Laboratory 9

1. Prove that the three medians of a triangle in general position (running from the center of an edge to the vertex on the opposite side) intersect in a single point (the center of gravity). Again, first design a helpful auxiliary Maple procedure. Instead of yesterday, please do this exercise with parametric definitions of straight lines.

2. Write a Maple procedure plotell(a,b) that plots an ellipse, whose implicit definition is \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 by using a rational parametric representation. Note that for the circle the rational parametric representation was (\frac{2t}{1+t^{2}} , \frac{1−t^{2}}{1+t^{2}} ). Experiment with restricted ranges for t.

3. Challenge problem: Design a Maple procedure that plots a linear map from 2D-space to 2D-space (given in parametric representation) by depicting the second value as a color hue. Your procedure is to plot only a patch of the plane delineated by parameter values t = 0..1. Seek help from Maple: ?plot[color] and ?plots. A circle depicting the hue values from 0 to 1 can be plotted by reading the Maple file piechart.mpl and running piechart(20). Study and understand the procedure piechart before proceeding.