- #1

phya

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The straight line parallel to each other is parallel. Concentric circles are parallel,too.

As shown in figure, There is a big circle,Oa，Another one is small, Oc.They are concentric circles. AB is a straight line. AB and Oa are intersections D, AB and Oc are intersections C. EF is a straight line. EF through point D. EF tangent and Oa. GH is a straight line. GH through point C. GH tangent and Oc. R is for Oa radius. r is for Oc radius.

set

β=∠BCH, α=∠BDF, G=CD

is

∠ADO=90-α

∠BCO=90+β

According to the cosine theorem:

G^2=r^2+R^2-2rRcos(180-(90-α+90+β)

After finishing to

G^2=r^2+R^2-2rRcos（α-β） （2）

After finishing (2)

cos（α-β）=（r^2+R^2-G^2）/2rR

If R →∞，r→∞, then

cos（α-β）→1

α→β

When R → ∞， Oa is a straight line， r → ∞，Oc is a straight line，too. This is straight line parallel!Therefore, straight line parallel to the curve of the parallel is special.

After the above discussion, I still have some conclusions are as follows:

Can mutually perpendicular lines, Curve can also mutually vertical.

Flat surface can be parallel, curved surface can also be parallel to each other.

...

As shown in figure, There is a big circle,Oa，Another one is small, Oc.They are concentric circles. AB is a straight line. AB and Oa are intersections D, AB and Oc are intersections C. EF is a straight line. EF through point D. EF tangent and Oa. GH is a straight line. GH through point C. GH tangent and Oc. R is for Oa radius. r is for Oc radius.

set

β=∠BCH, α=∠BDF, G=CD

is

∠ADO=90-α

∠BCO=90+β

According to the cosine theorem:

G^2=r^2+R^2-2rRcos(180-(90-α+90+β)

After finishing to

G^2=r^2+R^2-2rRcos（α-β） （2）

After finishing (2)

cos（α-β）=（r^2+R^2-G^2）/2rR

If R →∞，r→∞, then

cos（α-β）→1

α→β

When R → ∞， Oa is a straight line， r → ∞，Oc is a straight line，too. This is straight line parallel!Therefore, straight line parallel to the curve of the parallel is special.

After the above discussion, I still have some conclusions are as follows:

Can mutually perpendicular lines, Curve can also mutually vertical.

Flat surface can be parallel, curved surface can also be parallel to each other.

...