Circles and Squares

[wingnut]

Hi, have a problem that I am not sure on how to approach it exactly.

So given a circle with radius, r, would have a perimeter pi*2*r, and an area pi*r^2,

And given a square with similar dimensions to the circle, so the square has a "radius" of r with a perimeter of 8*r and an area of 4*r^2 (each side of the square is 2*r in length)

What I need to get is an equation that would describe the circle transforming into the square. Specifically I need the perimeter of the object as it changes from a circle into a square, preferably as a function of area. SO when I add a little bit of area to the circle, the circle can only grow into a square, it cant become a bigger circle.


Thanks for any help or direction you can offer.

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More background if you are curious as to why this matters:

In a confined pore, a blob of stuff will start as a sphere until it reaches the walls of the pore. As more material is added to the blob the blob must deform from a sphere to the shape of the pore. In the model I am using to describe porespace phenomena, the energy of the system is proportional to the surface area of the blob. In two dimensions this is the equivalent of the perimeter of a circle. So I am looking to determine when it is energetically favorable to deform the circle as opposed to create a new blob somewhere else in the porespace.

I am trying to solve the special case in two dimensions where you have 2 square shaped pores. so a circular (2D blob) will start in one of the pores, then grow until it reaches the walls. It prefers to be a circle because a circle is the minimum surface a 2D droplet can have. So the question is when the circle gets big enough to fill the pore, when is it more energetically favorable to deform the current blob, or form a brand new one.

Some specifics: the left pore is 30x30 units, the right pore is 20x20. This is done on a square lattice so any increase in volume is quantized (you have to add an entire lattice cell to increase the size of the blob), this is a Monte Carlo Lattice gas simulation.

 

[HallsofIvy]

Hi, have a problem that I am not sure on how to approach it exactly.

So given a circle with radius, r, would have a perimeter pi*2*r, and an area pi*r^2,

And given a square with similar dimensions to the circle, so the square has a "radius" of r with a perimeter of 8*r and an area of 4*r^2 (each side of the square is 2*r in length)

You need to specify what you mean by "radius" of a square. Apparently you mean half the side of the square so that a circle of that "radius" could be inscribed in it. It would make just as much sense to me to say that the radius is half the length of a diagonal so that a circle of that "radius" could be circumscribed about the square.

What I need to get is an equation that would describe the circle transforming into the square. Specifically I need the perimeter of the object as it changes from a circle into a square, preferably as a function of area. SO when I add a little bit of area to the circle, the circle can only grow into a square, it cant become a bigger circle.

There are an infinite number of such functions. Do you want a linear function or some other condition?

Thanks for any help or direction you can offer.

----------
More background if you are curious as to why this matters:

In a confined pore, a blob of stuff will start as a sphere until it reaches the walls of the pore. As more material is added to the blob the blob must deform from a sphere to the shape of the pore. In the model I am using to describe porespace phenomena, the energy of the system is proportional to the surface area of the blob. In two dimensions this is the equivalent of the perimeter of a circle. So I am looking to determine when it is energetically favorable to deform the circle as opposed to create a new blob somewhere else in the porespace.

I am trying to solve the special case in two dimensions where you have 2 square shaped pores. so a circular (2D blob) will start in one of the pores, then grow until it reaches the walls. It prefers to be a circle because a circle is the minimum surface a 2D droplet can have. So the question is when the circle gets big enough to fill the pore, when is it more energetically favorable to deform the current blob, or form a brand new one.

Some specifics: the left pore is 30x30 units, the right pore is 20x20. This is done on a square lattice so any increase in volume is quantized (you have to add an entire lattice cell to increase the size of the blob), this is a Monte Carlo Lattice gas simulation.

 

[wingnut]

An object starts as a circle with radius r, the circle then transforms into a square with sides of length 2r. As a circle it has the perimeter 2*r*pi, when it finishes transformation to a square it has the perimiter 8r. I need an equation that describes the perimeter of the object during the transition, preferably linear:

Perimeter = f(area)

So the whole thing looks like:
Start: P = 2*r*pi
Finish: P = 8r
Inbetween: P = f(area):

what is f(area)?

I hope this helps, not sure how much more clarification I can give, but the perimeter I need to solve for is in between the value of the circle's perimeter and the square's perimeter:

2*r*pi < "P = f(area)" < 8r

 

[wingnut]

So I just thought of something that might really help out. The object I am trying to figure out is a squircle, http://mathworld.wolfram.com/Squircle.html

I need an equation of the perimeter of a squircle as a function of area while the squareness parameter goes form 0 to 1, with the starting circle having a radius of r, and the resulting square having a side length of 2r.