Relative position of a point within a quadrilateral

Question

I am trying to find the easiest way to determine a relative position of a point within a quadrilateral. The known are (see figure) the positions of points 1, 2, 3, 4 and 5 in the xy-coordinate system: x1, y1, x2, y2, x3, y3, x4, y4, x5, y5.

Also known are the positions of points 1, 2, 3, and 4 in the ξ-η coordinate systems (see figure).

From this data, I want to determine what are the ξ and η for point 5.

Results

Thank you to all who anwsered! I find the solution by @dbc and @agentp similar. Also I find this solution better than the perspective transformation solution by @MBo, since I do not have to compute the inverse of a matrix (Ax=B --> x=inv(A)*B).

I get the following result for:

u = 0.5 * (ξ + 1)
v = 0.5 * (η + 1)

In my case all points are within the rectangle, therefore u>0 and v>0.

Answer 1

What you have here is a 2d bilinear blended surface. For simplicity, let's change its coordinates to range from zero to one:

u = 0.5 * (ξ + 1)
v = 0.5 * (η + 1)

In that case, the surface evaluator can be expressed as

F(u, v) = P1 + u * (P2 - P1) + v * ((P4 + u * (P3 - P4)) - (P1 + u * (P2 - P1)))

I.e., for a given u, construct a line passing through the following two points:

Pv0 = P1 + u * (P2 - P1);
Pv1 = P4 + u * (P3 - P4);

then interpolate between then for given v

F(u, v) = Pv0 + v * (Pv1 - Pv0)

What you seek are values (u,v) such that F(u, v) = P5. This will occur for given u when the line from Pv0 to Pv1 passes through P5, which will occur when P5 - Pv0 is parallel to Pv1 - Pv0 -- i.e. when their 2d cross is zero:

cross2d(P5 - Pv0, Pv1 - Pv0) = 0

⇒

cross2d(P5                 - (P1 + u * (P2 - P1)),  
        P4 + u * (P3 - P4) - (P1 + u * (P2 - P1))) = 0

Now, the 2d cross of two 2d vectors A ⨯ B is given by Ax*By - Ay*Bx, so that equation becomes

(x5 - (x1 + u * (x2 - x1))) * (y4 + u * (y3 - y4) - (y1 + u * (y2 - y1))) - (y5 - (y1 + u * (y2 - y1))) * (x4 + u * (x3 - x4) - (x1 + u * (x2 - x1))) = 0

Expanding this expression out and collecting collecting together terms in u, we get

    u^2 * (x1*y3 - x1*y4 - x2*y3 + x2*y4 + (-x3)*y1 + x3*y2 + x4*y1 - x4*y2)
  + u *   (-x1*y3 + 2*x1*y4 - x1*y5 - x2*y4 + x2*y5 + x3*y1 - x3*y5 - 2*x4*y1 + x4*y2 + x4*y5 + x5*y1 - x5*y2 + x5*y3 - x5*y4)
  +       (-x1*y4 + x1*y5 + x4*y1 - x4*y5 - x5*y1 + x5*y4)
= 0

This is now a quadratic equation over u, and can be solved as such. Note that in cases where the top and bottom edges of your quadrilateral are parallel then the quadratic devolves into a linear equation; your quadratic equation solver must needs handle this.

        double a = (x1 * y3 - x1 * y4 - x2 * y3 + x2 * y4 + (-x3) * y1 + x3 * y2 + x4 * y1 - x4 * y2);
        double b = (-x1 * y3 + 2 * x1 * y4 - x1 * y5 - x2 * y4 + x2 * y5 + x3 * y1 - x3 * y5 - 2 * x4 * y1 + x4 * y2 + x4 * y5 + x5 * y1 - x5 * y2 + x5 * y3 - x5 * y4);
        double c = (-x1 * y4 + x1 * y5 + x4 * y1 - x4 * y5 - x5 * y1 + x5 * y4);

        double[] solutions = Quadratic.Solve(a, b, c);

There may be more than one solution. There might also be no solutions for a degenerate quadrilateral.

Having solved for value(s) of u, finding the equivalent v is straightforward. Given points

Pv0 = P1 + u * (P2 - P1);
Pv1 = P4 + u * (P3 - P4);

you seek v such that

v * (Pv1 - Pv0) = P5 - Pv0;

Pick the coordinate index 0 or 1 such that |(Pv1 - Pv0)[index]| is maximized. (If both coordinates are almost zero, then give up -- there's no solution for this specific u. Then set

v = (P5 - Pv0)[index] / (Pv1 - Pv0)[index];

Finally, if you have more that one solution, prefer a solution inside the [u, v] boundaries of the blend. Then finally set

ξ = 2 * u - 1;
η = 2 * v - 1;