Objectives: to understand spline interpolation and integration to find the volum
ID: 3869292 • Letter: O
Question
Objectives: to understand spline interpolation and integration to find the volume of water that would be held by a champagne glass. Experiment how the data was collected: An odd-shaped champagne glass was chosen for this experiment. The outer diameter of the champagne glass was measured (see data on the back on this sheet) at different known locations along the height (1/2" markings along the height along the center of the revolution are already put on the glass). Then the thickness of the glass was measured, so that the inner radius of the champagne glass was found at the locations where the outer radius was measured. Use MATLAB to solve problems 1-3 (copy and paste the code, answers, tests, figures, etc.) into one Word file. Make sure that you use comments, display commands and fprintf statements, sensible variable names and units to explain your work In MATLAB: Find the spline interpolant that curve fits the radius vs height data. In MATLAB: Show the individual points and the spline interpolant of radius vs height on a single plot. In MATLAB: Find how much volume of the champagne glass would hold based on this equation: The volume of the champagne glass can be calculated as V = integtral^H_0 pi r^2 dh where r is the varying radius of the champagne glass as a function of height, h, H is the height of the champagne glass. Assume that the volume as measured by pouring water from top-filled glass into the graduated cylinder was 550 ml. Compare this with the result that you got from problem 3 using true % relative error. Discuss your findings in 100-200 words.Explanation / Answer
we conduct in class to compare spline and polynomial interpolation. If you do not want to conduct the experiment itself but want the (x,y) data to see for yourself how polynomial and spline interpolation compare, the data is given below.
Length of graduated flexible curve = 12
The points on the x-y graph are as follows
(-4.1,0), (-2.6,1), (-2.0,2,2), (-1.6, 3.0), (-1,3.6), (0,3.9), (1.6,2.8), (3.2,0.4), (4.1,0)In this experiment, we find the length of two curves generated from the same points – one curve is a polynomial interpolant and another one is a spline interpolant.Motivation behind the experiment: In 1901, Runge conducted a numerical experiment to show that higher order interpolation is a bad idea. It was shown that as you use higher order interpolants to approximate f(x)=1/(1+25x2) in [-1,1], the differences between the original function and the interpolants becomes worse. This concept also becomes the basis why we use splines rather than polynomial interpolation to find smooth paths to travel through several discrete points.
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