Page - 128 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

128 5 SomeMorePythonEssentials Filename:total_volume_boxes.py. Exercise5.6:AreaofaPolygon One of the most important mathematical problems through all times has been to find the area of a polygon, especially because real estate areas often had the shape of polygons, and it was necessary to pay tax for the area. We have a polygon as depictedbelow. The vertices (“corners”) of the polygon have coordinates (x1,y1), (x2,y2), . . ., (xn,yn),numberedeitherinaclockwiseorcounterclockwisefashion.TheareaAof thepolygoncanamazinglybecomputedbyjustknowingtheboundarycoordinates: A= 1 2 ∣ ∣(x1y2+x2y3+···+xn−1yn+xny1)−(y1x2+y2x3+···+yn−1xn+ynx1) ∣ ∣ . Writeafunctionpolyarea(x, y) that takestwocoordinatearrayswiththevertices asargumentsandreturns thearea. Test the function on a triangle, a quadrilateral, and a pentagon where you can calculate the areabyalternativemethodsforcomparison. Hint SincePythonlistsandarrayshave0as theirfirst index, it iswise torewrite the mathematical formula in termsof vertexcoordinatesnumberedasx0,x1,.. .,xn−1 andy0,y1,.. .,yn−1. Filename:polyarea.py. Exercise5.7:CountOccurrencesofa String in aString In the analysis of genesone encountersmanyproblemsettings involvingsearching for certain combinations of letters in a long string. For example, we may have a string like gene = ’AGTCAATGGAATAGGCCAAGCGAATATTTGGGCTACCA’ We maytraverse this string, letterby letter, by the for loopfor letter in gene. The length of the string is given by len(gene), so an alternative traversal over an indexi isfor i in range(len(gene)).Letter numberi is reached through