OK, I think there's no need to identify the intersection line as the intersecting shapes can differentiate from each other over that line ...

1. import peasy.*; // camera library
   PeasyCam cam;
   
   int shift = 0;
   
   PShape s1,s2,s3;
   
   void setup(){
     size(1000,1000,P3D);
     smooth(8);
     frameRate(60);
   
     cam = new PeasyCam(this, 1000);
   
     s1 = createShape(RECT, 0,0, 300,-300);
     s1.fill(255,0,0);
     s1.end();
2. 
     s2 = createShape(RECT, 0,0, 300,-300);
     s2.fill(0,255,0);
     s2.end();
3. 
     s3 = createShape(RECT, 0,0, 300,-300);
     s3.fill(0,0,255);
     s3.end();
   }
   
   void draw(){
     background(0);
     smooth(8);
    
     pushMatrix();
     translate(0,0,shift);
     shape(s1, 0,0);
     popMatrix();
   
     pushMatrix();
     translate(shift,0,0);
     rotateY(PI/(-2));
     shape(s2, 0,0);
     popMatrix();
    
     pushMatrix(); 
     translate(0,-shift,0);
     rotateX(PI/(-2));
     shape(s3, 0,0);
     popMatrix(); 
    
     shift += 1;
     if (shift>300){shift=0;}
    
   }
   
   
4. 
   

If the planes are perfectly vertical (or somehow otherwise aligned with the world coordinate system that makes the math easier) then all you have to do is a line to line intersection test as if you were looking down on them (making it a 2D problem). For arbitrary planes you'll have to do a google search for the required math, it is more of a geometry problem than a programming one.

thanks for your advices, asimes! will try to digest and do some tests accordingly.