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A correspondence between a Euclidean coordinate line and a semicircle.

Suppose a line is given with points P0 and P1 determining a correspondence between points on the line and the real numbers.

T assigns the point Px on the line to the unique point Qx on the semicircle that lies on the segment PxO where O is the center of the semicircle. So T(Px) = Qx

**Notes:**

**When x is a large negative real number, then Qx is
a point on the arc that is close to Q-.**
**Thus we can say that as x approaches infinity, Qx
approaches Q+, and similarly as x approaches negative infinity, Qx approaches
Q-.**

**Exercise: Find a formula for the length of this arc
as a function b and a.**

A real number x is close to infinity if Qx is close to Q+, and x is close to negative infinity if Qx is close to Q-.

P+ is on the ray P0P1 and every point on that ray lies
on the segment P0P+.

P- is on the ray P1P0 and every point on that ray lies
on the segment P-P0.

This extended line can be considered as the segment P-P+
. It corresponds to the semicircle with its endpoints Q- and Q+.

Suppose a line is given with points P0 and P1 determining
a correspondence between points on the line and the real numbers.

T assigns the point Px on the line to the unique point
Qx on the circle that lies on the segment PxQ* where Q* is the point on
the diameter of the circle opposite P0. So T(Px) = Qx.

**if a<x<b , so that Px is a point on the segment
PaPb,** **then Qx is a point on the arc of the semicircle determined
by QA and Qb.**

**Notes:When x is a large real number, then Qx is a point
on the circle that is close to Q*.**

**When x is a large negative real number, then Qx is
a point on the circle that is close to Q*.**

**Thus we can say that as x approaches infinity,
Qx approaches Q*, and similarly as x approaches negative infinity, Qx approaches
Q*.**

**Exercise: Find a formula for the length of this arc
as a function b and a.**

A real number x is close to infinity if Qx is close to Q* in the arc Q1Q*, and x is close to negative infinity if Qx is close to Q* in the arc Q-1Q*.

Exercise: Use correspondence of the extended line and the circle to visualize the statement: As x approaches 0 from above, 1/x approaches infinity while as x approaches 0 from below, 1/x approaches negative infinity.

Points and Lines: Notice that in the correspondence between
points on the line and points on the circle, every point Px on the
line corresponds uniquely to a line through Q* and a point on the circle
Qx and every line through Q* (with the exception of the line parallel to
the original line which is tangent to the circle at Q*) determines a unique
point on the line and the circle. Thus we can consider the pencil of lines
through the point Q* as being in correspondence with the extended line
with the tangent line at Q* corresponding to the point P*.

In this figure the coordinate circle can give coordinates to many lines at once- using a central similarity.

Consider a second line not parallel to the given coordinate line P0P1.

Draw a line perpendicular to *m* through the center
of the coordinate circle (constructed previously) to find a point on that
line and the coordinate circle. Call this point Q#. ( Q# is different
from Q*).

Now draw lines through Q# and Qx that meet *m *at
the point called Rx.

We can consider this as a projection, T, transforming
points on the circle Qx to points on the line *m,* so that T(Qx) =
Rx.

Continue in this fashion to give a correspondence
between points on the circle (with the exception of Q#) and points on *m*.
In particular label the point R* that corresponds to Q*.

Homogeneous Coordinates for Points on an Affine Line.

(a,b) gives homogeneous coordinates for (x,1) if and only if x = a/b as long as b is not 0.

Also, for a fixed a, when b is close to 0, Px = <a,b> is very large - i.e., close to P* and the corresponding line from (0,0) to (a,b) is close to the line {(a,0)}.

In this sense we say that** (a,0) gives homogeneous coordinates for P*,
or <a,0> = P*.**

Consider **two nonparallel lines,** ** l and
m,** in the ordinary Euclidean plane.

To each of these lines add respectively the distinct ideal points P* and Q* discussed previously.

Draw the lines

We will think of the line P*Q* determined by these two ideal points as an "ideal line" or the "horizon line."

Notes:

1. Any line containing an ordinary point has exactly one ideal point on it. Since the line P*Q* has at least two distinct ideal points on it, any point on the line P*Q* cannot be an ordinary point in the Euclidean plane. Thus any point on P*Q* must represent an ideal point for some ordinary line in the Euclidean plane.

2. A line in the Affine plane is still determined by exactly two points.

3. Distinct lines that meet at P* do not meet at another point in the Euclidean plane. Hence from the Euclidean point of view such lines are parallel lines. Since for any given point in the Euclidean plane there is only one line through that point that is parallel to

Thus **lines in the Euclidean plane are parallel if
and only if they meet at the same ideal point on the line P*Q*.**

Homogenous coordinates for points in the affine plane are given by triples: e.g. (2,5): = <2,5,1> = <6, 15,3> . Infinite points have the third coordinate `0`, such as <1,0,0> and <1,1,0>.