Blog Entries: 7

## About curve parallel and non-Euclid geometry mistake and absolute geometry completion

May say like this that the straight line parallel is the concentric circle parallel special row, but the concentric circle parallel is the curve parallel special row. Curve parallel is general parallel, but the straight line parallel is quite special being parallel.

Below we have a look at the concentric circle the nature.

As shown in Figure 1, great-circle and small circle concentric, straight line AB and the great-circle have point of intersection C, straight line AB and the small circle have point of intersection D, straight line EF are the great-circle tangents, and D is a tangential point, straight line GH is the small circle tangent, and C is a tangential point, R is the great-circle radius, r is the small circle radius.

Supposition

β=∠BCH, α=∠BDF, β and α is the corresponding angle mutually, β is the great-circle corresponding angle, α is the small circle corresponding angle, G=CD.

Then

90+β=∠BCO

According to sine law:

R/sin(90+β)=r/sin(90-α)

After the reorganization,

R/cosβ=r/cosα

Reorganizes again

Rcosα=rcosβ （1）

By (1) obviously, when R and r tend infinite, the great-circle is the straight line, the small circle is also a straight line, β=α, this is also straight line being parallel. Therefore the straight line parallel is only the concentric circle parallel extreme.

According to law of cosines:

G^2=r^2+R^2-2rRcos(180-(90-α+90+β)

After the reorganization,

G^2=r^2+R^2-2rRcos（α-β） （2）

After reorganization (2) formula, obtains

cos（α-β）=（r^2+R^2-G^2）/2rR （3）

By (3) obviously, when R and r tend infinite, the great-circle is the straight line, the small circle is also a straight line, β=α (corresponding angle equal), this is also straight line being parallel. Therefore the straight line parallel is only the concentric circle parallel extreme.

We may say, in a plane, so long as G^2=r^2+R^2-2rRcos (α-β), then two circles are parallel (concentric), otherwise is not parallel (not concentric).

After the above discussion, the conclusion which I obtain is:

When straight line the line, the curve is also the line, the line including the straight line and the curve. The line may not by the understanding be a straight line merely.

In the curve, the circle is a straight line. The straight line is that kind of radius infinitely great circle.

The straight line may the mutually perpendicular, the curve also be possible the mutually perpendicular. (i.e., straight line may mutually parallel, curve may also mutually parallel). Is vertical including the straight line between vertical and between curve vertical.

The angle side may be a straight line, may also be the curve, the angle including the linear angle and the curvilinear angle.

In the plane, the curve triangle's angle's summation may be bigger than or be smaller than π.

The surface may also be parallel mutually.The concentric spherical surface is mutually parallel.

The spherical surface is in the space these maintains with the fixed point the parallel distance is the r these spot set.The spherical surface and the center are parallel.

。。。。。。

Figure 1

Second, diagrammatic curve parallel

As shown in Figure 2:

In the plane, the rectangular two red side is parallel,

If we cause the rectangle to turn the cylinder, then that two red side is not parallel? I think them parallel, because between them the distance is constant.

If we cause the circular cylinder to turn the frustum, then that two red side is not parallel? I think them parallel, because between them the distance is constant.

If we cause the frustum to press the plane, then that two red side is not parallel? I think them parallel, because between them the distance is constant. (the attention, that two red side was precisely a concentric circle by now! ).

Therefore, once we acknowledge the straight line to be possible mutually parallel, then we can also not but acknowledge the curve also to be possible mutually parallel! The Euclid geometry only realized straight line parallel, but has neglected curve being parallel. He has received the historical limitation.

What is the parallel essence? This is the distance maintains constant invariable! No matter but whether is the straight line!

Figure 2

Third, about the non-Euclid geometry (including spherical geometry)

The spherical geometry thought that in the spherical surface does not have the parallel line. Actually this view is correct, is also wrong. In the spherical surface does not have the straight parallel line, but in the spherical surface actually has the curving parallel line. For instance, in the spherical surface latitude parallel is mutually parallel. Measures the grounding not mutually parallel, but this was not equal to said that in the spherical surface other curves may not be mutually parallel.

In the plane, through a known straight line's outside spot, has a straight line and the known straight line parallel, this is correct. In the spherical surface, through a known curve's outside spot, has a curve and the known curve parallel, this is also correct. Therefore may not remove the axiom of parallels in the spherical geometry! Certainly may not remove the axiom of parallels in the hyperbolic geometry!

The non-Euclid geometry thought: Crosses a straight line spot to be possible to make two straight lines and the known straight line at least parallel, actually this is the idea has the question. First, this confused has been parallel with the concept which did not intersect, because did not intersect was not equal to that was parallel, (in the Euclid geometry, because was parallel is limited in straight line, but in straight line, did not intersect as if is parallel, but in curve, did not intersect not necessarily is parallel); Second, this has confused the straight line and the curve difference, because the non-Euclid geometry said the straight line is a curve in fact!