How to efficiently solve the coordinates of the intersection point when the projections of two line segments overlap in three-dimensional space?

Efficiently solve the coordinates of intersection points of two line segments in three-dimensional space (projection overlaps)

This article introduces an efficient algorithm to calculate the intersection coordinates of two line segments in three-dimensional space, especially for the special case where line segments overlap at horizontal plane projections.

Suppose there are two line segments AB and CD, and their endpoint coordinates are A(x1, y1, z1), B(x2, y2, z2), C(x3, y3, z3) and D(x4, y4, z4), respectively. The known condition is that the projection of line segments AB and CD on the horizontal plane coincides, which means that the x and y coordinates of A and C are the same, and the x and y coordinates of B and D are the same.

Due to projection overlap, the x and y coordinates of intersection E can be directly determined as the x and y coordinates of A (or C). Therefore, we only need to calculate the z-coordinate of intersection E.

We can use the proportional relationship of the line segment in the z-axis direction to calculate the parameter t to obtain the z-coordinate of point E. The specific formula is as follows:

t = (z3 - z1) / ((z2 - z1) - (z4 - z3))

The z coordinates of point E are:

Ez = z1 t * (z2 - z1)

Therefore, the coordinates of intersection E are (x1, y1, Ez).

The improved algorithm is as follows:

 private double[] calculateIntersectionPoint(double x1, double y1, double z1, double x2, double y2, double z2, double x3, double y3, double z3, double x4, double y4, double z4) {
    double[] intersection = new double[3];
    intersection[0] = x1; // x coordinate of intersection intersection[1] = y1; // y coordinate of intersection double t = (z3 - z1) / ((z2 - z1) - (z4 - z3));
    intersection[2] = z1 t * (z2 - z1); // z-coordinate return intersection of intersection;
}

This algorithm directly utilizes the conditions of projection overlap, avoids redundant calculations, improves efficiency, and accurately calculates the three-dimensional coordinates of the intersection points. It should be noted that this algorithm assumes that the two line segments do intersect and the projections overlap at the horizontal plane. If the line segments do not intersect or the projections do not coincide, additional judgment and processing are required.

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