C=4(1+¾²)r

Why?

I don't think anyone understood your post where you explained that. Can you clarify what you wrote there?

Analysis of a quarter of the frame square:

In 1 m radius scale, total distance of the ¼ square frame equals 2m. It has two chambers: 2m=1m+1m each.

So, the two chambers give full distance.

The ¼ circle within has two chambers as well: 1m+0.5625m each. The average=(1m+0.5625m):2=0.78125.

This is the proportion between the circle & the frame.

So the curve within ¼square will be (0.78125:1)×2 m = 1.5625m.

Total circle=¼circle×4=1.5625m×4=6.25 m.

Phi=½circle AKA C:d=6.25m:2m=3.125.

0.78125 comes from quadratic value of ¾ s>> ¾²=9/16 AKA 0.5625. It is gaining of one chamber for one ¼ circle.

It is added with another chamber which get full gaining.

Total : 1+0.5625=1.5625 (either by meters or by second).

Comparion:

1.5625m:2m=0.78125.

This value is correcting the previous 1½:2 AKA ¾.

It's 1.5625, not 1.5, coz we play in 2 dimention calculation (slope). Not 1 dimention (straight distance AKA d=½.a.t²) which gets 1 part of velocity+ ½ part of acceleration, whose distance is compared to 2 max, resulting pure ¾)

Equation:

C=(¾²+1²):2×8m=0.78125×8m=6.25 m.

Conclussion:

frame:curve=1:0.78125

Radius:curve=0.5:0.78125=1:1.5625