if i know the start and end coordinates of a line and i have a arbitrary point
X next to the line, what is the mathematical formula to calculate closest
point on the line from the arbtrary point X
The mathematical formula to calculate the closest point on a line from an
arbitrary point is called the orthogonal projection formula. It is given by:

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p = a + dot(x - a, b - a) / dot(b - a, b - a) * (b - a)
where a and b are the start and end coordinates of the line, x is the
arbitrary point, and p is the closest point on the line to x.

To use the formula, you first need to calculate the vector b - a that
represents the direction of the line. Then you can use the dot product to find
the scalar projection of the vector x - a onto the line. Finally, you can use
this scalar projection to find the closest point on the line to x by adding it
to the vector a.

Here is an example to illustrate how to use the formula:

Suppose we have a line with start and end coordinates (1, 2) and (4, 6),
respectively, and an arbitrary point (2, 3). We can use the formula above to
find the closest point on the line to this point as follows:

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b - a = (4, 6) - (1, 2) = (3, 4)
x - a = (2, 3) - (1, 2) = (1, 1)
dot(x - a, b - a) = (1, 1) * (3, 4) = 3 + 4 = 7
dot(b - a, b - a) = (3, 4) * (3, 4) = 9 + 16 = 25
p = (1, 2) + 7 / 25 * (3, 4) = (1, 2) + (7/25) * (3, 4) = (1.28, 2.32)
Therefore, the closest point on the line to the arbitrary point (2, 3) is
(1.28, 2.32).

I hope this helps. Let me know if you have any other questions.