Goal
I want to determine if a test point is within a defined quadrilateral. I'm probably going to implement the solution in Matlab so I only need pseudo-code.
Inputs
Corners of quadrilateral : (x1,y1) (x2,y2) (x3,y3) (x4,y4)
Test point : (xt, yt)
Output
1 - If within quadrilateral
0 - Otherwise
Update
It was pointed out that identifying the vertices of the quadrilateral is not enough to uniquely identify it. You can assume that the order of the points determines the sides of the quadrilateral (point 1 connects 2, 2 connects to 3, 3 connects to 4, 4 connects to 1)
Draw a horizontal line to the right of each point and extend it to infinity. Count the number of times the line intersects with polygon edges. A point is inside the polygon if either count of intersections is odd or point lies on an edge of polygon.
To determine the status of a point (xp,yp) consider a horizontal ray emanating from (xp,yp) and to the right. If the number of times this ray intersects the line segments making up the polygon is even then the point is outside the polygon.
A simple way is to: find the vectors connecting the point to each of the triangle's three vertices and sum the angles between those vectors. If the sum of the angles is 2*pi then the point is inside the triangle.
You can test the Point with this condition. Also you can treat quadrilateral as 2 triangles to calculate its area.
Use inpolygon
. Usage would be inpolygon(xt,yt,[x1 x2 x3 x4],[y1 y2 y3 y4])
Since it's a simple quadrilateral you can test for a point in triangle for each end and a point in rectangle for the middle.
EDIT Here is some pseudo code for point in triangle:
function SameSide(p1,p2, a,b)
cp1 = CrossProduct(b-a, p1-a)
cp2 = CrossProduct(b-a, p2-a)
if DotProduct(cp1, cp2) >= 0 then return true
else return false
function PointInTriangle(p, a,b,c)
if SameSide(p,a, b,c) and SameSide(p,b, a,c)
and SameSide(p,c, a,b) then return true
else return false
Or using Barycentric technique:
A, B, and C are the triangle end points, P is the point under test
// Compute vectors
v0 = C - A
v1 = B - A
v2 = P - A
// Compute dot products
dot00 = dot(v0, v0)
dot01 = dot(v0, v1)
dot02 = dot(v0, v2)
dot11 = dot(v1, v1)
dot12 = dot(v1, v2)
// Compute barycentric coordinates
invDenom = 1 / (dot00 * dot11 - dot01 * dot01)
u = (dot11 * dot02 - dot01 * dot12) * invDenom
v = (dot00 * dot12 - dot01 * dot02) * invDenom
// Check if point is in triangle
return (u > 0) && (v > 0) && (u + v < 1)
If the aim is to code your own test, then pick any classic point in polygon test to implement. Otherwise do what Jacob suggests.
assuming you the given coordinates are arranged s.t. (x1,y1) = rightmost coordinate (x2,y2) = uppermost coordinate (x3,y3) = leftmost coordinate (x4,y4) = botoom-most coordinate
You can do the following:
1. calculate the 4 lines of the quadrilateral (we'll call these quad lines)
2. calculate 4 lines, from the (xt, yt) to every other coordinate (we'll call these new lines)
3. if any new line intersects any of the quad lines, then the coordinate is outside of the quadrilateral, otherwise it is inside.
Assume A,B,C,D
are the vertices of the quadrilateral and P
is the point.
If P
is inside the quadrilateral then all dot products dot(BP,BA), dot(BP,BC), dot(AP,AB), dot(AP,AD), dot(DP,DC), dot(DP,DA), dot(CP,CB)
and dot(CP,CD)
will be positive.
If P
is outside the quadrilateral at least one of these products will be negative.
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