Mathematical formula solving

aallian

Mathematical formula solving [Copy link]

A point p (x, y, z) in space coordinates is L from three other points p1 (x1, y1, z1), p2 (x2, y2, z2) and p3 (x3, y3, z3). Find the coordinates of point p. 1) Derive the solution formula 2) Implement it in high-level programming languages such as c/c++, python, etc. PS: I remember it was high school math content, I have almost forgotten it:) Point p should have two points that can be determined.

maychang

"Point p should have two points that can be determined."

Not necessarily. This problem may have two solutions, one solution, or no solution.

maychang

This is the content of the analytic geometry course. This question is actually about finding the coordinates (x, y, z) of point p. You can use the distance formula between two points to find it.

aallian

maychang posted on 2020-1-6 19:13 "Point p should have two points that can be determined." Not necessarily. This problem may have two solutions, one solution, or no solution.

Point p is on the vertical line passing through the center of the triangle formed by p1, p2, and p3. If it is exactly in the center of the triangle, what is the situation where there is one solution and no solution?

aallian

maychang published on 2020-1-6 19:15 This is the content of the analytic geometry course. This question is actually to find the coordinates of point p (x, y, z). You can use the distance formula between two points to find it.

Based on the distance between two points (p, p1), I can list the formula (x-x1)^2 + (y-y1)^2+(z-z1)^2=L^2. There are 3 sets of three-variable quadratic equations. But I can't solve them

maychang

aallian posted on 2020-1-6 20:38 Point p is on the vertical line passing through the center of the triangle formed by p1, p2, and p3. If it is exactly in the center of the triangle, what is the situation where there is only one solution and no solution?

The three points p1p2p3 define a circle. The distance L from point p to the three points p1p2p3 is given in the question. If L is less than the radius of the circle defined by the three points p1p2p3, then there is no solution. If L is equal to the radius of the circle, then there is a unique solution. If L is greater than the radius of the circle, then there are two solutions.

maychang

aallian posted on 2020-1-6 20:38 Point p is on the vertical line passing through the center of the triangle formed by p1, p2, and p3. If it is exactly in the center of the triangle, what is the situation where there is only one solution and no solution?

The three points p1p2p3 determine a circle. In the second solution, point p is at the vertex of a regular cone with this circle as its base. In the first solution, point p is at the center of this circle. There is no solution because L is too small and point p "cannot reach" the three points p1p2p3.

littleshrimp

Matlab solves the following equation, no solution

[x,y,z]=solve('sqrt((x-x1)^2+(y-y1)^2+(z-z1)^2)=L','sqrt((x-x2)^ 2+(y-y2)^2+(z-z2)^2)=L','sqrt((x-x3)^2+(y-y3)^2+(z-z3)^2)= L','x,y,z')