motions include
adding a constant vector a to all "points"
rotating all vectors through a specified angle.

Model 17:
"Points" are vectors in R3 with unit length meeting the condition
z > 0 or (z = 0 and y > 0) or (z= 0 and y = 0 and x = 1)
A "line" is defined by a "point" p and a unit "direction vector" v that is orthogonal to p. It is the set of "points" of the form 
p cos q + v sin q  that satisfy the condition that
z > 0 or (z = 0 and y > 0) or (z= 0 and y = 0 and x = 1)
You can check that all such vectors have unit length.
Distance between points on a line is defined by parameterizing the line as p cos q + v sin q  
and taking the difference q2 - q1 for the two points.
The angle between two intersecting lines is defined as the angle between their direction vectors where they intersect.
Motions are all linear transformations of R3 that preserve length.
Model 18:
Introduce a Lorentz scalar product in R3 so that the scalar product of (x1, y1, z1) and (x2, y2, z2)
is x1x2 + y1y2 - z1z2.
"Points" are unit "timelike" vectors in R3 with z2 = x2 + y2 + 1 and z > 0.
A "line" is defined by a "point" p and a unit "spacelike" direction vector v with vx2 + vy2 -vz2 = 1 that is Lorentz-orthogonal to p. It is the set of "points" of the form 
p cosh t + v sinh t for all real values of t.  
You can check that all such vectors satisfy z2 = x2 + y2 + 1 and z > 0
Distance between points on a line is defined by parameterizing the line as p cosh t + v sinh t 
and taking the difference t2 - t1 for the two points.
The angle between two intersecting lines is defined as the angle between their direction vectors where they intersect.
Motions are all linear transformations of R3 that preserve Lorentz length; i.e. Lorentz transformations.