```{-# LANGUAGE PatternGuards #-}

-- | Geometric functions concerning lines and segments.
--
--   A @Line@ is taken to be infinite in length, while a @Seg@ is finite length line segment represented by its two endpoints.
module Graphics.Gloss.Geometry.Line
( segClearsBox

-- * Closest points
, closestPointOnLine
, closestPointOnLineParam

-- * Line-Line intersection
, intersectLineLine

-- * Seg-Line intersection
, intersectSegLine
, intersectSegHorzLine
, intersectSegVertLine

-- * Seg-Seg intersection
, intersectSegSeg
, intersectSegHorzSeg
, intersectSegVertSeg)

where
import Graphics.Gloss.Data.Point
import Graphics.Gloss.Data.Vector

-- | Check if line segment (P1-P2) clears a box (P3-P4) by being well outside it.
segClearsBox
:: Point	-- ^ P1 First point of segment.
-> Point 	-- ^ P2 Second point of segment.
-> Point 	-- ^ P3 Lower left point of box.
-> Point	-- ^ P4 Upper right point of box.
-> Bool

segClearsBox (x1, y1) (x2, y2) (xa, ya) (xb, yb)
| x1 < xa, x2 < xa	= True
| x1 > xb, x2 > xb	= True
| y1 < ya, y2 < ya	= True
| y1 > yb, y2 > yb	= True
| otherwise		= False

-- | Given an infinite line which intersects `P1` and `P1`,
--	return the point on that line that is closest to `P3`
closestPointOnLine
:: Point 	-- ^ `P1`
-> Point	-- ^ `P2`
-> Point	-- ^ `P3`
-> Point	-- ^ the point on the line P1-P2 that is closest to `P3`

{-# INLINE closestPointOnLine #-}

closestPointOnLine p1 p2 p3
= p1 + (u `mulSV` (p2 - p1))
where	u	= closestPointOnLineParam p1 p2 p3

-- | Given an infinite line which intersects P1 and P2,
--	let P4 be the point on the line that is closest to P3.
--
--	Return an indication of where on the line P4 is relative to P1 and P2.
--
-- @
--      if P4 == P1 then 0
--      if P4 == P2 then 1
--      if P4 is halfway between P1 and P2 then 0.5
-- @
--
-- @
--        |
--       P1
--        |
--     P4 +---- P3
--        |
--       P2
--        |
-- @
--
{-# INLINE closestPointOnLineParam #-}
closestPointOnLineParam
:: Point 	-- ^ `P1`
-> Point 	-- ^ `P2`
-> Point 	-- ^ `P3`
-> Float

closestPointOnLineParam p1 p2 p3
= (p3 - p1) `dotV` (p2 - p1)
/ (p2 - p1) `dotV` (p2 - p1)

-- Line-Line intersection -------------------------------------------------------------------------

-- | Given four points specifying two lines, get the point where the two lines cross, if any.
--   Note that the lines extend off to infinity, so the intersection point might not lie
--   between either of the two pairs of points.
--
-- @
--     \\      /
--      P1  P4
--       \\ /
--        +
--       / \\
--      P3  P2
--     /     \\
-- @
--
intersectLineLine
:: Point	-- ^ `P1`
-> Point	-- ^ `P2`
-> Point	-- ^ `P3`
-> Point	-- ^ `P4`
-> Maybe Point

intersectLineLine (x1, y1) (x2, y2) (x3, y3) (x4, y4)
= let	dx12	= x1 - x2
dx34	= x3 - x4

dy12	= y1 - y2
dy34	= y3 - y4

den	= dx12 * dy34  - dy12 * dx34

in if den == 0
then Nothing
else let
det12	= x1*y2 - y1*x2
det34	= x3*y4 - y3*x4

numx	= det12 * dx34 - dx12 * det34
numy	= det12 * dy34 - dy12 * det34
in	Just (numx / den, numy / den)

-- Segment-Line intersection ----------------------------------------------------------------------

-- | Get the point where a segment @P1-P2@ crosses an infinite line @P3-P4@, if any.
--
intersectSegLine
:: Point	-- ^ `P1`
-> Point	-- ^ `P2`
-> Point	-- ^ `P3`
-> Point	-- ^ `P4`
-> Maybe Point

intersectSegLine p1 p2 p3 p4
-- TODO: merge closest point check with intersection, reuse subterms.
| Just p0	<- intersectLineLine p1 p2 p3 p4
, t12		<- closestPointOnLineParam p1 p2 p0
, t12 >= 0 && t12 <= 1
= Just p0

| otherwise
= Nothing

-- | Get the point where a segment crosses a horizontal line, if any.
