Methodology

Let there exist two different spaces in the two-dimensional Euclidean space E². One space S is contained by a circle with the origin as its center and a radius of $\rho$. The space can be defined as $S = \{(x,y):x^2 + y^2 \leq \rho^2 \}$. Another space T is contained by a square of equal area, the center of which is also at the origin. The area of space T is equal to the area of S and therefore each side has a length of $\sqrt{\pi \rho^2} = \rho \sqrt{\pi}$. A formal definition of this space is $T = \{(x,y): \vert x\vert \leq {{\rho \sqrt{\pi}} \over 2}, \vert y\vert \leq {{\rho \sqrt{\pi}} \over 2} \}$

The morphing transformation maps any point $(x_0,y_0) \in S$ into a point $(x_1,y_1) \in T$. Of course, there is an entire family of bijective functions which satisfy this property, but few transformations can guarantee a small distance between (x₀,y₀) and (x₁,y₁). Another important criterion is that the transformation preserves Lebesgue measure. In other words, if a subspace $A \subset S$ is transformed into A', then the area of A` is equal to the area of A.

Two premises were used to define the morphing transformation. First, if the point (x₀,y₀) lies on a circle with the origin as its center and a radius r then its transformed point (x₁,y₁) lies on a square with the origin as its center and with an area ($\pi r^2$) equal to that of the circle with radius r. Second, the ratio of the distance along this circle from (r,0) to (x₀,y₀) divided by the circumference $2 \pi r$ is equal to the ratio of the distance along the perimeter of the square from $({{r \sqrt{\pi}} \over 2},0)$ to (x₁,y₁) divided by perimeter of the square $4r{\pi}^2$. 12pt

The equations for this transformation are

$$x_1 = \left [ {\rm I}_{(0,{{3 \pi} \over 4}]} (\theta) + {\rm... ...{7 \pi} \over 4}]} (\theta) \right ] \left ({s \over 2}\right )$$

$$- {\rm I}_{({\pi \over 4},{{3 \pi}\over 4}]}(\theta) \left [{... ...t [{{2s(\theta - {{5 \pi}\over 4})}\over\pi} \right ] \eqno(1)$$

$$y_1 = \left [ {\rm I}_{({\pi\over 4},{{5\pi}\over 4}]}(\theta... ...(0,{\pi\over 4}]}(\theta)\left [{{2s \theta} \over\pi}\right ]$$

$$- {\rm I}_{({{3\pi}\over 4},{{5\pi} \over 4}]} (\theta)\left ... ...ft [{{2s(\theta - {{7\pi}\over 4})} \over\pi}\right ] \eqno(2)$$

where the following substitutions are defined:

$$s = \sqrt{\pi(x_0^2+y_0^2)} \eqno(3)$$

$$\theta = \tan^{-1}{{y_0}\over{x_0}}+\pi {\rm I}_{( - \infty,0)}(x_0) \eqno(4)$$

These equations are fragmented with many indicator functions because, though the points along the circumference of a circle are easily defined ( $x=r\cos\theta$, $y=r\sin\theta$), a point along the perimeter of a square uses different equations depending on which of the four sides that point falls on. Without loss of generality, we will focus exclusively on those points which fall on the first eighth of the circle, where $0 \le \theta \le {\pi\over 4}$. We will call this set S', and we will analyze the transformation from S' to a set T'. The remaining points can be studied using the same procedure.