Curves Bounding a Convex Body in the Plane
That is, `d=1`, `M` is "convex"- Distance is [Fejes Tóth][McClure/Vitale]
- Here `mu_{II}(M)` is a size measure on `M`, based on curvature
- `mu_{II}(M) : = int_M {:sqrt kappa_x dx :}`
- `kappa_x :=`curvature at point `x`
- For smoother curves, distance known up to `o(1//n^4)`[Ludwig]
`{:1/8:}(mu_{II}(M)//n)^2(1+o(1))`, as `n\rightarrow\infty`
Curvature of Curves
- Curvature `kappa_p = 1//r`, where
- Curve `M` (that is, `d=1`)
- `p in M`
- `r=` radius of circle osculating curve at `p`
- If `M` is a circular arc of angular span `theta`,
`int_M kappa_p dp = int_M {:1/r:} = (2 pi theta r){:1/r:} = 2 pi theta`
- `int_M kappa_p dp approx` turning angle of `M`
What Metric?
- Could use arc length `D_I(x, y) := \int_{x..y} ds` but bad at tight turns
- Could use turning angle `D_{III}(x,y) := \int_{x..y} kappa_s ds`, but misses scale
- Sum `\int_{x..y} (1 + kappa)ds` works ok, `D_{II} := int_{x..y} sqrt kappa_s ds` turns out to be better
Why `D_{II}`, `d=1`
- For a point `p` on a curve, put `p` at the origin, make tangent horizontal
- Close to `p=0`, curve looks like `y=f(x)`
- Consider approximating a function `f(x)`, `x in RR` near `0`
- By Taylor's Theorem,
`f(x) = f(0) + f'(0)x + f''(0)x^2//2 + O(x^3)`
- Here the coordinate system makes `f(0) = f'(0) = 0`, so
`f(x) = f''(0)x^2//2 + O(x^3)`
Why `D_{II}`, `d=1`, More
- When `f(x) approx x^2 f''(0)//2` (assume `f''(0)>0`)
- Max error of linear interp. is at midpoint `x//2`, and is `f(x)//4`
- Curvature `kappa_x approx f''(x) approx f''(0)`,
- `f(x) approx hat x^2`, where
- `hat x := x sqrt{:{:f''(0):}//2:} approx \int_{0..x} sqrt{:{:f''(y):}//2:} dy approx int_{0..x} sqrt kappa_y = D_{II}(x,0)`
- So: error `approx hat x^2 approx D_{II}(0,x)^2
