The coordinates of P0 = (0, 0) and of P4 = (1020, 0) are known but the coordinates of the points P1, P2 and P3 must be determined.
To determine the coordinates of a point P, measure the distances from P = (xP, yP) to two points with known coordinates,
A = (xA, yA) and B = (xB, yB).
If we call the distances between P and A and P and B LA and LB respectively, we get the following equation system where the equations describe two circles with centers in A and B and with radii given by LA and LB.
This system of equations has two solutions (intersection points) where there are two points, P and Q, which are both at the same distance from points A and B. In this case it is point P, the right intersection we are interested in. See Figure 2.
Figure 2: Determination of coordinates for the point P with binding. The coordinates of the points A and B and the lengths LA and LB are known.
1a) Start by determining the coordinates (with the appropriate tolerance) for points P1, P2 and P3 using binding. For each point, a non-linear system of equations must correspond (1) - (2) is solved by Newton's method.
Initial values are determined by drawing the circles.
The table below shows points A and B with known coordinates and measured distances between A and P and B and P for points P1, P2 and P3.
The coordinates are given in meters from the point (0, 0).
The program should not use code repetition. Use a for-loop for the three points. Report the coordinates and that Newton's method converges squarely.
1b) Determine the fourth degree polynomial, p (x), which passes through the five points P0, P1, P2, P3 and P4. Draw the path (graph of the polynomial) for x ∈ (0, 1020). Also draw the five the interpolation points and mark them with ‘o’. Report the coefficients in the polynomial and the figure with the plotted path and the interpolation points.
1c) Calculate the length of the new road from the point (0, 0) to the point (1020, 0). Use the trapezoidal rule with the appropriate step length.
Report the length of the road and that your implementation of the trapezoidal rule has an accuracy order 2. Tip: how do you calculate the arc length of a curve?
1d) Assume the lengths LA and LB have an uncertainty of ± 5 m. Calculate the uncertainty length on the road with experimental disturbance calculation. Report the uncertainty in the length calculation.
Tip: For the experimental disturbance count, the code that solves a) -c) is packaged in a “black box ”(a Matlab function) that takes LA and LB for the three points that input and gives the length of the path as output.
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