Surely the white rectangle with a hole in it is no more white than the background?!?

Here's the white rectangle with a hole in it solution:

Thanks a lot for the responce! It's ment for a logo generator and therefore i want the output file to be transparent and not have a white background. So i ment to remove the loop and background in the final static version. My bad that i pasted the looping version with a background redrawing.

@ amnon.owed

That seems very much like what i'm looking for but there is a problem with it. When i draw a line outside side the ellipse area it will move it inside the ellipse area. 

I tried to play around with it and got the following result. In this code the blue line should be outside the visible ellipse area but it moves the line in to the area.

I'm very thankfull for your help!

2. int rad = 200;
3. PVector ab, r, ac, ad;
4.  
5. void setup() {
6.   size(600, 600);
7.   ab = new PVector();
8.   ac = new PVector();
9.   ad = new PVector();
10.   r = new PVector();
11.   smooth();
12. }
13.  
14. void draw() {
15.   background(255);
16.   translate(300, 300);
17.   rotate(radians(frameCount));
18.   
19.   ab.set(20, 40, 0);
20.   r.set(4300,-2300, 0);
21.   ac.set(5000, 3000,0);
22.   ad.set(4000, 7000,0);
23.   
24.   ab.limit(rad);
25.   ac.limit(rad);
26.   ad.limit(rad);
27.   r.limit(rad);
28.   
29.   strokeWeight(5);
30.   
31.   stroke(255,0,0);
32.   line(ab.x, ab.y, r.x, r.y);
33.   
34.   stroke(0,0,255);
35.   line(ac.x, ac.y, ad.x, ad.y);
36. 
    
37. }

Option 1 is by far the easiest. Both points are inside the circle. You don't have to do anything, just draw the line.

Option 2 may be easy, but could provide tricky. If the line does not intersect with the circle, it's easy. Because similarly to option 1, you do not have to do anything, just draw the line. If however, both points are outside the circle, but the line between them intersects the circle, you have part of the line inside the circle. Which means you need to 'do stuff' and then draw this line. To be continued...

Option 3 is the hardest, because you always have to do stuff, knowing that the line is partially in and partially outside the circle. For option 3 - and as stated in some cases for option 2 - you will need to find the intersection point(s) between the line and the circle. Unfortunately this is not as easy as you might hope. You will need some real math do find the right points and make sure everything goes smoothly in all situations.

Note that just limiting the magnitude of a PVector is a very easy method to limit points, but will provide incorrect results when it comes to lines! Because it limits the magnitude from center to point A or B and does not limit of the line as such between A and B. In other words, this solution will create a different line instead of the same line but limited.

Now for the good news! Here is a code example that can handle and shows all possible situations. A line outside the circle (these are drawn red). A line inside the circle (these are drawn green). And a moving point plus mouse point combination that allows the user to interactively create all the possible line types (A out, B out / A in, B in / A out, B in / A in, B out). When you have an in/out combination, you also need to choose the right line segments aka the line within the circle to draw (these are drawn blue).

Code Example

1. int rad = 200;
2. PVector origin, mouse, moving;
3. PVector inA, inB, outA, outB;
4.  
5. void setup() {
6.   size(900, 900);
7.   origin = new PVector();
8.   mouse = new PVector();
9.   moving = new PVector();
10.   inA = new PVector(100, -50);
11.   inB = new PVector(-100, 100);
12.   outA = new PVector(-400, -350);
13.   outB = new PVector(300, -200);
14.   noFill();
15.   smooth();
16. }
17.  
18. void draw() {
19.   background(255);
20.   translate(width/2, height/2);
21.  
22.   moving.set(sin(frameCount*0.01)*width/2, 0, 0); 
23.   mouse.set(mouseX-width/2, mouseY-height/2, 0);
24.  
25.   drawLine(inA, inB); // completely in
26.   drawLine(outA, outB); // completely out
27.   drawLine(mouse, moving); // interactive
    
28. 
    
29.   stroke(0);
30.   ellipse(origin.x, origin.y, rad*2, rad*2);
31. }
32.  
33. void drawLine(PVector a, PVector b) {
34.   stroke(0);
35.   ellipse(a.x, a.y, 10, 10);
36.   ellipse(b.x, b.y, 10, 10);
37.   ArrayList <PVector> points = lineIntersectCircle(a, b, origin, rad);
38.   if (points.size()>0) {
39.     strokeWeight(1);
40.     line(a.x, a.y, b.x, b.y);
41.     stroke(0, 0, 255);
42.     for (PVector p : points) {
43.       ellipse(p.x, p.y, 20, 20);
44.     }
45.     if (points.size()==2) {
46.       a = points.get(0);
47.       b = points.get(1);
48.     } else {
49.       boolean inA = a.mag()<rad;
50.       if (!inA) {
51.         a = points.get(0);
52.       } else {
53.         b = points.get(0);
54.       }
55.     }
56.   } else {
57.     boolean inA = a.mag()<rad;
58.     boolean inB = b.mag()<rad;
59.     if (inA&&inB) { stroke(0, 255, 0); } else if (!inA&&!inB) { stroke(255, 0, 0); }
60.   }
61.   strokeWeight(5);
62.   line(a.x, a.y, b.x, b.y);
63. }
64.  
65. ArrayList <PVector> lineIntersectCircle(PVector a1, PVector a2, PVector c, float r) {
66.   ArrayList <PVector> points = new ArrayList <PVector> ();
67.  
68.   float a  = (a2.x - a1.x) * (a2.x - a1.x) + (a2.y - a1.y) * (a2.y - a1.y);
69.   float b  = 2 * ( (a2.x - a1.x) * (a1.x - c.x) + (a2.y - a1.y) * (a1.y - c.y)   );
70.   float cc = c.x*c.x + c.y*c.y + a1.x*a1.x + a1.y*a1.y - 2 * (c.x * a1.x + c.y * a1.y) - r*r;
71.   float d  = b*b - 4*a*cc;
72.   float e  = sqrt(d);
73.   float u1 = ( -b + e ) / ( 2*a );
74.   float u2 = ( -b - e ) / ( 2*a );
75.  
76.   if ( !( (u1 < 0 || u1 > 1) && (u2 < 0 || u2 > 1) ) ) {
77.     if ( 0 <= u1 && u1 <= 1) { points.add( new PVector( lerp(a1.x, a2.x, u1), lerp(a1.y, a2.y, u1) ) ); }
78.     if ( 0 <= u2 && u2 <= 1) { points.add( new PVector( lerp(a1.x, a2.x, u2), lerp(a1.y, a2.y, u2) ) ); }
79.   }
80.  
81.   return points;
82. }

