Mathematics 138
Parabolas in Galilean Geometry

Reading (all in Yaglom):
Pages 77-83, 91-98, and 104-109.  It's worth skimming the sections in between, but don't worry about the details. The remaining pages on Galilean geometry, up to page 157, contain some good ideas for term projects.
Since I have no technology for drawing parabolas, the diagrams in this handout are somewhat deficient, but there are plenty of good ones in Yaglom!)
In Euclidean geometry a circle has the following properties:
1. It is the set of all points with a fixed directed angle between the lines joining the point to the ends of a specified segment.
2. Its radius is invariant under all Euclidean isometries.
3. There is a Euclidean isometry that maps a given circle into itself, carrying one specified point on the circle into another specified point on the circle.
4. Through three given noncollinear points there is a unique circle.
In Galilean geometry a parabola has all the same properties!  The "radius" is closely related to the constant acceleration of a moving particle whose position-vs-time graph is a parabola.
The "directed angle" between an ordered pair of lines (not two rays) is the smallest positive angle through which the first line must be rotated to bring it into coincidence with the second.  It cannot be negative, and it cannot exceed a right angle. This diagram shows the circle generated by a horizontal segment and a directed angle of f = 45 degrees. 

At M1, M2, or M3 the directed angle between the lines MxA and MxB is 45 degrees. At A or B the tangent to the circle makes a 45 degree angle with the segment AB.  The segment AB subtends an angle of 90 degrees at the center of the circle (twice 45 degrees).
In general, if the directed angle between the lines to A and B is f, the angle subtended by AB at the center of the circle is 2f and the arc AB is 2rf.  The radius of the circle is therefore
r = length of arc AB/(2f)

Now let's define a "cycle" of Galilean geometry in the same way.
Choose any segment AB that does not lie on a special line.  For a point M on the cycle, the lines from M to A and B have slopes k and k1 respectively.  The angle between these lines, k1 - k, must have a fixed value a.  Points M and M1 in the diagram both have a approximately 1.

To find the equation satisfied by all points M for which k1 - k  = a, just introduce coordinates for A (a1, a2), B (b1, b2), and M (x, y).  Then

.   Multiply by (x - b1) (x - a1)
xy - a1y - b2 x + a1 b2 - xy + b1 y + a2 x - a2 b1 = ax2 - a b1 x - a a1 x + a a1b1.
Collect terms, and the equation takes the form

y = a x2 + 2 b x + c, where
a =   b =    c = 
 Since AB is an ordinary line, not a special line, the denominator b1 - a1 is not zero.

This is the equation of a parabola.
To find the radius of the parabola, use the same formula as for the circle.  The length of the arc AB is the same as the length of the segment AB, simply b1 - a1.
So r = length of arc AB/ (2a) = = (where a is the coefficient of x2)
The inverse of r (twice the coefficient a) can be regarded as the "curvature" of the parabola.

Cyclic rotations:
In the Euclidean plane, any direct isometry is either a translation or a rotation about some point in the plane.  A rotation holds fixed any circle whose center is the fixed point of the rotation, and its angle can be chosen to carry any point on the circle into any other.  in fact, the rotation holds fixed a whole family of concentric circles.

Similarly, in the Galilean plane, any direct isometry that is neither a pure translation nor a shear is a cyclic rotation that carries a family of parabolas into themselves.

Suppose we have a parabola with equation y = a x2 + c.
Carry out a shear with parameter v, followed by a translation of v/2a along the horizontal axis and a translation of v2/4a along the vertical axis.

Then x' = x +  v/2a
        y' = vx + y + v2/4a
So y' = vx + ax2 + c + v2/4a  
and ax'2 = a(x2 + vx/a +  v2/4a2) = ax2 + vx + v2/4a

Thus y' = a x'2 + c, so one point on the parabola has been carried into another point on the same parabola.  Although a is fixed for a given parabola,  v can still be chosen to move a point as far to the left or the right as may be desired, so any point on the parabola can be moved into any other.

Notice that the entire family of parabolas (fixed a but any value of c) is mapped into itself by a cyclic rotation.  Thus "parallel parabolas" related by a "special translation" along the vertical axis play the same role in Galilean geometry as do concentric circles in Euclidean geometry.
Here is the matrix that represents a cyclic rotation. C(v) =   Notice that
=
This means that for fixed a, the cyclic rotations form a one-parameter subgroup of the
group of Galilean isometries.  In a similar manner (though the analogy is not perfect) the 
rotations about a fixed point form a one-parameter subgroup of the
group of Euclidean isometries.

Just as the midpoints of parallel chords of a circle lie on a line (a diameter of the circle) in Euclidean geometry, so also the midpoints of parallel chords of a parabola lie on a line in Galilean geometry.  This line is always vertical -- a special line, hence perpendicular to the parabola just as a diameter is perpendicular to a circle.

This provides a nice way of thinking about cyclic rotations (see diagrams 84b and 84c in Yaglom.) When parallel chords are all horizontal, their midpoints lie on the special line that passes through the vertex of the parabola.  After a shear carries the parallel chords into "sloping" segments,  the special line through these chords no longer intersects the parabola at the vertex; instead, it intersects it where the tangent is parallel to all the chords.  The translation part of a cyclic rotation restores the vertex to its original location.

The diagrams 84b and 84c in Yaglom present the same phenomenon in an equivalent way.  If you consider parallel chords of a parabola with positive slope, then the diameter through their midpoints lies to the right of the vertex.  A cyclic rotation carries each original horizontal chord into a sloping chord but also moves the midpoint by an appropriate translation so that the parabola stays fixed.
Finally, consider the question of finding a cycle that passes through the three vertices of a triangle.  In Euclidean geometry, the cycle is a circle, and its diameter 2R is the ratio that appears in the Law of Sines:

This is obvious in the case where
angle A is a right angle,
as in the diagram.

In Galilean geometry the cycle is a parabola.  Again, twice its "radius" is the ratio that appears in the Galilean Law of Sines:

   (here a is the length of a side)
But the radius of a Galilean cycle is related to the coefficient a of the quadratic term in the equation for the parabola:

2R = (here a is the coefficient of the quadratic term)
But for motion described by
y = at2 + b t + c (using t rather than x for the horizontal coordinate)

the acceleration is = 2a (Yaglom calls this w)
So the coefficient of the quadratic term is half the acceleration, and we have a nice mechanical interpretation of the Law of Sines:
acceleration/2 = a = 2/2R = A/a = B/b = C/c

This also resolves some of the mystery of area and its dual, both of which are invariant under Galilean isometries. The area of a triangle is measured physically in units of meters times seconds, since it is an ordinary distance times a special distance.  The dual quantity,co-area, is a Galilean angle (measured in meters per second) times a special distance.  The ratio of co-area to area is measured in meters per second per second and is equal to B/b or C/c: it is half the acceleration of the motion determined by the three events that define the triangle.  Since this acceleration is an invariant property of the triangle, it is reasonable to have two invariant area-like properties of the triangle.