I don't know... But P3D uses OpenGL and default mode uses Java2D, so it is normal to have a different behavior.
And curve() is just passed to the corresponding PGraphics, and their implementation probably differ, hence the difference.

The trouble is that the P3D behaviour is the 'correct', desired behaviour. If you think about the code, the first rendering really doesn't make sense. This becomes especially problematic when I want to generate PDF files, which use the 2D rendering.

In JAVA2D the default fill is white. In OpenGL there is not default fill.

Solution: use noFill();

Thanks, amnon.owed. That does solve the problem for drawing to screen. However, it doesn't seem to solve the issue for pdf, which is where I'm ultimately trying to get - this code, for example, produces an image like the first I posted above (I also tried fill(0, 0), which also doesn't work - so it seems it's not possible to override  the PDF renderer's fill settings (?)):

The problem is your code contains errors. Just take a look at the reference and examples. You are currently mixing up two different ways to create a PDF, so the code example you posted doesn't even run. There are two ways to fix your code example.

Adapted Code 1

1. import processing.pdf.*;
2. 
   
3. void setup() {
4.   size(300, 300, PDF, "output.pdf");
5.   background(255);
6.   smooth();
7.   curveDetail(100);
8.   noFill();
9.   for (int i=0; i<50; i++) {
10.     curve(0-(i*i), height/4, 0, 0, width, height, width+(i*i), ((height/4)*3));
11.   }
12.   exit();
13. }

Adapted Code 2

1. import processing.pdf.*;
2.  
3. void setup() {
4.   size(300, 300);
5.   beginRecord(PDF, "output.pdf");
6.   background(255);
7.   smooth();
8.   curveDetail(100);
9.   noFill();
10.   for (int i=0; i<50; i++) {
11.     curve(0-(i*i), height/4, 0, 0, width, height, width+(i*i), ((height/4)*3));
12.   }
13.   endRecord();
14.   exit();
15. }

Ah, yes, I accidentally mixed the two different ways here, but I hadn't done so working with pdf on my own. The error I made, though, which I can see from your example, is that I placed the beginRecord() function too late in the process, after the setup functions (i.e. noFill(), etc).