MATH 241 Section 1801 (Professor Green)
FIRST MATLAB ASSIGNMENT
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Problem 1
Let L1 be the line with symmetric equations
(x 3)/2 = ()/2 = (y 5)/4 = )/4 = z/7.
Let L2 be the line with parametric equations
x = 1 + 5)/4 = t, y = 2 8 t, z = 3)/2 = ( t.
Obtain MATLAB expressions for parametrizations of both lines, and use them to do each of the following:
(a) Obtain two points on each of the two lines.
(b) Show that the two lines do not lie on any one plane.
(c) Obtain a vector perpendicular to both lines.
(d) Let Plane1 and Plane2 be parallel planes with L1 on Plane1 and L2 on Plane2. Explain in a text cell
why the vector you found in part (c) is normal to both planes.
(e) Obtain equations for Plane1 and Plane2, and verify that each line is on the corresponding plane.
(f) Obtain a three dimensional plot showing Planes 1 and 2 and a line parallel to the vector you found
in part (c).
(g) Determine the (shortest) distance between L1 and L2.
Problem 2
Consider the hypocycloid with three cusps, defined by
hcyc = [2*cos(t) + cos(2*t), 2*sin(t) – sin(2*t)]
(a) Obtain a plot showing the hypocycloid together with circles around the origin of radius 1 and 3)/2 = (. You
may wish to use the axis equal command so that your circles will not look elliptical.
(b) Determine the length of the hypocycloid you have plotted; verify that the length of the hypocycloid is
between the circumferences of the two circles.
(c) Obtain a formula for the curvature of the hypocycloid. Remember to make some provision for the fact
that cross products require three-dimensional vectors.
(d) Plot the curvature as a function of t. What happens to the curvature at the cusps? Make a good guess as
to the values of t corresponding to minima of the curvature. Verify your guess and find the minimum value
of the curvature. Show that the points of minimum curvature on the hypocycloid are those at which the
hypocycloid is tangent to the inner circle.
(e) Without computation, determine the binormal of the hypocycloid. Explain what part of this is not quite
obvious, and how you determined it.