I'm almost there, as you can see but now i'm having another little issue.

I need the lines to rotate around one point that is not similar to the "visible ellipse area"s center point. The ellipse area should stay at the same position/not rotate. I's there a quick fix for this or? Sorry if i'm starting to get on your nerves but your help has really been useful!

2. import processing.pdf.*;
3. 
   
4. //radius af ellipse området
5. int rad = 300;
6. 
   
7. //PVector variabler for stjerner og cirkelområde
8. PVector origin, s1, s2, s3, s4, s5, s6, s7, s8;
9. 
   
10. //opsætning variabler 
11. void setup() {
12.   size(900, 900);
13.   //noLoop();
14.   
15.   //begynd pdf
16.   beginRecord(PDF, "a.PDF");
17.   
18.   //ellipsens midtpunkt
19.   origin = new PVector(100,100);
20.   
21.   //punkternes vaerdier
22.   s1 = new PVector(100, 100);
23.   s2 = new PVector(500, 500);
24.   s3 = new PVector(-100, -0);
25.   s4 = new PVector(100, 0);
26.   s5 = new PVector(0, -350);
27.   s6 = new PVector(300, -200);
28.   s7 = new PVector(0, 0);
29.   s8 = new PVector(0, 700);
30.     
31. }
32. 
    
33. ArrayList <PVector> lineIntersectCircle(PVector a1, PVector a2, PVector c, float r) {
34.   
35.   ArrayList <PVector> points = new ArrayList <PVector> ();
36.   
37.   float a  = (a2.x - a1.x) * (a2.x - a1.x) + (a2.y - a1.y) * (a2.y - a1.y);
38.   float b  = 2 * ( (a2.x - a1.x) * (a1.x - c.x) + (a2.y - a1.y) * (a1.y - c.y)   );
39.   float cc = c.x*c.x + c.y*c.y + a1.x*a1.x + a1.y*a1.y - 2 * (c.x * a1.x + c.y * a1.y) - r*r;
40.   float d  = b*b - 4*a*cc;
41.   float e  = sqrt(d);
42.   float u1 = ( - b + e ) / ( 2*a );
43.   float u2 = ( -b - e ) / ( 2*a );
44.      
45.   if (!( (u1 < 0 || u1 > 1) && (u2 < 0 || u2 > 1) )) {
46.     
47.     if ( 0 <= u1 && u1 <= 1) { points.add( new PVector( lerp(a1.x, a2.x, u1), lerp(a1.y, a2.y, u1) ) );}
48.     if ( 0 <= u2 && u2 <= 1) { points.add( new PVector( lerp(a1.x, a2.x, u2), lerp(a1.y, a2.y, u2) ) );}
49.    
50. }
51.     
52.   return points;
53.   
54. }
55.  
56. void draw() {
57.   
58.   translate(width/2, height/2);
59.   rotate(radians(frameCount));
60.   background(255,255,255);
61.     
62.   //tegn streger imellem variabler  
63.   drawLine(s1, s2);
64.   drawLine(s3, s4);
65.   drawLine(s5, s6);
66.   drawLine(s7, s8);
67.   
68.   rotate(radians(0));
69.   
70.   //translate(0, 0);
71.   
72.   //markering af ellipse
73.   noFill();
74.   ellipse(origin.x, origin.y, rad*2, rad*2);
75. 
    
76. }
77.    
78. 
    
79. void drawLine(PVector a, PVector b) {
80.   
81.   noStroke();
82.    
83.   ArrayList <PVector> points = lineIntersectCircle(a, b, origin, rad);
84.   
85. if (points.size()>0) {
86.     stroke(0,0,0);
87.     
88.   for (PVector p : points) {
89.     }
90.     
91.   if (points.size()==2) {
92.       a = points.get(0);
93.       b = points.get(0);
94.     }
95.   
96.   else {
97.       boolean inA = a.mag()<rad;
98.   
99.       if (!inA) {
100.           a = points.get(0);
101.           }
102.        else {
103.           b = points.get(0);
104.           }
105.     }
106. } 
107. 
     
108.   else {
109.     boolean inA = a.mag()<rad;
110.     boolean inB = b.mag()<rad;
111.         
112.     if (inA&&inB) { stroke(0,0,0); } else if (!inA&&!inB) {}
113. 
     
114.   }
115.   line(a.x, a.y, b.x, b.y);
116. }
117. 
     
118. void mouseReleased() {
119.   endRecord();
120. }