Re: Connecting semi-circles

Dear Bram,

My apologies for not being so familiar with your work, but allow me to
please take a stab at this.

What you are doing is something closely akin to a 'quadrature' or
'squaring the circle', or its inverse 'circling the square'
with different variations.

It is like taking a hypothetical string forming a circle which has a
continuous surface area and then moving it to a square with a perfect,
yet highly discrete perimeter area. They are antithetical structures.

Their surface areas could transform into each other, but their
structures are internally different. In other words, a square is a
square and a circle is a circle. But that does not mean they cannot
internally change into one another.

One of the great distinctions between them is that a square can have
discrete cutting, while the surface area of a circle cannot do that. It
is continuous. You will hit the Banach Tarski Paradox.

Supposedly, these two structures are paradoxical to one another.

There is no way to get a 'smooth corner.' That would be an oxymoron
in math. :-)

However, I am working on a transformation device to do just that, but
it is still on the chalkboard. It involves math of distinctions and not
exclusions.

Distinction does not necessarily mean exclusion.

B.T. Pittsburgh, USA

b.fokke@xxxxxxxxx wrote:
Hello everyone!

I'm programming a piece of software that simulates a trajectory between
a couple of points. I need smooth corners and therefore I'm using a
combination of straight pieces and semicircles. Say I've got two pieces
of track leading up to two points p1 and p2, with directions d1 and d2.
Is it possible to create two semicircles so that:

- Those two semicircles connect the two points
- There is no angle between the semicircle and the straight track at
the place where they meet (p1 and p2). i.e. the track continues
smoothly.
- There is no angle between the semicircles at the place where they
meet i.e. the track continues smoothly.

And if so: how do I compute theseand how do I compute these?

Thanks in advance,
Bram Fokke
Utrecht, the Netherlands

.

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