# Notes for Math 371 by M. Flashman (Work in Progress).

• The Line and a Semicircle
• The Line and a Circle
• A Circle and many Parallel Coordinate Lines.
• Two lines and a Circle
• Homogeneous Coordinates for Points on an Affine Line.
• Infinite Points in an "Affine/Projective" Plane.
• Coordinates in the Affine Plane.

• 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.

• There is a one to one correspondence between the points this line and the points on a semicircle tangent at its midpoint to the point P0 on the line.
• This correspondence can be considered as a transformation T.

• 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
• It is not difficult to show that T is one to one and onto all the points of the semicircle with the exception of  the two end points of the semicircle, Q+ and Q-.
• The transformation T preserves the order of the points on the line and the semicircle in the sense that

• 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.

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Notes:

• When x is a large real number, then Qx is a point on the arc that is close to Q+.

•

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-.

• The length of the arc between QA and Qb is not |b-a|.

•

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

• We can consider the semicircle with Q+ and Q- as a way to visualize the concepts of real numbers approaching infinity.

•

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-.

• We can consider the line extended with two additional "ideal" points, P+ and P-.

•

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+.

• The extended line allows us to use geometric language to say that a real number x approaches infinity if Px (and thus Qx) approaches P+ ( corresponding to Q+) and x approaches negative infinity if Px (and thus Qx) approaches P- (corresponding to Q-).

•
• Exercise: Use correspondence of the extended line and the semicircle 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.

• A correspondence between a Euclidean coordinate line and a circle.

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

• There is a one to one correspondence between the points this line and the points on a circle tangent to the point P0 on the line.

•
• This correspondence can be considered as a transformation T.

•

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.

• It is not difficult to show that T is one to one and onto all the points of the circle with the exception of  the point Q*.

•
• The transformation T preserves the order of the points on the line and the semicircle in the sense that

•

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.

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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*.

• The length of the arc between QA and Qb is not |BA|.

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

• We can consider the circle with Q* as a way to visualize the concepts of real numbers approaching infinity.

• 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*.

• We can consider the line extended with one additional "ideal" points, P*. This extended line corresponds to the circle.

•
• The extended line allows us to use geometric language to say that a real number x approaches infinity if Px (and thus Qx) approaches P* ( corresponding to Q*) and P1 is in the segment P0PX, and x approaches negative infinity if Px (and thus Qx) approaches P* (corresponding to Q*)  and P-1 is in the segment PXP0.

• 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*.

A circle and many Parallel Coordinate Lines.
In
this figure the coordinate circle can give coordinates to many lines at once- using a central similarity.
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How to see a point at infinity for one coordinate line:
Projecting the coordinate circle onto a second line m.

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*.

#### 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)}.

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

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How to see infinite points in an "affine/projective" plane.

#### 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*.

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

#### Sorry, this page requires a Java-compatible web browser.  Coordinates for points on an affine plane.

Here are two sketches of the affine plane showing coordinates for some key points.
In the first sketch the controlling points are P(0,0), P(1,0), P(0,1) P(*,0) and P(0,*).

In the second sketch, the coordinates of all points are determined by just four points, P(0,0), P(1,1), P(*,0) and P(0,*). In this sketch  is also shown how to find P(1,0), P(0,1), P(2,0) and P(0,2) from the four points- as well as some of the homogeneous coordinates for these points.
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>.