A spiral may be considered to be the figure described by the motion of a point on an imaginary line as that line pivots around an origin at constant angular velocity. If the point is fixed on the line, then the figure described is a circle.

a. If the point on the rotating line moves from the origin with constant speed, its position describes an Archimedean spiral. In polar coordinates the equation of this spiral is r = a + b. Use pylab to plot the spiral defined by a = 0, b = 2 for 0 8.

b. If the point moves along the rotating line with a velocity that increases in pro- portion to its distance from the origin, the result is a logarithmic spiral, which may be written as r = a . Plot the logarithmic spiral defined by a = 0.8 for 0 8. The logarithmic spiral has the property of self-similarity: with each 2 whorl, the spiral grows but maintains its shape.3 Logarithmic spirals occur frequently in nature, from the arrangements of the chambers of nautilus shells to the shapes of galaxies.

In Python