Non-overlapping rectangles constrained to a boundary

Question

I am trying to model placement of parts on a circuit board. Without any meaningful constraints, my basic schema looks like this:

create table part (
    part_id bigserial primary key,
    name text not null,
    width double precision not null,
    height double precision not null
);
create table board (
    board_id bigserial primary key,
    width double precision not null,
    height double precision not null
);
create table board_part (
    board_id bigint not null references board,
    part_id bigint not null references part,
    position point not null
);

^{(SQL Fiddle, Visualization)}

For b and b2 any board_parts, I want to enforce the following constraints:

1. b lies on the board:

   box(b.position, point(b.part.width,b.part.height))
       <@ box(point(0,0), point(b.board.width,b.board.height))
   

2. b and b2 do not overlap if they lie on the same board:

   b.board_id != b2.board_id or
   not (box(b.position, point(b.part.width,b.part.height))
           && box(b2.position, point(b2.part.width,b2.part.height)))
   

How can I achieve this (without too much data duplication)? Changing the schema is fine.

Here is my best attempt (SQL Fiddle), taking inspiration from Erwin's answer to my previous question. It enforces the constraints I wanted, but has a lot of duplicate data in the board_part table. I imagine I could write a function to fill in the board_width, board_height, part_width, and part_height fields automatically, but it still feels wrong having so much duplicate data around. Also, keying to the width/height fields feels like a hack.