更新： 我下面的术语是错误的。我在“ Lerp2D”中描述的“正向”算法（我需要逆运算）需要四个任意角。它沿着每个边都是线性的，但是 所有
4个 边_都可以独立拉伸。它 _不是双线性的 。

我已经在标题中保留了双线性-如果您来这里是寻找“双线性的逆”，例如in
x和中的独立拉伸y。

如果需要更一般的情况（由任意四边形定义拉伸），请参见接受的答案。

在此问题的评论中，另请参阅人们给出的链接。

原始问题：

双线性插值的计算很简单。但是我需要一种执行逆运算的算法。（算法对我来说可以是伪代码或任何广泛使用的计算机语言）

例如，这是双线性插值的Visual Basic实现。

' xyWgt ranges (0..1) in x and y. (0,0) will return X0Y0,
(0,1) will return X0Y1, etc.
' For example, if xyWgt is relative location within an image,
' and the XnYn values are GPS coords at the 4 corners of the image,
' The result is GPS coord corresponding to xyWgt.
' E.g. given (0.5, 0.5), the result will be the GPS coord at center of image.
Public Function Lerp2D(xyWgt As Point2D, X0Y0 As Point2D, X1Y0 As Point2D, X0Y1 As Point2D, X1Y1 As Point2D) As Point2D
    Dim xY0 As Point2D = Lerp(X0Y0, X1Y0, xyWgt.X)
    Dim xY1 As Point2D = Lerp(X0Y1, X1Y1, xyWgt.X)

    Dim xy As Point2D = Lerp(xY0, xY1, xyWgt.Y)
    Return xy
End Function

哪里

' Weighted Average of two points.
Public Function Lerp(ByVal a As Point2D, ByVal b As Point2D, ByVal wgtB As Double) As Point2D
    Return New Point2D(Lerp(a.X, b.X, wgtB), Lerp(a.Y, b.Y, wgtB))
End Function

和

' Weighted Average of two numbers.
' When wgtB==0, returns a, when wgtB==1, returns b.
' Implicitly, wgtA = 1 - wgtB. That is, the weights are normalized.
Public Function Lerp(ByVal a As Double, ByVal b As Double, ByVal wgtB As Double) As Double
    Return a + (wgtB * (b - a))
End Function

在1-D中，我确定了Lerp的反函数：

' Calculate wgtB that would return result, if did Lerp(a, b, wgtB).
' That is, where result is, w.r.t. a and b.
' < 0 is before a, > 1 is after b.
Public Function WgtFromResult(ByVal a As Double, ByVal b As Double, ByVal result As Double) As Double

    Dim denominator As Double = b - a

    If Math.Abs(denominator) < 0.00000001 Then
        ' Avoid divide-by-zero (a & b are nearly equal).
        If Math.Abs(result - a) < 0.00000001 Then
            ' Result is close to a (but also to b): Give simplest answer: average them.
            Return 0.5
        End If
        ' Cannot compute.
        Return Double.NaN
    End If

    ' result = a + (wgt * (b - a))   =>
    ' wgt * (b - a) = (result - a)   =>
    Dim wgt As Double = (result - a) / denominator

    'Dim verify As Double = Lerp(a, b, wgt)
    'If Not NearlyEqual(result, verify) Then
    '    Dim test = 0    ' test
    'End If

    Return wgt
End Function

现在，我需要在二维中执行相同的操作：

' Returns xyWgt, which if given to Lerp2D, would return this "xy".
' So if xy = X0Y0, will return (0, 0). if xy = X1Y0, will return (1, 0), etc.
' For example, if 4 corners are GPS coordinates in corners of an image,
' and pass in a GPS coordinate,
' returns relative location within the image.
Public Function InverseLerp2D(xy As Point2D, X0Y0 As Point2D, X1Y0 As Point2D, X0Y1 As Point2D, X1Y1 As Point2D) As Point2D
    ' TODO ????
End Function