Undefined variable $realm in openid_authentication_request

I'm marking this as critical, because although it's only a notice there is an underlying logic error and the error message is very visible to new users on a site.

This code produces the error:

function openid_authentication_request($claimed_id, $identity, $return_to = '', $assoc_handle = '', $version = 2) {
  module_load_include('inc', 'openid');

  $ns = ($version == 2) ? OPENID_NS_2_0 : OPENID_NS_1_0;
  $request =  array(
    'openid.ns' => $ns,
    'openid.mode' => 'checkid_setup',
    'openid.identity' => $identity,
    'openid.claimed_id' => $claimed_id,
    'openid.assoc_handle' => $assoc_handle,
    'openid.return_to' => $return_to,
  );

  if ($version == 2) {
    $request['openid.realm'] = url('', array('absolute' => TRUE));
  }
  else {
    $request['openid.trust_root'] = $realm; // NOTE: $realm is NOT DEFINED anywhere in this scope, under ANY conditions.
  }

  // Simple Registration
  $request['openid.sreg.required'] = 'nickname,email';
  $request['openid.ns.sreg'] = "http://openid.net/extensions/sreg/1.1";

  $request = array_merge($request, module_invoke_all('openid', 'request', $request));

  return $request;
}

I don't know what variable the coder meant to use instead of $realm, or what value it is supposed to contain. But a brief scan over the function code shows that unless openid.inc defines $realm in its global code (and it doesn't), the variable scope of this function does not ever contain a $realm variable.