CODEWITHSUNDEEP
javascriptmathgeometry2dcoordinates
Niall McKenna
Niall McKenna

Reputation: 181

Finding an eqauivalent point in a quadrilateral

I am trying to find the rough equivalent point from one quadrilateral to another.

quadrilateral equivalent point

What is a good method to find this point?

Any info in the right direction would be great.

Thanks

Upvotes: 2

Views: 196

Answers (2)

fang
fang

Reputation: 3633

You can find the general barycentric coordinates of the point against the first quadrilateral, then applies the barycentric coordinates against the 2nd quadrilateral to find the "equivalent" point.

There are many different ways to compute the general barycentric coordinates for a point against a quadrilateral (or against a n-sided polygon). One of them is the Wachpress coordinates, which only works when the polygon is convex. For a convex polygon with vertices V0, V1,...Vn:

enter image description here

we can compute the Wachpress coordinates for point P within the polygon as

enter image description here

where A(a,b,c) is the signed area of triangle abc.

Then, we can compute the barycentric coordinates of P as

enter image description here

The equivalent point P* against the new polygon with vertices V*i (or quadrilateral in your case) can be computed as

P* = \summation(Wi * V*i) for i=0~n.

Upvotes: 0

Leandro Caniglia
Leandro Caniglia

Reputation: 14868

Let the points be [P1, P2, P3, P4] and assume they are transformed into [Q1, Q2, Q3, Q4] by means of an affine transformation of the form x -> Mx + b where M is a 2x2 matrix and b is a constant vector. The idea is to find M and apply the transformation to P to get Q.

Let v1 = P2 - P1, v2 = P3 - P1, w1 = Q2 - Q1, w2 = Q3 - Q1. Then M is the linear transformation that applies [v1, v2] onto [w1, w2]. One way to find M is by calculating the matrix product

M = S * T

where S is the 2x2 matrix whose columns are the vectors w1 and w2 and T is the inverse of the matrix whose columns are v1 and v2.

Regarding the displacement vector b, it can be calculated as

b = Q1 - M * P1

The clarification here is that all of this is correct only if the fourth points are consistent with the transformation, i.e., if

M * P4 + b = Q4

otherwise the Q-quadrilateral is not an affine transformation of the P-quadrilateral.

Upvotes: 1

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