--
-- @
--                + P1
--               /
--       -------+---------
--             /        y0
--         P2 +
-- @
--
intersectSegHorzLine
:: Point 	-- ^ P1 First point of segment.
-> Point 	-- ^ P2 Second point of segment.
-> Float 	-- ^ y value of line.
-> Maybe Point
intersectSegHorzLine (x1, y1) (x2, y2) y0

-- seg is on line
| y1 == y0, y2 == y0	= Nothing

-- seg is above line
| y1 > y0,  y2 > y0	= Nothing

-- seg is below line
| y1 < y0,  y2 < y0	= Nothing

-- seg is a single point on the line.
-- this should be caught by the first case,
-- but we'll test for it anyway.
| y2 - y1 == 0
= Just (x1, y1)

| otherwise
= Just ( (y0 - y1) * (x2 - x1) / (y2 - y1) + x1
, y0)

-- | Get the point where a segment crosses a vertical line, if any.
--
-- @
--              |
--              |   + P1
--              | /
--              +
--            / |
--       P2 +   |
--              | x0
-- @
--
intersectSegVertLine
:: Point 	-- ^ P1 First point of segment.
-> Point 	-- ^ P2 Second point of segment.
-> Float 	-- ^ x value of line.
-> Maybe Point

intersectSegVertLine (x1, y1) (x2, y2) x0

-- seg is on line
| x1 == x0, x2 == x0	= Nothing

-- seg is to right of line
| x1 > x0,  x2 > x0	= Nothing

-- seg is to left of line
| x1 < x0,  x2 < x0	= Nothing

-- seg is a single point on the line.
-- this should be caught by the first case,
-- but we'll test for it anyway.
| x2 - x1 == 0
= Just (x1, y1)

| otherwise
= Just (  x0
, (x0 - x1) * (y2 - y1) / (x2 - x1) + y1)

-- Segment-Segment intersection -------------------------------------------------------------------

-- | Get the point where a segment @P1-P2@ crosses another segement @P3-P4@, if any.
intersectSegSeg
:: Point	-- ^ `P1`
-> Point	-- ^ `P2`
-> Point	-- ^ `P3`
-> Point	-- ^ `P4`
-> Maybe Point

intersectSegSeg p1 p2 p3 p4
-- TODO: merge closest point checks with intersection, reuse subterms.
| Just p0	<- intersectLineLine p1 p2 p3 p4
, t12		<- closestPointOnLineParam p1 p2 p0
, t23		<- closestPointOnLineParam p3 p4 p0
, t12 >= 0 && t12 <= 1
, t23 >= 0 && t23 <= 1
= Just p0

| otherwise
= Nothing

-- | Check if an arbitrary segment intersects a horizontal segment.
--
-- @
--                 + P2
--                /
-- (xa, y3)  +---+----+ (xb, y3)
--              /
--          P1 +
-- @

intersectSegHorzSeg
:: Point 	-- ^ P1 First point of segment.
-> Point 	-- ^ P2 Second point of segment.
-> Float 	-- ^ (y3) y value of horizontal segment.
-> Float        -- ^ (xa) Leftmost x value of horizontal segment.
-> Float 	-- ^ (xb) Rightmost x value of horizontal segment.
-> Maybe Point	-- ^ (x3, y3) Intersection point, if any.

intersectSegHorzSeg p1@(x1, y1) p2@(x2, y2) y0 xa xb
| segClearsBox p1 p2 (xa, y0) (xb, y0)
= Nothing

| x0 < xa	= Nothing
| x0 > xb	= Nothing
| otherwise	= Just (x0, y0)

where x0 | (y2 - y1) == 0 	= x1
| otherwise		= (y0 - y1) * (x2 - x1) / (y2 - y1) + x1

-- | Check if an arbitrary segment intersects a vertical segment.
--
-- @
--      (x3, yb) +
--               |   + P1
--               | /
--               +
--             / |
--        P2 +   |
--               + (x3, ya)
-- @

intersectSegVertSeg
:: Point	-- ^ P1 First point of segment.
-> Point 	-- ^ P2 Second point of segment.
-> Float 	-- ^ (x3) x value of vertical segment
-> Float	-- ^ (ya) Lowest y value of vertical segment.
-> Float	-- ^ (yb) Highest y value of vertical segment.
-> Maybe Point	-- ^ (x3, y3) Intersection point, if any.

intersectSegVertSeg p1@(x1, y1) p2@(x2, y2) x0 ya yb
| segClearsBox p1 p2 (x0, ya) (x0, yb)
= Nothing

| y0 < ya	= Nothing
| y0 > yb	= Nothing
| otherwise	= Just (x0, y0)

where y0 | (x2 - x1) == 0	= y1
| otherwise		= (x0 - x1) * (y2 - y1) / (x2 - x1) + y1

```