math - 两条线段相交或重叠

Question

给定两条线段，每条线段由两点定义，我怎样才能最有效地确定它们的交点或重叠？

相交被定义为两个线段在实际接触中相互交叉。重叠被定义为在每个端点都具有相同的 x 值或相同的 y 值，但至少一端在另一条线的点之间。

我问这个是因为我有兴趣找到一个例程来计算两者并将交点作为线段返回，即使它的两个点都在同一位置(交点而不是重叠)。

使用 Lua，我计算重叠的函数是:

local function getParallelLineOverlap( ax, ay, bx, by, cx, cy, dx, dy )
    local function sortAB( a, b )
        return math.min( a, b ), math.max( a, b )
    end

    ax, bx = sortAB( ax, bx )
    cx, dx = sortAB( cx, dx )

    local OverlapInterval = nil
    if (bx - cx >= 0 and dx - ax >=0 ) then
        OverlapInterval = { math.max(ax, cx), math.min(bx, dx) }
    end

    return OverlapInterval
end

同样使用 Lua，我计算交集的函数是:

local function doLinesIntersect( a, b, c, d )
    -- parameter conversion
    local L1 = {X1=a.x,Y1=a.y,X2=b.x,Y2=b.y}
    local L2 = {X1=c.x,Y1=c.y,X2=d.x,Y2=d.y}

    -- Denominator for ua and ub are the same, so store this calculation
    local _d = (L2.Y2 - L2.Y1) * (L1.X2 - L1.X1) - (L2.X2 - L2.X1) * (L1.Y2 - L1.Y1)

    -- Make sure there is not a division by zero - this also indicates that the lines are parallel.
    -- If n_a and n_b were both equal to zero the lines would be on top of each
    -- other (coincidental).  This check is not done because it is not
    -- necessary for this implementation (the parallel check accounts for this).
    if (_d == 0) then
        return false
    end

    -- n_a and n_b are calculated as seperate values for readability
    local n_a = (L2.X2 - L2.X1) * (L1.Y1 - L2.Y1) - (L2.Y2 - L2.Y1) * (L1.X1 - L2.X1)
    local n_b = (L1.X2 - L1.X1) * (L1.Y1 - L2.Y1) - (L1.Y2 - L1.Y1) * (L1.X1 - L2.X1)

    -- Calculate the intermediate fractional point that the lines potentially intersect.
    local ua = n_a / _d
    local ub = n_b / _d

    -- The fractional point will be between 0 and 1 inclusive if the lines
    -- intersect.  If the fractional calculation is larger than 1 or smaller
    -- than 0 the lines would need to be longer to intersect.
    if (ua >= 0 and ua <= 1 and ub >= 0 and ub <= 1) then
        local x = L1.X1 + (ua * (L1.X2 - L1.X1))
        local y = L1.Y1 + (ua * (L1.Y2 - L1.Y1))
        return {x=x, y=y}
    end

    return false
end

Answer 1

这是我的解决方案:

function get_intersection (ax, ay, bx, by, cx, cy, dx, dy) -- start end start end
    local d = (ax-bx)*(cy-dy)-(ay-by)*(cx-dx)
    if d == 0 then return end  -- they are parallel
    local a, b = ax*by-ay*bx, cx*dy-cy*dx
    local x = (a*(cx-dx) - b*(ax-bx))/d
    local y = (a*(cy-dy) - b*(ay-by))/d
    if x <= math.max(ax, bx) and x >= math.min(ax, bx) and
        x <= math.max(cx, dx) and x >= math.min(cx, dx) then
        -- between start and end of both lines
        return {x=x, y=y}
    end
end