Does the 256-bit integer secret exponent define the X, Y pair? I'm trying to understand better how bitcoin works and I'm stuck. I've read a blog showing how to construct the WIF from the 256-bit secret integer and up to that point, I think I follow.

However I'm "lost" when it comes to the (X,Y) point.

In the following facetious example (where the 256-bit secret exponent as been set to '1' on purpose), what defines the value of X and Y?

A related question: for the same secret exponent ('1' in this case), could we have ended up with a different (X,Y)?

secret exponent: 1
  hex:           1
WIF:             KwDiBf89QgGbjEhKnhXJuH7LrciVrZi3qYjgd9M7rFU73sVHnoWn
  uncompressed:  5HpHagT65TZzG1PH3CSu63k8DbpvD8s5ip4nEB3kEsreAnchuDf
public pair x:   55066263022277343669578718895168534326250603453777594175500187360389116729240
public pair y:   32670510020758816978083085130507043184471273380659243275938904335757337482424
  x as hex:      79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798
  y as hex:      483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8
y parity:        even
key pair as sec: 0279be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798
  uncompressed:  0479be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798\
                   483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8
hash160:         751e76e8199196d454941c45d1b3a323f1433bd6
  uncompressed:  91b24bf9f5288532960ac687abb035127b1d28a5
Bitcoin address: 1BgGZ9tcN4rm9KBzDn7KprQz87SZ26SAMH
  uncompressed:  1EHNa6Q4Jz2uvNExL497mE43ikXhwF6kZm


That example has been generated with an utility you can find here:

http://blog.richardkiss.com/?p=371
I usually use the following analogy to oversimplify things: The secret key is how far you walk along a known curve starting from a known point and the public point is where on the curve you wind up when you finish. If you repeat the same walk, you will always wind up at the same place. The operation is irreversible because the curve is complex, you can only figure out where you wind up by taking steps. Given a destination, you cannot figure out how to get there because you must walk in integer steps.