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.

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.
 

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*.

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*.

The correspondence of the points on P0P1 to those on m gives a coordinate system for m in which the points for large values of x (either positive or negative) will be close to the point R*, so we can think of R* as being a point on m that represents infinity.

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Homogeneous Coordinates for Points on an Affine Line.

From the previous work we can see that giving coordinates to points on an affine line also gives coordinates to points on a circle and coordinates to lines through a point on a circle.

For convenience we consider the ordinary points for an affine line as points on the line in the Euclidean plane given by { (x,1)}.

We say that a point (a,b) [different from (0,0) ] gives homogeneous coordinates for the ordinary point Px = (x,1) if the line though (a,b) and (0,0) passes through the point (x,1).

From similar triangles or a vector interpretation we can see that 
(a,b) gives homogeneous coordinates for (x,1) if and only if  x = a/b as long as b is not 0. 

We use <a,b> to denote the ordinary point on the affine line which corresponds to the homogeneous coordinates (a,b) when b  is not 0. 

Notice that <a,b> = <c,d>  where b and d are not zero if and only if a/b = c/d. 
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)}.

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 l and m showing these ideal points in view.
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 l, all lines parallel to l must pass through P*.

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Coordinates for points on an affine plane.

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"Continuity, inverses, and visualizing the infinite" with the circle, a single infinite point. [To come...]