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intersections_2D: THEORY
%--------------------------------------------------------------------------
%
% Derived from paper "Intersection of Linear and Circular Components in 2D"
% by David Eberly. Available at:
%
% www.magic-software.com
%
% Author: Ricky Butler NASA Langley Research Center
%
%--------------------------------------------------------------------------
BEGIN
IMPORTING distance_2D,
det_2D,
lines_2D, parallel_2D
p,p0,p1,p2,pnot,u,v,v0,v1,v2: VAR Vect2
s,t,t1,t2: VAR real
nza: VAR nzreal
cross(p0,p1): MACRO real = det(p0,p1)
% | Basic | Parametric
% |------------------|-----------------
% Line : TYPE = [# p: Vect2, | point on the line| position at time 0
% v: Nz_vect2 #] | direction vector | velocity vector
% Let L0 = p0 + s*v == L0`p + s*L0`v
% Let L1 = p0 + s*v1 == L1`p + s*L1`v
L0,L1: VAR Line2D
det_line: LEMMA LET DELTA = L1`p - L0`p IN
L0`p + s*L0`v = L1`p + t*L1`v IMPLIES
det(L0`v,L1`v)*s = det(DELTA,L1`v) AND
det(L0`v,L1`v)*t = det(DELTA,L0`v)
% Three cases
% intersecting : det(L0`v,L1`v) /= 0
% parallel : det(L0`v,L1`v) = 0 AND det(DELTA,L0`v) /= 0
% same_line : det(L0`v,L1`v) = 0 AND det(DELTA,L0`v) = 0
intersect?(L0,L1): bool = det(L0`v,L1`v) /= 0
same_line?(L0,L1): bool = LET DELTA = L1`p - L0`p IN
det(L0`v,L1`v) = 0 AND det(DELTA,L0`v) = 0
% ----------- function to return intersection point --------------
intersect_pt(L0:Line2D,L1: Line2D | det(L0`v,L1`v) /= 0): Vect2 =
LET DELTA = L1`p - L0`p,
ss = det(DELTA,L1`v)/det(L0`v,L1`v) IN
L0`p + ss*L0`v
% ----------- basic properties ---------------
IMPORTING parallel_2D
same_line_sym: LEMMA same_line?(L0,L1) IMPLIES same_line?(L1,L0)
intersect_not_parallel: LEMMA intersect?(L0,L1) IMPLIES
NOT parallel?(L0`v,L1`v)
intersection_lem : LEMMA det(L0`v,L1`v) /= 0 IMPLIES
LET DELTA = L1`p - L0`p,
ss = det(DELTA,L1`v)/det(L0`v,L1`v),
tt = det(DELTA,L0`v)/det(L0`v,L1`v)
IN
L0`p + ss*L0`v = L1`p + tt*L1`v
lines_intersection(so:Vect2,vo:Nz_vect2,si:Vect2,vi:Nz_vect2|det(vo,vi) /= 0) : [real,real] =
LET ns = si-so,
d = det(vo,vi) IN
(det(ns,vi)/d, det(ns,vo)/d)
lines_intersection_sound : LEMMA
FORALL(so:Vect2,vo:Nz_vect2,si:Vect2,vi:Nz_vect2|det(vo,vi) /= 0):
LET (ko,ki) = lines_intersection(so,vo,si,vi) IN
so+ko*vo = si+ki*vi
pt_intersect : LEMMA on_line?(p,L0) AND on_line?(p,L1) AND
NOT same_line?(L0,L1) IMPLIES
intersect?(L0,L1)
intersect_pt_unique : LEMMA intersect?(L0,L1) IMPLIES
pnot /= intersect_pt(L0,L1) AND
on_line?(pnot,L0)
IMPLIES
NOT on_line?(pnot,L1)
same_line_lem : LEMMA p0 /= p1 AND
( on_line?(p0,L0) AND on_line?(p0,L1) AND
on_line?(p1,L0) AND on_line?(p1,L1) )
IMPLIES same_line?(L0,L1)
not_same_line : LEMMA on_line?(p,L0) AND NOT on_line?(p,L1) IMPLIES
NOT same_line?(L0,L1)
intersect_pt_lem : LEMMA NOT same_line?(L0,L1) AND
on_line?(pnot,L0) AND
on_line?(pnot,L1)
IMPLIES
intersect_pt(L0,L1) = pnot
END intersections_2D
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