Research Article
Geometry of Ovals in R2 in Terms of the Support Function
Department of Mathematics and Statistics, Mu`tah University,Al-Karak, P.O. Box 7, Jordan
We deal here with a smooth closed convex curve in R2, called an oval, which contains the origin in its interior. The support function is the function that measures the perpendicular distance from the tangent line at a point on the oval to the origin (Eggleston, 1958; Fillmore, 1969; Yaglom and Boltyanskii, 1961). The line from the origin that is perpendicular on that tangent is called the line of the support function of the oval. We assume that our oval is parametrized by θ, the angle between the line of the support function of the oval and the positive x-axis. If the oval is defined by f(θ) = (f1(θ), f2 (θ), then it is natural to say that, θε[0, 2π] and f is regular. It has been proved that if α represents the support function, then
(1) |
The previous formula for f in terms of the support function α is derived using the idea of the envelope of the tangent lines of a smooth closed convex curve in R2 (Hsiung, 1981; Struik, 1950). Moreover, the orientation of f is naturally counterclockwise. We will use such a formula to derive new formulas concerning the differential geometry of ovals in R2 for a detailed proof of Eq. 1, (Al-Banawi, 2004).
Curvature and focal points: We start with the following theorem concerning curvature and total curvature of ovals.
Theorem 1: Let f = (f1, f2) be an oval in R2 with α as a support function. Let κ be the curvature of f. Let Κ be the total curvature of f. Then
(1a) |
(1b) |
(1c) |
Proof:
a: Observe that,
f` = (α+α``)(–sinθ, cosθ)
and
f`` = (α`+α```)(–sinθ, cosθ)+(α+α``)(–cosθ, –sinθ)
Thus,
b: Now
f1 = α cosθ–α`sinθ
and
f2 = α sinθ+α`cosθ
So
(2) |
Now differentiate Eq. 2 twice and substitute in (a) to get the formula for the curvature in (b).
c: Recall that the total curvature of a curve f1 on [a, b] is
Thus,
Corollary 1: Let f be an oval in R2 with α as a support function. Then the trace of f is a circle iff
Where, c1, c2, c are constants.
Proof: Now if the trace of f is a circle, then κ(θ) is constant,
Now if, α(θ) = c1 cosθ+c2 sinθ+c, then,
∀θε[a, b]
Thus, the trace of f is a circle.
Now recall that the curve of focal points of a curve f is
(3) |
Where, v is a unit normal of f. Now substitute in Eq. 3 for f as in Eq. 1,
and choose a unit normal of the oval f to be v(θ) = (–cosθ, –sinθ). Then the curve of focal points of f is the essence of the next theorem.
Theorem 2: Let f be an oval in R2 with α as a support function. Then the curve of focal points of f is
(4) |
Corollary 2: Let f be an oval in R2. Let g be the curve of focal points of f. Then, g(θ) ≠ f(θ), ∀θε[0, 2π]
Proof: First of all, the regularity of f implies that the curvature of f is bounded, hence, α(θ)+α``(θ)≠ 0, ∀θε[0, 2π]. Now if g(θ1) = f(θ1) for some θ1ε[0, 2π], then by Eq. 1 and 4, we will have:
and
which is true only if α(θ1)+α``(θ1) = 0, which contradicts the regularity of f.
Calculation of the support function: It looks very easy to calculate the support function using Eq. 2 Also the support function is the solution of the second order differential equation
Nevertheless, The next theorem gives a good formula in the language of first order differential equations for calculating α.
Theorem 3. Let f be an oval in R2 with α as a support function. Then α is the solution of the first order differential equation
(5) |
Proof: Observe that
Also |f`| = α+α`` Thus
So,
We give two examples for using Eq. 5 to calculate α.
Example 1: For the unit circle f = (cosθ, sinθ), θε[0,2π], Eq. 5 becomes α` = 0. So α = c constant. But α(0) = 1. Thus, α = 1.
Example 2: Consider the embedding f: [0, 2π]→ R2 defined by:
which was constructed by Fillmore (1969). First of all we show that the trace of f is an oval. For, observe that
and
Thus,
Since κ(θ)>0, ∀θε[0, 2π], f is convex and its trace is an oval. Using Eq. 5, we have α` = –3sin3θ. Since α(0) = 10 we have α = cos3θ+9.
Ovals of constant width in R2: Ovals of constant width in R2 were studied by Mellish (1931). Mellish`s work was explained by Robertson (1984). First we introduce the definition.
Definition 1: An oval f in R2 is of constant width a if the perpendicular distance between support tangent lines at opposite points is always a. That is,
Theorem 4: Let f be an oval in R2. Then f is of constant width a iff
Proof: If f is of constant width a, then
Now assume that
Then
Thus,
(6) |
Where, c1, c2 are constants. Now replace θ by θ+π in Eq. 6 to get
(7) |
Add Eq. 6 to 7, with the fact that α(θ+2π) = α(θ) to get
Thus, f is of constant width α.
The next theorem is known historically as Barbier`s theorem, 1860 and had been proved in different methods. Even though, we use the idea of the support function to introduce a simple proof.
Theorem 5: All ovals in R2 of constant width a have the same length aπ.
Proof: Let f be an oval in R2 with constant width a and length L. Then
Thus, L = aπ.
Now we prove that there is a special curve conjugate to an oval of constant width in R2.
Theorem 6: Let f be an oval in R2 with constant width a. Then there exists a periodic curve w (with period π), parallel to f and at a distance from f.
Proof: For θε[0, 2π], define w by
(8) |
Using Eq. 1, with the fact that α(θ+π) = α–α(θ), α`(θ+π) = –α`(θ) we have:
Thus, w(θ+π) = w(θ) and so w is periodic with period π. Now
So if v(θ) = (–cosθ, –sinθ) is a unit normal of f, then v.w` = 0. Thus, v is also a unit normal of w and so w is parallel to f.
Now
So for θε[0, 2π], the distance between f(θ) and w(θ)
is always .
Corollary 3: Let f be an oval in R2 with constant width a. Let g be the curve of focal points of f and let w be the curve as in Eq. 8. Then the trace of f is a circle iff g(θ) = w(θ), ∀ θ ∈[0,2π]
Proof: If the trace of f is a circle, then
Where, c1, c2, c are constants, Corollary 1. Now substitute in Eq. 4 to get g(θ) = (c1, c2). Similarly, substitute in Eq. 8 to get
Now solving α(θ)+α(θ+π) = a with, α(θ) = c1 cosθ+c2 sinθ+c, we have and so w(θ) = (c1, c2). Thus, g(θ) = w(θ), ∀θε[0, 2π].
Now assume that g(θ) = w(θ),
We finish this work with the following argument. Since α is continuous with a continuous derivative and period 2π, it has a Fourier expansion
(9) |
Where, a0, a1, a2,...,b1, b2,... are constants. The above series converges to α(θ) for all θεR.
Since a(θ)+α(θ+π) = a, then
a0 = a
a2k = b2k = 0, k = 1, 2,...
Thus,
(10) |
The expansion in Eq. 10 allows a huge number of different curves of constant width in R2 by just assigning nonzero values, which preserves convexity, to a finite number of the constants and ignoring the rest.