I am working on a very similar project as you are now. You must first create the Basecamp application. The lower the code should be connected:
$appName = 'your app name';
$appContact = 'your app email;
$basecampAccountId = 'xxxxx';
$basecampUsername = 'youremailhere';
$basecampPassword = 'yourpassword here';
$baseUrl = "https://basecamp.com/$basecampAccountId/api/v1";
$url= $baseUrl.'/projects.json';
$credentials = "$basecampUsername:$basecampPassword";
$helloHeader = "User-Agent: $appName ($appContact)";
echo $url.'<br>';
echo $credentials.'<br>';
echo $helloHeader.'<br>';
$ch = curl_init($url);
curl_setopt($ch, CURLOPT_USERPWD, $credentials);
curl_setopt($ch, CURLOPT_HEADER, true);
curl_setopt($ch, CURLOPT_RETURNTRANSFER, true);
curl_setopt($ch, CURLOPT_FOLLOWLOCATION, true);
curl_setopt($ch, CURLOPT_MAXREDIRS, 3);
curl_setopt($ch, CURLOPT_SSL_VERIFYPEER, true);
curl_setopt($ch, CURLOPT_SSL_VERIFYHOST, 2);
curl_setopt($ch, CURLOPT_CONNECTTIMEOUT, 30);
curl_setopt($ch, CURLOPT_TIMEOUT, 30);
curl_setopt($ch, CURLOPT_HTTPGET, true);
curl_setopt($ch, CURLOPT_HTTPHEADER, array($helloHeader));
$response = curl_exec($ch);
$errno = curl_errno($ch);
$error = curl_error($ch);
curl_close($ch);
print_r($response);
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